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Applied and Numerical Harmonic Analysis
Kristian Bredies
Dirk Lorenz
Mathematical
Image
Processing

Series Editor
John J. Benedetto
University of Maryland
College Park, MD, USA
Editorial Advisory Board
Akram Aldroubi Gitta Kutyniok
Vanderbilt University Technische Universität Berlin
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Arizona State University Duke University
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University of Paris XII Michigan State University
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Carnegie Mellon University
Pittsburgh, PA, USA
More information about this series at http://www.springer.com/series/4968

Kristian Bredies • Dirk Lorenz
Mathematical Image
Processing

Kristian Bredies
Institute for Mathematics and Scientific
University of Graz
Graz, Austria
Dirk Lorenz
Braunschweig, Germany
ISSN 2296-5009 ISSN 2296-5017 (electronic)
ISBN 978-3-030-01457-5 ISBN 978-3-030-01458-2 (eBook)
https://doi.org/10.1007/978-3-030-01458-2
Library of Congress Control Number: 2018961010
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ANHA Series Preface
The Applied and Numerical Harmonic Analysis (ANHA) book series aims to
provide the engineering, mathematical, and scientific communities with significant
developments in harmonic analysis, ranging from abstract harmonic analysis to
basic applications. The title of the series reflects the importance of applications
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v

vi ANHA Series Preface
Along with our commitment to publish mathematically significant works at the
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ANHA Series Preface vii
adaptive modeling inherent in time-frequency-scale methods such as wavelet theory.
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University of Maryland John J. Benedetto
College Park, MD, USA Series Editor

Preface
Mathematical imaging is the treatment of mathematical objects that stand for images
where an “image” is just what is meant in everyday conversation, i.e., a picture of
a real scene, a photograph, or a scan. In this book, we treat images as continuous
objects, i.e., as image of a continuous scene or, put differently, as a function of a
continuous variable. The closely related field of digital imaging, on the other hand,
treats discrete images, i.e., images that are described by a finite number of values
or pixels. Mathematical imaging is a subfield of computer vision where one tries
to understand how information is stored in images and how information can be
extracted from images in an automatic way. Methods of computer vision usually
use underlying mathematical models for images and the information therein. To
apply methods for continuous images in practice, i.e., to digital images, one has
to discretize the methods. Hence, mathematical imaging and digital imaging are
closely related and often methods in both fields are developed simultaneously. A
method based on a mathematical model is useful only if it can be implemented in an
efficient way and the mathematical treatment of a digital method often reveals the
underlying assumptions and may explain observed effects.
This book emphasizes the mathematical character of imaging and as such is
geared toward students of mathematical subjects. However, students of computer
science, engineering, or natural sciences who have a knack for mathematics may
also find this book useful. We assume knowledge of introductory courses like
linear algebra, calculus, and numerical analysis; some basics of real analysis and
functional analysis are advantageous. The book should be suited for students in their
third year; however, later chapters of the book use some advanced mathematics. In
this book, we give an overview of mathematical imaging; we describe methods and
solutions for standard problems in imaging. We will also introduce elementary tools
as histograms and linear and morphological filters since they often suffice to solve
a given task. A special focus is on methods based on multiscale representations,
partial differential equations, and variational methods In most cases, we illustrate
how the methods can be realized practically, i.e., we derive applicable algorithms.
This book can serve as the basis for a lecture on mathematical imaging, but is also
possible to use parts in lectures on applied mathematics or advanced seminars.
ix

x Preface
The introduction of the book outlines the mathematical framework and intro-
duces the basic problems of mathematical imaging. Since we will need mathematics
from quite different fields, there is a chapter on mathematical basics. Advanced
readers may skip this chapter, just use it to brush up their knowledge, or use it as a
reference for the terminology used in this book. The chapter on mathematical basics
does not cover the basics we will need. Many mathematics facts and concepts are
introduced when they are needed for specific methods. Mathematical imaging itself
is treated in Chaps. 3–6. We organized the chapters according to the methods, and
not according to the problems. Somehow we present a box of tools that shall serve
as a reservoir of methods so that the user can pick, combine, and develop tools
that seem to be best suited for the problem at hand. These mathematical chapters
conclude with exercises which shall help to develop a deeper understanding of
the methods and techniques. Some exercises involve programming, and we would
like to encourage all readers to try to implement the method in their favorite
programming language. As with every book, there are a lot of topics which did
not find their way into the book. We would still like to mention some of these topics
in the sections called “Further developments.”
Finally, we would like to thank all the people who contributed to this book in one
way or another: Matthias Bremer, Jan Hendrik Kobarg, Christian Kruschel, Rainer
Löwen, Peter Maaß, Markus Müller (who did a large part of the translation from the
German edition), Tobias Preusser, and Nadja Worliczek.
Graz, Austria Kristian Bredies
Braunschweig, Germany Dirk Lorenz
August 2018

Contents
1 Introduction .................................................................. 1
1.1 What Are Images? ...................................................... 1
1.2 The Basic Tasks of Imaging ............................................ 5
2 Mathematical Preliminaries ................................................ 15
2.1 Fundamentals of Functional Analysis.................................. 15
2.1.1 Analysis on Normed Spaces.................................... 16
2.1.2 Banach Spaces and Duality .................................... 23
2.1.3 Aspects of Hilbert Space Theory............................... 29
2.2 Elements of Measure and Integration Theory ......................... 32
2.2.1 Measure and Integral ........................................... 32
2.2.2 Lebesgue Spaces and Vector Spaces of Measures............. 38
2.2.3 Operations on Measures ........................................ 45
2.3 Weak Differentiability and Distributions .............................. 49
3 Basic Tools .................................................................... 55
3.1 Continuous and Discrete Images ....................................... 55
3.1.1 Interpolation..................................................... 55
3.1.2 Sampling ........................................................ 59
3.1.3 Error Measures.................................................. 61
3.2 Histograms .............................................................. 62
3.3 Linear Filters ............................................................ 68
3.3.1 Definition and Properties ....................................... 69
3.3.2 Applications..................................................... 75
3.3.3 Discretization of Convolutions ................................. 81
3.4 Morphological Filters ................................................... 86
3.4.1 Fundamental Operations: Dilation and Erosion ............... 88
3.4.2 Concatenated Operations ....................................... 92
3.4.3 Applications..................................................... 95
3.4.4 Discretization of Morphological Operators.................... 97
3.5 Further Developments .................................................. 101
3.6 Exercises................................................................. 105
xi

xii Contents
4 Frequency and Multiscale Methods........................................ 109
4.1 The Fourier Transform.................................................. 109
4.1.1 The Fourier Transform on L1(Rd)............................. 109
4.1.2 The Fourier Transform on L2(Rd)............................. 112
4.1.3 The Fourier Transform for Measures and Tempered
Distributions .................................................... 120
4.2 Fourier Series and the Sampling Theorem ............................. 125
4.2.1 Fourier Series ................................................... 125
4.2.2 The Sampling Theorem......................................... 126
4.2.3 Aliasing ......................................................... 128
4.3 The Discrete Fourier Transform ........................................ 135
4.4 The Wavelet Transform ................................................. 141
4.4.1 The Windowed Fourier Transform............................. 141
4.4.2 The Continuous Wavelet Transform ........................... 144
4.4.3 The Discrete Wavelet Transform............................... 149
4.4.4 Fast Wavelet Transforms ....................................... 156
4.4.5 The Two-Dimensional Discrete Wavelet Transform .......... 161
4.6 Exercises................................................................. 166
5 Partial Differential Equations in Image Processing ...................... 171
5.1 Axiomatic Derivation of Partial Differential Equations ............... 172
5.1.1 Scale Space Axioms ............................................ 173
5.1.2 Examples of Scale Spaces ...................................... 176
5.1.3 Existence of an Infinitesimal Generator ....................... 186
5.1.4 Viscosity Solutions ............................................. 191
5.2 Standard Models Based on Partial Differential Equations ............ 196
5.2.1 Linear Scale Spaces: The Heat Equation ...................... 196
5.2.2 Morphological Scale Space .................................... 199
5.3 Nonlinear Diffusion ..................................................... 206
5.3.1 The Perona-Malik Equation .................................... 207
5.3.2 Anisotropic Diffusion .......................................... 222
5.4 Numerical Solutions of Partial Differential Equations ................ 229
5.4.1 Diffusion Equations ............................................ 234
5.4.2 Transport Equations ............................................ 240
5.6 Exercises................................................................. 247
6 Variational Methods ......................................................... 251
6.1 Introduction and Motivation ............................................ 251
6.2 Foundations of the Calculus of Variations and Convex Analysis ..... 263
6.2.1 The Direct Method.............................................. 263
6.2.2 Convex Analysis ................................................ 270
6.2.3 Subdifferential Calculus ........................................ 285
6.2.4 Fenchel Duality ................................................. 301

Contents xiii
6.3 Minimization in Sobolev Spaces and BV ............................. 316
6.3.1 Functionals with Sobolev Penalty.............................. 316
6.3.2 Practical Applications .......................................... 334
6.3.3 The Total Variation Penalty .................................... 351
6.3.4 Generalization to Color Images ................................ 385
6.4 Numerical Methods ..................................................... 391
6.4.1 Solving a Partial Differential Equation ........................ 392
6.4.2 Primal-Dual Methods........................................... 396
6.4.3 Application of the Primal-Dual Methods ...................... 415
6.6 Exercises................................................................. 432
References......................................................................... 445
Picture Credits.................................................................... 453
Notation ........................................................................... 455
Index ............................................................................... 461
Applied and Numerical Harmonic Analysis (81 Volumes) .................. 469

Chapter 1
Introduction
1.1 What Are Images?
We omit the philosophical aspect of the question “What are images?” and aim
to answer the question “What kind of images are there?” instead. Images can be
produced in many different ways:
Photography: Photography produces two-dimensional images by projecting a
scene of the real world through some optics onto a two-dimensional image plane.
The optics are focused onto some plane, called the focal plane, and objects appear
more blurred the farther they are from the focal plane. Hence, photos usually have
both sharp and blurred regions.
At first, photography was based on chemical reactions to map the different
values of brightness and color onto photographic film. Then some other chemical
reactions were used to develop the film and to produce photoprints. Each of the
different chemical reactions happens with some slight uncontrolled variations,
and hence the photoprint does not exactly correspond to the real brightness and
color values. In particular, photographic film has a certain granularity, which
amounts to a certain noise in the picture.
Nowadays, most photos are obtained digitally. Here, the brightness and color
are measured digitally at certain places—the pixels, or picture elements. This
results in a matrix of brightness or color values. The process of digital picture
acquisition also results in some noise in the picture.
Scans: To digitize photos one may use a scanner. The scanner illuminates the
photo row by row and measures the brightness or color along the lines. Usually
this does not result in some additional blur. However, a scanner operates at some
resolution, which results in a reduction of information. Moreover, the scanning
process may result in some additional artifacts. Older scans are often pale and
may contain some contamination. The correction of such errors is an important
problem in image processing.
K. Bredies, D. Lorenz, Mathematical Image Processing,
Applied and Numerical Harmonic Analysis,
https://doi.org/10.1007/978-3-030-01458-2_1
1

2 1 Introduction
Microscopy: Digital and analog microscopy is a kind of mixture of photography
and scanning. One uses measurement technology similar to photography but
since the light cone is very wide in this case, the depth of field is very low.
Consequently, objects that are not in the focal plane appear very blurred and
are virtually invisible. This results in images that are almost two-dimensional.
Confocal microscopy exacerbates this effect in that only the focal plane is
illuminated, which suppresses objects not in the focal plane even further.
Indirect imaging: In some scenarios one cannot measure the image data directly.
Prominent examples are computerized tomography(CT) and ultrasound imaging.
In the case of CT, for example, one acquires X-ray scans of the object from
different directions. These scans are then used to reconstruct a three-dimensional
image of the density of the object.
Here one needs some mathematics to reconstruct the image in the first
place [103]. The reconstruction often generates some artifacts, and due to noise
from the measurement process, the reconstructed images are also not noise-free.
Other indirect imaging modalities are single-photon emission computerized
tomography (SPECT), positron-emission-tomography (PET), seismic tomogra-
phy, and also holography.
Generalized images: Data different from the above cases can also be treated as
images. In industrial engineering, for example, one measures surfaces to check
the smoothness of some workpiece. This results in a two-dimensional “elevation
profile” that can be treated just like an image. In other cases one may have
measured a chemical concentration or the magnetization.
A slightly different example is liquid chromatography-mass spectrometry
(LC/MS). Here one measures time-dependent one-dimensional mass spectra.
The resulting two-dimensional data has a mass and a time dimension and can
be treated by imaging techniques.
Some examples of different kinds of images can be seen in Fig. 1.1.
As we have seen, images do not have to be two-dimensional. Hence, we will work
with d-dimensional images in general (which includes volume data and movies). In
special cases we will restrict ourselves to one- or two-dimensional data.
Let us return to the question of what images are. We take a down-to-earth
viewpoint and formulate an answer mathematically: an image is a function that
maps every point in some domain of definition to a certain color value. In other
words: an image u is a map from an image domain to some color space F:
u : → F.
We distinguish between discrete and continuous image domains:
• discrete d-dimensional images, for example = {1, . . ., N1} × · · · ×
{1, . . . , Nd}.
• continuous d-dimensional images, for example ⊂ Rd, or specifically =
[0, a1] × · · · × [0, ad].

1.1 What Are Images? 3
Fig. 1.1 Different types of images. First row: Photos. Second row: A scan and a microscopy image
of cells. Third row: An image from indirect measurements (holography image of droplets) and a
generalized image (“elevation profile” of a surface produced by a turning process)
Different color spaces are, for example:
• Black-and-white images (also binary images): F = {0, 1}.
• Grayscale images with discrete color space with k-bit depth: F =
{0, . . . , 2k − 1}.
• Color images with k-bit depth for each of N color channels: F =
{0, . . . , 2k − 1}N .
• Images with continuous gray values: F = [0, 1] or F = R.
• Images with continuous colors: F = [0, 1]3 or F = R3.

4 1 Introduction
The field of digital image processing treats mostly discrete images, often also
with discrete color space. This is reasonable in the sense that images are most often
generated in discrete form or have to be transformed to a discrete image before
further automatic processing. The methods that are used are often motivated by
continuous considerations. In this book we take the viewpoint that our images are
continuous objects ( ⊂ Rd). Hence, we will derive methods for continuous images
with continuous color space. Moreover, we will deal mostly with grayscale images
(F = R or F = [0, 1]).
The mathematical treatment of color images is a delicate subject. For example,
one has to be aware of the question of how to measure distances in the color space:
is the distance from red to blue larger than that from red to yellow? Moreover, the
perception of color is very complex and also subjective. Colors can be represented
in different color spaces and usually they are encoded in different color channels.
For example, there are the RGB space, where colors are mixed additively from the
red, green, and blue channels (as on screens and monitors) and the CMYK space,
where colors are mixed subtractively from the cyan (C), magenta (M), yellow (Y),
and black (K for black) channels (as is common in print). In the RGB space, color
values are encoded by triplets (R, G, B) ∈ [0, 1]3, where the components represent
the amount of the respective color; (0, 0, 0) represents the color black, (1, 1, 1)
stands for white. This is visualized in the so-called RGB cube; see Fig. 1.2. Also the
colors cyan, magenta, and yellow appear as corners of the color cube. To process
color images one often uses the so-called HSV space: a color is described by the
channels Hue, Saturation, and Value. In the HSV space a color is encoded by a
triplet (H, S, V ) ∈ [0, 360[ × [0, 100] × [0, 100]. The hue H is interpreted as an
angle, the saturation S and the value V as percentages. The HSV space is visualized
as a cylinder; see Fig. 1.3. Processing only the V-channel for the value (and leaving
the other channels untouched) often leads to fairly good results in practice.
The goal of image processing is to automate or facilitate the evaluation and
interpretation of images. One speaks of high-level methods if one obtains certain
information from the images (e.g., the number of objects, the viewpoint of the
B=(0,0,1)
R=(1,0,0)
G=(0,1,0)
K=(0,0,0)
C=(0,1,1)
M=(1,0,1)
Y=(1,1,0)
W=(1,1,1)
Fig. 1.2 RGB space, visualized as a cube

1.2 The Basic Tasks of Imaging 5
Fig. 1.3 HSV space,
visualized as a cylinder
V
S
H
camera, the size of an object, and even the meaning of the scene). Low-level methods
are methods that produce new and improved images out of given images. This book
treats mainly low-level methods.
For the automatic processing of images one usually focuses on certain properties
and structures of interest. These may be, for example:
Edges, corners: An edge describes the boundary between two different struc-
tures, e.g., between different objects. However, a region in the shade may also be
separated from a lighter area by an edge.
Smooth regions: Objects with uniform color appear as smooth regions. If the
object is curved, the illumination creates a smooth transition of the brightness.
Textures: The word “texture” mostly stands for something like a pattern. This
refers, for example, to the fabric of a cloth, the structure of wallpapers, or fur.
Periodic structures: Textures may feature some periodic structures. These struc-
tures may have different directions and different frequencies and also occur as
the superposition of different periodic structures.
Coherent regions: Coherent regions are regions with a similar orientation of
objects as, for example, in the structure of wood or hair.
If such structures are to be detected or processed automatically, one needs good
models for the structures. Is an edge adequately described by a sudden change of
brightness? Does texture occur where the gray values have a high local variance?
The choice of the model then influences how methods are derived.
1.2 The Basic Tasks of Imaging
Many problems in imaging can be reduced to only a few basic tasks. This section
presents some of the classical basic tasks. The following chapters of this book will
introduce several tools that can be used for these basic tasks. For a specific real-

6 1 Introduction
Fig. 1.4 Unfavorable light conditions lead to noisy images. Left: Photo taken in dim light. Right:
Gray values along the depicted line
world application one usually deals with several problems and tasks and has to
combine and adapt methods or even invent new methods.
Denoising: Digital images contain erroneous information. Modern cameras that
can record images with several megapixels still produce noisy images, see
Fig. 1.4; in fact, it is usually the case that an increase of resolution also results
in a higher noise level. The camera’s chip uses the photon count to measure the
brightness. Since the emission of photons is fundamentally a random process,
the measurement is also a random variable and hence contains some noise.The
presence of noise is an inherent problem in imaging. The task of denoising is:
• Identify and remove the noise but at the same time preserve all important
information and structure.
Noise does not pose a serious problem for the human eye. We have no problems
with images with high noise level, but computers are different. To successfully
denoise an image, one needs a good model for the noise and the image. Some
reasonable assumptions are, for example:
• The noise is additive.
• The noise is independent of the pixel and comes from some distribution.
• The image consists of piecewise smooth regions that are separated by lines.
In this book we will treat denoising at the following places: Example 3.12,
Example 3.25, Sect. 3.4.4, Sect. 3.5, Example 4.19, Exam-
ple 5.5, Remark 5.21, Example 5.39, Example 5.40, Example 6.1,
Application 6.94, and Example 6.124.
Image decomposition: This usually refers to an additive decomposition of an
image into different components. The underlying assumption is that an image is

a superposition of different parts, e.g.,
image = cartoon + texture + noise.
Here, “cartoon” refers to a rough sketch of the image in which textures and
similar components are omitted, and “texture” refers to these textures and other
fine structure.
A decomposition of an image can be successful if one has good models for the
different components.
In this book we will treat image decomposition at these places: Example 4.20,
Sect. 6.5.
Enhancement, deblurring: Besides noise, there are other errors that may be
present in images:
• Blur due to wrong focus: If the focus is not adjusted properly, one point in the
image is mapped to an entire region on the film or chip.
• Blur due to camera motion: The object or the camera may move during
exposure time. One point of the object is mapped to a line on the film or
chip.
• Blur due to turbulence: This occurs, e.g., as “shimmering” of the air above a
hot street but also is present in the observation of astronomic objects.
• Blur due to erroneous optics: One of the most famous examples is the Hubble
Telescope. Only after the launch of the telescope was it recognized that one
mirror had not been made properly. Since a fix in orbit was not considered
appropriate at the beginning, elaborate digital methods to correct the resulting
errors were developed.
See Fig. 1.5 for illustrations of these defects.
The goal of enhancement is to reduce the blur in images. The more is known
about the type of blur, the better. Noise is a severe problem for enhancement and
deblurring, since usually, noise is also amplified during deblurring or sharpening.
Methods for deblurring are developed at the following places in this book:
Application 3.24, Remark 4.21, Example 6.2, Appplication 6.97,
and Example 6.127.
Edge detection: One key component to the understanding of images is the
detection of edges:
• Edges separate different objects or an object from the background.
• Edges help to infer the geometry of a scene.
• Edges describe the shape of an object.
Edges pose different questions:
• How to define an edge mathematically?
• Edges exist at different scales (e.g., fine edges describe the shape of bricks,
while coarse edges describe the shape of a house). Which edges are important
and should be detected?

8 1 Introduction
Fig. 1.5 Blur in images. Top left: Blur due to wrong focus; top right: motion blur due to camera
shake; bottom left: shimmering; bottom right: an image from the Hubble Telescope before the error
was corrected
We cover edge detection at the following places: Application 3.23 and
Application 5.33.
Segmentation: The goal of segmentation is to decompose an image into different
objects. In its simplest form, one object is to be separated from the background.
At first glance, this sounds very similar to edge detection. However, in segmen-
tation one focuses on decomposing the whole image into regions, and it may be
that different objects are not separated by an edge. However, if all objects are

separated by edges, edge detection is a method for segmentation. In other cases,
there are other methods to separate objects without focusing on edges.
There are several problems:
• While object boundaries are easy to see for the human eye, they are not easy
to detect automatically if they are not sharp but blurred due to conditions as
described above.
• Objects may have different color values, and often these are distorted by noise.
• Edges are distorted by noise, too, and may be rough, even without noise.
Segmentation is treated at the following places in this book: Application 3.11,
Application 3.40, and in Sect. 6.5.
Optical flow computation: Movement is another attribute that can be used to
identify and characterize objects and help to understand a scene. If one considers
image sequences instead of images one can infer information about movements
from digital data.
The movement of an object may result in a change of the gray value of a specific
pixel. However, a change of the gray value may also be caused by other means,
e.g., a change in illumination. Hence, one distinguishes between the real field of
motion and the so-called optical flow. The real field of motion of a scene is the
projection of the motion in the three-dimensional scene onto the image plane.
One aims to extract this information from an image scene. The optical flow is
the pattern of apparent motion, i.e., the change that can be seen. The real field of
motion and the optical flow coincide only in special cases:
• A single-colored ball rotates. The optical flow is zero, but the real field of
motion is not.
• A ball at rest is illuminated by a moving light source. The real field of motion
is zero, but the optical flow is not.

10 1 Introduction
The correspondence problem is a consequence of the fact that the optical flow
and the real field of motion do not coincide. In some cases different fields of
motion may cause the same difference between two images. Also there may be
some points in one image that may have moved to more than one place in the
other image:
• A regular grid is translated. If we observe only a small part of the grid, we
cannot detect a translation that is approximately a multiple of the grid size.
• Different particles move. If the mean distance between particles is larger than
their movements, we may find a correspondence; if the movement is too large,
we cannot.
• A balloon is inflated. Since the surface of the object increases, one point does
not have a single trajectory, but splits up into multiple points.

The aperture problem is related the correspondence problem. Since we see only
a part of the whole scene, we may not be able to trace the motion of some object
correctly. If a straight edge moves through the picture, we cannot detect any
motion along the direction of the edge. Similarly, we are unable to detect whether
a circle is rotating. This problem does not occur if the edges of the object have a
varying curvature, as illustrated by this picture:
To solve these problems, we make additional assumptions, for example:
• The illumination does not change.
• The objects do change their shape.
• The motion fields are smooth (e.g., differentiable).
In this book we will not treat methods to determine the optical flow. In Chap. 6,
however, we will introduce a class of methods that can be adapted for this task,
see also Sect. 6.5. Moreover, the articles [17, 24, 78] and the book [8] may be
consulted.
Registration: In registration one aims to map one image onto another one. This
is used in medical contexts, for example: If a patient is examined, e.g., by CT
at different times, the images should show the same information (apart from
local variations), but the images will not be aligned similarly. A similar problem
occurs if the patient is examined with both a CT scan and a scan with magnetic
resonance tomography. The images show different information, but may be
aligned similarly.
The task in registration is to find a deformation of the image domains such that
the content of one image is mapped to the content of the other one. Hence, the

12 1 Introduction
problem is related to that of determining the optical flow. Thus, there are similar
problems, but there are some differences:
• Both images may come from different imaging modalities with different
properties (e.g., the images may have a different range of gray values or
different characteristics of the noise).
• There is no notion of time regularity, since there are only two images.
• In practice, the objects may not be rigid.
As for optical flow, we will not treat registration in this book. Again, the methods
from Chap. 6 can be adapted for this problem, too; see Sect. 6.5. Moreover,
one may consult the book [100].
Restoration (inpainting): Inpainting means the reconstruction of destroyed parts
of an image. Reasons for missing parts of an image may be:
• Scratches in old photos.
• Occlusion of objects by other objects.
• Destroyed artwork.
• Occlusion of an image by text.
• Failure of sensors.
• Errors during transmission of images.
There may be several problems:
• If a line is covered, it is not clear whether there may have been two different,
separated, objects.
• If there is an occluded crossing, one cannot tell which line is in front and
which is in back.

We are going to treat inpainting in this book at the following places: Sect. 5.5,
Example 6.4, Application 6.98, and Example 6.128.
Compression: “A picture is worth a thousand words.” However, it needs even
more disk space:
1 letter = 1bByte
1 word ≈ 8 letters = 8 bytes
1000 words ≈ 8 KB.
1 pixel = 3 bytes
1 picture ≈ 4,000,000 pixels ≈ 12 MB
So one picture is worth about 1,500,000 words!
To transmit an uncompressed image with, say, four megapixels via email with an
upstream capacity of 128 KB/s, the upload will take about 12 min. However,
image data is usually somewhat redundant, and an appropriate compression
allows for significant reduction of this time. Several different compression
methods have entered our daily lives, e.g. JPEG, PNG, and JPEG2000.
One distinguishes between lossless and lossy compression. Lossless compression
allows for a reconstruction of the image that is accurate bit by bit. Lossy
compression, on the other hand, allows for a reconstruction of an image that
is very similar to the original image. Inaccuracies and artifacts are allowed as
long as they are not disturbing to the human observer.
• How to measure the “similarity” of images?
• Compression should work for a large class of images. However, a simple
reduction of the color values works well for simple graphics or diagrams, but
not for photos.
We will treat compression of images in this book at the following places:
Sect. 3.1.3, Application 4.53, Application 4.73, and Remark 6.5.

Chapter 2
Mathematical Preliminaries
Abstract Mathematical image processing, as a branch of applied mathematics, is
not a self-contained theory of its own, but rather builds on a variety of different
fields, such as Fourier analysis, the theory of partial differential equations, and
inverse problems. In this chapter, we deal with some of those fundamentals that
commonly are beyond the scope of introductory lectures on analysis and linear
algebra. In particular, we introduce several notions of functional analysis and
briefly touch upon measure theory in order to study classes of Lebesgue spaces.
Furthermore, we give an introduction to the theory of weak derivatives as well as
Sobolev spaces. The following presentation is of reference character, focusing on
the development of key concepts and results, omitting proofs where possible. We
also give references for further studies of the respective issues.
Mathematical image processing, as a branch of applied mathematics, is not a self-
contained theory of its own, but rather builds on a variety of different fields,
such as Fourier analysis, the theory of partial differential equations, and inverse
problems. In this chapter, we deal with some of those fundamentals that commonly
are beyond the scope of introductory lectures on analysis and linear algebra. In
particular, we introduce several notions of functional analysis and briefly touch upon
measure theory in order to study classes of Lebesgue spaces. Furthermore, we give
an introduction to the theory of weak derivatives as well as Sobolev spaces. The
following presentation is of reference character, focusing on the development of
key concepts and results, omitting proofs where possible. We also give references
for further studies of the respective issues.
2.1 Fundamentals of Functional Analysis
For image processing, mainly those aspects of functional analysis are of interest that
deal with function spaces (as mathematically, images are modeled as functions).
Later, we shall see that, depending on the space in which an image is contained,
K. Bredies, D. Lorenz, Mathematical Image Processing,
Applied and Numerical Harmonic Analysis,
https://doi.org/10.1007/978-3-030-01458-2_2
15

16 2 Mathematical Preliminaries
it exhibits different analytical properties. Functional analysis allows us to abstract
from concrete spaces and obtain assertions based on these abstractions. For this
purpose, the notions of normed spaces, Banach- and Hilbert spaces are essential.
2.1.1 Analysis on Normed Spaces
Let K denote either the field of real numbers R or complex numbers C. For complex
numbers, the real part, the imaginary part, the conjugate, and the absolute value are
respectively defined by
z = a + ib with a, b ∈ R : Re z = a, Im z = b, z = a − ib, |z| =
√
zz.
Definition 2.1 (Normed Space) Let X be a vector space over K. A function · :
X → [0, ∞[ is called a norm if it exhibits the following properties:
1. λx = |λ|x for λ ∈ K and x ∈ X, (positive homogeneity)
2. x + y ≤ x + y for x, y ∈ X, (triangle inequality)
3. x = 0 ⇔ x = 0. (positive definiteness)
The pair (X, · ) is then called a normed space.
Two norms · 1, · 2 on X are called equivalent if there exist constants
0 c C such that
cx1 ≤ x2 ≤ Cx1 for all x ∈ X.
In order to distinguish norms, we may add the name of the underlying vector
space to it, for instance, · X for the norm on X. It is also common to refer to X
itself as the normed space if the norm used is obvious due to the context. Norms
on finite-dimensional spaces will be denoted by | · | in many cases. Since in finite-
dimensional spaces all norms are equivalent, they play a different role from that in
infinite-dimensional spaces.
Example 2.2 (Normed Spaces) Obviously, the pair (K, | · |) is a normed space. For
N ≥ 1 and 1 ≤ p ∞,
|x|p =
N

i=1
|xi|p
1/p
and |x|∞ = max
i∈{1,...,N}
|xi|
define equivalent norms on KN . The triangle inequality for | · |p is also known as
the Minkowski inequality. For p = 2, we call | · |2 the Euclidean vector norm and
normally abbreviate | · | = | · |2.
The norm on a vector space directly implies a topology on X, the norm topology
or “strong” topology: for x ∈ X and r 0, define the open r-ball around x as

2.1 Fundamentals of Functional Analysis 17
the set
Br(x) = {y ∈ X

x − y r}.
A subset U ⊂ X is
• open if it consists of interior points only, i.e., for every x ∈ U, there exists an
ε 0 such that Bε(x) ⊂ U,
• a neighborhood of x ∈ X if x is an interior point of U,
• closed if it consists of limit points only, i.e., for every x ∈ X for which for every
ε 0, the sets Bε(x) intersect the set U, one has also x ∈ U,
• compact if every covering of U by a family of open sets has a finite subcover,
i.e.,
Vi open, i ∈ I with U ⊂

i∈I
Vi ⇒ ∃J ⊂ I, J finite with U ⊂

j∈J
Vj .
The fact that the set U is a compact subset of X we denote by U ⊂⊂ X.
Furthermore, let the interior of U, abbreviated by int(U), be the set of all interior
points, and the closure of U, denoted by U, the set of all limit points. The set
difference ∂U = U int(U) is called the boundary. For open r-balls, one has
Br (x) = {x ∈ X

x − yX ≤ r},
which is why the latter is also referred to as a closed r-ball. We say that a subset
U ⊂ X is dense in X if U = X. In particular, X is called separable if it possesses a
countable and dense subset.
Normed spaces are first countable (i.e., each point has a countable neighborhood
base, cf. [122]). That is why we can also describe the terms closed and compact also
by means of sequences and their convergence properties:
• We say that a sequence (xn) : N → X converges to x ∈ X if (xn − x) is a null
sequence. This is also denoted by xn → x for n → ∞ or x = limn→∞ xn.
• The subset U is closed if and only if for every sequence (xn) in U with xn → x,
the limit x lies in U as well (sequential closedness).
• The subset U is compact if and only if every sequence (xn) in U has a convergent
subsequence (sequential compactness).
For nonempty subsets V ⊂ X, we naturally obtain a topology on V through
restriction, which is referred to as the relative topology. The notions introduced
above result simply through substituting X by the subset V in the respective
definitions.

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husband so constantly invited him, without letting him see she
mistrusted herself—more especially as he never failed in his respect
for her.
He became her only comforter and friend; and if from her youth and
inexperience she was likely to fall into even the appearance of any
error, it was this kind, this friendly monitor that guarded her from it.
His attentions became as necessary to her soul, as aliments of food
are to the support of the body, while the respectful distance of his
behaviour proved to her his passion was controlled by his respect.
Some surmises were at length insinuated to Mr Lee, to his wife's
dishonour. He paid little attention to them—but coming home one
night flushed with wine, and finding Mr Villars alone at supper with
her, (no unusual matter, and by his own request) he drew his sword,
and wounded him before he had time to defend himself! Mrs Lee
fainted away——on her recovery she removed herself from a house
to which no entreaties on his part could prevail on her to return—
declaring she would live no longer with a man who could at once
suspect her virtue, endanger his friend's life, and ruin her reputation.
The world talked differently about this affair. Should not the example
of the law be followed, which is so tender in criminal cases, that
delinquents are often found not guilty, for want of legal evidence, at
the same time that the court, the jury, and every one present at the
trial feel the strongest moral conviction of their guilt? Scandal on the
contrary always gives its most important and fatal decisions from
appearances and suppositions, though reputation is dearer to a
woman of honor than life itself. Mrs Lee experienced the
malevolence of her own sex particularly. What, said they, could
engage Mr Villars to devote all his time to her? is not friendship
between a man and a woman a chimera, the mark of a passion
which honor or self-interest bids them conceal? But whilst the world
represented this affair in the worst colours, Lady Frances wrote her
an affectionate letter, offering her assistance, and begging she would
communicate her real situation, that she might the more effectually

be enabled to serve her; to which Mrs Lee returned the following
answer.
'Dear Madam.
I received the honor of your letter, and find myself elevated by
your notice—if there can be pride that ranks with virtues, it is
that we feel from friendships with the worthy. The liberal
sentiments you express, are a proof of the goodness of your
heart——I have ever thought that to believe the worst is a mark
of a mean spirit, and a wicked soul; at least I am sure, that the
contrary quality, when it is not due to weakness of
understanding, is the fruit of a generous temper. In return for
your generosity, I will lay open my whole heart to you; and if in
consequence I lose your esteem, I shall at least have the
satisfaction resulting from a consciousness of my candour. This
is a liberty I should have taken before, had it not proceeded
from the timidity I felt in unbosoming myself to one whose
virtues I dreaded, and in discovering my weakness to one who I
think has none of her own. Your ladyship knows the trials I
suffered for many years; my conduct under the severest
mortifications human nature could sustain. I was wounded in
my affections, condemned and insulted in my person,
impoverished in my circumstances: I still had strength of mind
to regulate myself so as to meet your approbation: no species
of calamity was unknown to me, nor were there wanting those
of the other sex, who judged from my situation they might have
a chance of succeeding with me, if I was weak enough to listen
to them—but they soon gave up the pursuit, judging the excess
of my misfortunes had hardened my heart entirely against
certain impressions. But this was so far from being the case,
that my sorrows, my sufferings, rendered my heart (naturally
tender) more susceptible of that refined passion, which, when
dignified by respect, and softened by tenderness, found so
ready access to it[24].

In short, circumstanced as I was, if it is a crime to love, I am
very culpable! but had I unfortunately proceeded to any act
contrary to my engagements with Mr Lee, I myself would have
acquainted him with it, though, in the opinion of many, he
would not have deserved so much candour from me.
This being the real state of the case, I flatter myself your
ladyship will think me more weak than wicked, more frail than
culpable, more unfortunate than indiscreet. And I must now
acquaint you, that I am determined never to return to my
husband—I have consulted my reason on this subject, and
when we have done so, whatever the decision be, whether in
favour of our prejudices, or against them, we must rest
satisfied, since nothing can be more certain than this, that he
who follows that guide in the search of truth, as that was given
to direct him, will have a much better plea to make for his
conduct, than he who has resigned himself implicitly to the
guidance of others. My maxim is, our understanding, properly
exercised, is the medium by which God makes known his will to
us; and that in all cases, the voice of impartial reason is the
voice of God. Were my marriage even to be annulled, all the
theologians in the world could not prove the least impiety in it.
—Milton wrote the doctrine and discipline of divorce; wherein he
proves, that a contrariety of mind, destructive of felicity, peace,
and happiness, are greater reasons of divorce than adultery,
especially if there be no children, and there be a mutual consent
for separation.
He dedicated the second edition to the parliament of England,
with the assembly of divines——The latter summoned him
before the house of Lords, who, whether approving his doctrine,
or not favouring his accusers, dismissed him. Necessary and just
causes have necessary and just consequences: what error and
disaster joined, reason and equity should disjoin.
I see no reason why those who upon the evidence of more than
fourteen years experience are unsuited to each other, joined not

matched, should live disagreeably together, and exist miserably
—merely for the inadequate satisfaction of exulting upon the
degree of their patience in having to say they did not part. A
person may mistake in fixing love without knowledge of the
party, but he cannot err that finds cause to dislike from woeful
experience. It is, indeed, convenient for the lords of the creation
to inculcate another doctrine, upon the same principles that the
extreme and timorous attention to his own security made James
I. very anxious to infuse into his subjects the belief of divine
hereditary right, and a scrupulous unreserved obedience to the
power which God had set over them. Mr Villars, who is now
reconciled with my husband, has written to intercede in his
behalf, assuring me of his penitence and affection. Boileau has
observed, that it is an easy matter in a Christian poem for God
to bring the devil to reason. Could I believe that all my husband
did, were the effects of love, it would not in the least alter my
resolution, since I should consider a person whose affection had
such dreadful effects, as dangerous to my repose, as one whose
anger was implacable.——What signifies it to me whether it be
love or hatred by which I suffer, if the danger and inconvenience
be the same? I am certain were we to live together again,
whenever we met we should as naturally quarrel as the
elephant and the rhinoceros. Reconciliations in the marriage
state, after violent breaches, are seldom lasting, and after what
has passed between us, like the father of the gods and the
queen of heaven, we shall be the best company when asunder.
He says his conduct proceeded from an excess of love! I desire
to be subject no more to such excesses! I am content to be
moderately beloved; nor shall I ever again give occasion for
such extraordinary proofs of affection. Were I to act otherwise,
it would afford too much encouragement for the men to use
their wives ill. Too good subjects are apt to make bad kings. He
has my consent to live with any woman who can delight in such
a loving husband, while I will force him to esteem my conduct,
and irritate his animosity by declining a reconciliation. We are

tired with perpetual gratitude, and perpetual hatred.——He
wishes to be reconciled to me, not from any religious motive, or
return of affection, his animosity being still the same—but
because he is tired of acting the part of a provoked husband.
I am piqued at Mr Villars's interesting himself in this matter. I
shall not answer his letter for a week; I mistrust my own
vivacity.
Our imagination is often our greatest enemy: I am striving to
weary mine before I act. Business like fruit hath its time of
maturity, and we should not think of dispatching it while it is
half ripe. The Cardinal de Retz said, 'I have all my life-time held
men in greater esteem for what they forbore to do on some
occasions than for what they did.'
I have here a most delightful dwelling——It is thatched, and
covered on every side with roses, wood-bines, and honey-
suckles, surrounded with a garden of the most artful confusion.
The streams all around murmur, and fall a thousand ways. A
great variety of birds are here collected, and are in high
harmony on the sprays. The ruins of an abbey enhance the
beauties of this place: they appear at the distance of four
hundred yards from the house; and as some great trees are
now grown up among the remains, and a river winds among the
broken walls, the view is solemn, the picture fine. Here I often
meditate on my misfortunes.
'There is a joy in grief when peace dwells
in the breast of the sad.'
Ossian's Poems.
Sadness receives so many eulogiums in the scripture, that it is
easy to judge, that if it be not of the number of the virtues, it
may be usefully employed in their service——and it may be truly
observed, that without experiencing sorrow, we should never
know life's true value.

About a mile above the house is a range of very high hills, the
sight of which renders me less incredulous of the accounts of
Olympus, and mount Athos. Hygeia resides here, and dispenses
the chief blessings of life, ease and health. I will pass my days
in sweet tranquillity and study.
'In either place 'tis folly to complain,
The mind, and not the place, creates the pain.'
Horace, lib. i. epist. 14.
Could I flatter myself I should ever be honored by your
presence, how happy I should be!—--Your eye, I am sure, would
catch pleasure while it measures the surrounding landscape
(even at this season of the year) of russet lawns and grey
fallows, on which stray the nibbling flocks: the mountains too,
which seem to support the labouring clouds, add sublimity to
the charming scene. When I take a walk after a sedentary
occupation, I feel a sensible pleasure; rest in its turn becomes
agreeable, if it has been preceded by a moderate fatigue. Every
action of our lives may be converted into a kind of pleasure, if it
is but well timed: Life owes all its joys to this well-adapted
succession; and he will never enjoy its true relish, who does not
know to blend pleasure with dissipation. I ask pardon for
detaining your ladyship so long—My cousin Lord Darnley has
been to see, and admires my cottage.—I perceive plainly he
flatters himself that you will one day make him happy. I do not
presume to offer my advice; it would be imitating the savage
chief, who marks out to the sun the course it is to take——but
surely his respectful, uninterrupted attachment deserves your
consideration. Were I not perfectly convinced of his worth and
sincerity, I should be the last person to speak in his behalf. The
bitterness of conjugal repentance, which I have experienced, is
beyond all others poignant; and happy it is if disunion, rather
than perpetual disagreement, results from it.

I ever am your ladyship's
Obliged and affectionate friend,
Lucy Lee.'
Lady Frances returned Mrs Lee immediately the following answer.
'Dear Madam, Munster-house.
I return you many thanks for the confidence you honored me
with; and I sincerely sympathize with you on the many
disagreeable events that have occurred to you. If my
approbation can confer on you any satisfaction, you possess it
in a very eminent degree: for though I cannot approve of your
sentiments concerning divorce, etc. yet your conduct in your
family was exemplary.
There is no reasoning about the motions of the heart. Reflection
and sensation are extremely different—our affections are not in
our own power, though yours seem to have been under proper
regulations.
I am not surprised at the calumny you met with. Many people
stoop to the baseness of discovering in a person distinguished
by eminent qualities, the weaknesses of humanity, while there is
scarcely to be found an honest heart, who knows how to render
a noble and sincere homage to another's superiority. I
acknowledge myself guilty with respect to you, of a too common
instance of injustice, that of desiring that others would always
conduct themselves by our maxims! I am the more culpable, as
I entirely agree with you in thinking that all our actions should
proceed from the fixed principles we have adopted. I never pay
a blind deference to the judgment of any man, or any body of
men whatever. I cannot acquiesce in a decision, however
formidable made by numbers, where my own reason is not
satisfied. When the mind has no data, no settled principles to
which it may recur as the rule of action, the agent can feel little

or no satisfaction within himself, and society can have no moral
security whatever against him.
The most permanent, the most pleasing enjoyment the human
soul is capable of entertaining, is that which arises from a
consciousness of having acted up to that standard of rectitude
which we conceive to be the proper measure of our duty: and
the best grounds on which we can expect others to place
confidence in us, is the assurance we give them that we act
under the influence of such moral obligations. This principle has
influenced my conduct: and as you say you are absolutely
determined never to live with your husband again; although my
sentiments do not correspond with yours on that head, I will
add nothing further on that subject, but refer you to certain
passages in scripture, which I think on sober reflection must
invalidate your present opinion[25].
The caprice you have often tacitly blamed me for respecting
Lord Darnley, had you known the motives for, you would have
approved—I will now in reward for your candour to me be
equally sincere with you—trusting to your honor, that you will
not divulge what is it so material to me to conceal.
At the time I agreed to give Lord Darnley my hand, I was at
liberty to indulge my inclinations, and to devote myself entirely
to him: But on my father's death, when I found the estate in my
possession, I considered myself as mother to my brother's
children. This was my motive for rejecting the man I (did, and
do now) fondly love: who by his generous and friendly, his
respectful and tender behaviour, deserves every thing from me.
Whoever pretends to be without passions, censures the wisdom
of that Power which made him; and if men of sense (for they
alone are capable of refined pleasure) would so far admit love,
as not to exclude their necessary and more important duties,
they need not be ashamed to indulge one of the most valuable
blessings of an innocent life. I honor the married state: and
have high ideas of the happiness resulting from an union of

hearts. Domestic society is founded on the union betwixt
husband and wife. Among all the civilized nations, this union
hath been esteemed sacred and honorable; and from it are
derived those exquisite joys, or sorrows, which can embitter all
the pleasures, or alleviate all the pains in human life. The heart
has but a certain degree of sensibility, which we ought to be
economists of. Lord Darnley engrossed my whole soul; nothing
could afford me any pleasure which had no reference to him.—
He was ever uppermost in my thoughts, and I bestowed only a
secondary reflection on all other subjects.
I could have cheerfully, for his conversation, abandoned all
society on earth beside, and have been more blessed, than if,
for them, I had been deprived of his. But if we suffer one
particular duty (even the worship of the Deity) to engross us
entirely, or even to encroach upon the rest, we make but a very
imperfect essay towards religion, or virtue; and are still at a
considerable distance from the business of a moral agent. The
dial that mistells one hour, of consequence is false through the
whole round of day.
Virtue, in my acceptation, is nothing else than that principle by
which our actions are intentionally directed, to produce good, to
the several objects of our free agency. I was aware, that it was
not only necessary that I should mean to act a right part, and
take the best way which could direct me to effect it, but that I
should previously take those measures which were in my power
to acquire the knowledge of my duty, and of the weaknesses I
had to guard against. I was sensible, that, had I given my hand
to Lord Darnley, I would have been defective in the duties
incumbent on me to my own family:—Love would have taken
entire possession of my soul, and shut up the avenues of my
heart against every other sentiment. Upon this occasion I felt
how justly the sacrifice of our own happiness is placed among
the highest virtues. How painful must it be to the most
generous heart! Men lose their lives to honor—I relinquished my
love—the life of life. I am sensible I have been condemned for

permitting him to be so much with me: but what recompence
can the world bestow on me, for relinquishing the society of a
real and tender friend? Common attachments, the shadows of
friendship, the issue of chance, or fantastic likings, rashly
cemented, may as hastily be dissolved: but mine has had the
purest virtue for its basis, and will subsist whilst vital breath in
me remains. My affections are founded on those amiable
qualities, which are seldom united, and therefore but little liable
to be displaced. My partiality is founded on esteem: take away
the cause, the effect will cease. The dread of the world has
never yet withheld me from following the bent of my own
inclinations, and the dictates of my own heart, not the dread of
censure ever influenced my conduct.
Your mention of his continued attachment is highly flattering,
and very pleasing—There you touched the tenderest springs of
my heart, bring me down to all the softness of my sex, and
press upon me a crowd of tender, lovely, ideas—
If the consciousness of good-will to others, though inactive, be
highly delightful, what a superior joy have I not experienced,
my dear friend, in exerting this disposition, in acts of
beneficence! Is not this the supreme enjoyment in nature? It is
true, the great works I have carried on, the encouragement I
have given to learning, the manufactories I have introduced into
this kingdom, etc. etc. have procured me the suffrage of the
world, and may transmit my name down to posterity. But what
flatters me most is, that if I have acquired any fame, it is
derived from the man I love. My acquaintance with him, has
been a happiness to my mind, because it has improved and
exalted its powers. The epithet of great, so liberally bestowed
on princes, would, in most cases, if narrowly scanned, belong
rather to their ministers. Unassisted by Agrippa and Mecænas,
where should we have placed Augustus? What is the history of
Lewis XIII. but the shining acts of Richelieu? Lewis XIV. was
indeed a great king; but the Condés, the Turennes, as well as
the Luvois, and Colberts, had no small share in acquiring the

glories of his reign. In all situations of life, it is of great
consequence to make a right choice of those we confide in—It
is on that choice our own glory and peace depend.—But it is still
more so to princes, or persons of large property. A private man
will find a thousand persons ready to open his eyes, by
reproaching him with the wrong steps into which bad advice
drew him; whereas courtiers, or those who are interested,
approve and applaud whatever the prince or the great person
does. An ingenious courtier replied to his friend, who upbraided
him with his too great complaisance for the emperor who had
made bad verses, which he commended; Would you have me
have more sense than a man who commands twelve legions,
and can banish me?
That day my nephew is of age, I shall assign over his estate,
and acquaint him of his obligations to Lord Darnley, to whom, at
the same time, I shall offer my hand, if I have reason then to
think it shall be agreeable to him. If it should not, I shall be
mortified, though I shall not deck my brow with the plaintive
willow. I need not tell you how agreeable it will be for me to see
you at this place, which is considerably improved since you were
here last. This day month I give a feast, in imitation of the
Saturnalia[26]; make me happy by your presence on that
occasion.
I remain, with great esteem,
Your affectionate friend,
Frances Finlay.
Mrs Lee, soon after the receipt of the above letter, came to Munster-
house, where she generally resided during the winter months, (after
her separation from her husband) retiring to her cottage in Wales, in
the summer.
Lady Frances had always a select number of friends with her.
Notwithstanding her passion for music, she kept the performers in
their own line; and though she venerated the liberal sciences, and

contributed so largely to their cultivation, their several professors
only waited on her by invitation: by this means she had it always in
her power to suit her company, and never to be intruded on; as the
best things are irksome to those whose inclinations, tastes, and
humours, they do not suit.
I have already mentioned Mrs Norden, who had the care of Lady
Frances's education, and who now continued to reside with her: this
Lady's seriousness was happily contrasted with Lady Eliza's
sprightliness, while Lady Frances's scientifical knowledge was
agreeably relieved by the strokes of nature observable in Mrs Lee—
who declared she had never read, or studied, any more than to
assist her decyphering what was incumbent her to understand. 'I
hate your wise ones,' said she, 'there is no opinion so absurd but it
has been mentioned by some philosopher.' She is nature itself,
without disguise, quite original disdaining all imitation, even in her
dress, which is simple but unaffected. She plays most divinely on the
fiddle. Her genius for music is sublime and universal. She holds the
fiddle like a man, and produces music in all its genuine charms,
raising the soul into the finest affections.
An aunt and sister of Sir Harry Bingley's were also much at Munster-
house. Miss Bingley was of the same age with Lady Eliza: to the
charms of a regular beauty she joins all those of a cultivated mind,
together with a disposition replete with candor, and a turn for
ridicule; two things rarely joined together—as a calm dispassionate
love of truth, with a disposition to examine carefully, and judge
impartially, with a love of diverting one's self at other people's
expense, seldom meet together in the same mind. Mrs Dorothea
Bingley is a maiden lady of fifty, possessed of a large independent
fortune, which she proposes to bestow on her niece. She was in her
youth very handsome: but having lived all her life in the country, she
derived all her ideas of love from the heroic romance. To talk to her
of love was a capital offence. Her rigour must be melted by the
blood of giants, necromancers, and paynim knights. She expected,
that, for her sake, they would retire to desarts, mourn her cruelty,
subsist on nothing, and make light of scampering over impassable

mountains, and riding through unfordable rivers, without
recollecting, that, while the imagination of the lover is linked to this
muddy vesture of decay, she must now and then condescend to
partake of the carnality of the vivres of the shambles.
Those of the other sex who were mostly at Munster-house, were,
Lord Darnley, Sir Harry Bingley, Sir James Mordaunt, etc. etc. etc.
Great marriages had been proposed to Lady Frances; but she had
ceased long to be importuned on that head. When Lord Munster was
of age she gave a splendid entertainment to the neighbourhood,
which finished with a ball. The day after she shewed her nephew the
state of her affairs, when she succeeded to the estate: and that,
exclusive of the buildings, etc. etc. she had already doubled it: that
the perpetual burdens she had entailed on it, did not amount to one
quarter of the advanced rents, which would continue to encrease:
that she had put aside for Lady Eliza's fortune fifty thousand pounds,
and an equivalent sum for herself, and then with great pleasure
resigned the remainder to his Lordship, who she was happy to find
so worthy of filling the place of his ancestors. She at the same time
acquainted him with her motives for concealing her intentions in his
favor, and that, had she seen him addicted to any irregularities, she
would not have assigned over the property so soon to him—as the
law of this country does not interfere like that of France, where, if a
person, before he attains the age of twenty-five, wastes his fortune
by anticipation, or other means, and is in a fair way of ruining
himself, and, perhaps, his family; the government interposes:
guardians of his estate are appointed, and his person may be
detained in custody till he arrives at that age; but there the
jurisdiction stops. The acknowledgments of Lord Munster are easier
to be conceived, than expressed—he concluded by saying, 'he hoped
Lady Frances would always consider Munster-house as still her own,
and make it her principle residence!' She smiled, and looking to Lord
Darnley, said, 'Having my lord performed my duty to this family; it is
now in my power to make myself happy by conforming to your
wishes—Sixteen years ago, I had singly an engagement to fulfil; but
I have now a breach of it to repair.' Lord Darnley's joy may easily be

supposed great on this occasion, who had maintained for Lady
Frances, for so long a time, an uninterrupted attachment.—They
were married a few days afterwards. Never did Phœbus gild a more
auspicious day; never did Cupid inspire two lovers with a higher
sense of each other's merit; and never did Hymen light his torch
with a greater complacency, than to reward that constancy which
remained invincible in Lord Darnley, without even being supported
by hope.
The part Lady Darnley performed would have been difficult for
another; but the club which a man of ordinary size could but lift, was
but a walking-stick to Hercules.
No one enjoyed this wedding more than Mrs Dorothea Bingley. A
sixteen years courtship corresponded entirely with her ideas of the
right and fitness of things. She harangued her niece and Lady Eliza
on this subject, telling them that Lady Darnley is the only woman
she knows in this degenerate age, that has acted up to the propriety
of the ancients—that she respected the sublimity of her ideas. She
was very desirous of her niece's marrying a Mr Bennet, because he
made love in heroics, was inebriated in his science, and thought all
the world considered him as a Phoenix of wit. Miss Bingley would
often reason with her aunt on this subject? 'Of what use in the world
(said she) is an erudition so savage, and so full of presumption?'

One moral, or a mere well-natur'd deed,
Does all desert in sciences exceed.'
Sheffield
But Mrs Dorothea always insisted that he was a classical scholar, and
a fine gentleman! The niece declared he was a Pagan, and ought to
have lived two centuries ago, as he spoke a language she did not
understand! 'He may be learned (said she) but he has no passion!'
'No passion (replied Mrs Dorothea) how comes he then to write such
fine letters?'
'The fine letters (replied Miss Bingley) show memory and fancy, but
no sensations of the heart! lovers who make use of extravagant
tropes are reduced to that expedient, to supply the defect of passion
by the deceitful counterfeit of hyperbolical language. The passions of
the heart depend not on the deductions of the understanding—but it
was necessary he should have a Corinna, because Ovid had one;
and he makes me inconstant, although I never gave him any
encouragement, because Gallus's favourite run away with a soldier.
He seems to be intimately acquainted with the history of Cupid and
Venus, but knows nothing of love: and would be sooner applauded
for writing a good elegy, than have his mistress smile on him.'
Mrs Dorothea told her, that she was exceedingly perverse, but she
would give her leave to talk, as she had the power to do.
Miss Bingley said, 'Since Mr Bennet was so much in her good graces,
she made no doubt but he would pay her his homage, on the
smallest hint, would transfer his affections—as the foundation of his
passion was the same for both, built on that of her mansion, would
grow with her trees and increase with her estate——Increase, you
know, my dear aunt, is the end of marriage; and your fortune is
better than Medea's charm, for that only made an old man young
again; but your riches will make a young man enamoured of an old
woman! He will swear you are not only wiser than Minerva, but

fairer than the Paphian queen! Though you are old, your trees are
green; and though you have lost the roses in your cheeks, there are
great plenty of them on your pleasure-grounds.'
Mrs Dorothea with great good-humour laughed at her niece's sallies,
saying, 'You remember what Martial says;
'Fain would kind Paula wed me if she could:
I won't, she's old; if older yet, I would.'
'But seriously, niece (said she) you will never make a choice that I
shall so much approve of—he has so much wit.'
Miss Bingley replied, that all the credit he has for wit is owing to the
gratification he gives to others ill-nature: and said she would be very
happy to accommodate herself to her aunt's wishes; but was not
upon such a religious strain, and so desirous of canonization
hereafter (if sufferings can make a saint) as to marry a man of his
character, that she might have her mortifications and punishments in
this life: but at the same time would faithfully promise never to
marry any man she disapproved of.
There were great rejoicings for some weeks at Munster-house:—at
which time Lord and Lady Darnley set out for their estate in
Dorsetshire, and Lady Eliza accompanied Lord Munster to London.
As a correspondence commenced at this period between the parties
I have already introduced to the reader, the sequel of this history will
appear from their letters. I shall only observe, that Lord Munster's
figure was remarkably agreeable, his address engaging; he first
attracted, and then commanded the admiration of all who knew him.
On the slightest acquaintance with him, a most exact regard to all
the proprieties and decencies of life were observable in his conduct;
and such an evident desire to oblige, and to make all about him
easy, as became a good mind and a liberal education. An agreeable
chearfulness made his conversation as lively and agreeable as it was
useful and instructing. But the discerning eye of friendship could
discover that he was not happy, and that delicacy to the feelings of
his friends restrained him from giving way to an uneasiness, which it

was too apparent he laboured under. His general behaviour bore the
genuine stamp of true politeness, the result of an overflowing
humanity and benevolence of heart. Such qualities very justly and
forcibly recommend, lying obvious to almost every observer; but to
the more discerning, a nearer view of him quickly discovered
endowments far above the common standard. He had, in truth,
endowments of mind to have honored any station.
As Lady Darnley's breast glowed with that exalted fervent charity
which embraces the wide extended interests of men, of
communities, of the species itself; it is easy to conceive how her
heart exulted at finding her nephew so deserving of all she had done
for him. But though she felt the greatest satisfaction at his being so
conformable to her wishes, and his fortune so adequate to his
beneficence; the same sensibility rendered her wretched for the
evident melancholy in which he was plunged. Her social affections
ever awake, even on those whose objects lie beyond the nearer ties
of nature, on many occasions gave her most painful sympathetic
feelings; so deeply was she interested in the fortunes of all with
whom she had any connection. How then must she mourn to
observe, that, notwithstanding the possession of every advantage of
person and wealth, her nephew was miserable!—If men would but
consider how many things there are that riches cannot buy, they
would not be so fond of them—for all the outward advantages Lord
Munster had, were, to a man in his situation of mind, landscapes
before a blind man, or music to one that is deaf.
Delicacy kept Lady Darnley from interrogating her nephew on the
subject of his grief; sensible that the remotest desire from her, must
amount to a command to him. She only, at parting, insinuated the
happiness it would afford her to see him ally himself suitably to
some lady of merit: and, as Lady Eliza was to accompany him to
town, requested him to moderate her liveliness, and to be a careful
observer of her conduct.
'I never see (said she) a single man, who hath passed middle age in
celibacy, where no particular security arises from his profession or

character; but I think I see an unsafe subject, and a very dangerous
instrument for any mischief that his own parts may inspire, or other
men's may prompt him to: As to other achievements of virtue, a
distinction ought, I think, to be made; because, in common
acceptation, there is a variety of things which pass under that name,
and are generally applauded, which, properly estimated, would not
deserve it. A regard to posterity hath carried arms, arts, and
literature, further than any other motive ever did or could. Who is so
likely to be influenced by this regard as they who are to leave behind
them the darling pledges of their affection, in whom they hope to
have their names continued, and all the fruits of their study, toil, and
exploits, abiding and permanent?' Lord Munster assured the
Countess, that he would ever think it his glory to conform to her
wishes in ever respect.
End of the First Volume

VOLUME II
Soon after Lord Munster's arrival in London, he wrote Lady Darnley
the following letter.
From the Earl of Munster to the
Countess of Darnley.
'My Dear Aunt,
Over powered as I am with a weight of obligations, I should
think myself highly wanting to my own feelings, were I in any
one instance in my future life to leave you dubious of my
gratitude, or the earnest desire I have of conforming to your
wishes.
You have, my dear Madam, expressed your desire I should
marry; but that, my dear aunt, is impossible at present. But I
revere that state: men who laugh at a serious engagement,
have never known the allurements of modesty when blended
with affability; nor felt the power of beauty, when innocence has
increased its force. This has been my case, and my heart is
already a prey to a hopeless passion. But it is necessary to carry
you back some years, in order to give a recital of its
commencement.

The amiable character of Mr Vanhagen, my landlord at
Rotterdam, you are already acquainted with: his humanity and
benevolence inspired me with the greatest respect. The
advantages his countrymen have over us, are their industry,
vigilance, and wariness: But they in general exert them to
excess, by which means they turn their virtues into vices. Their
industry becomes rapine, their vigilance fraud, their wariness
cunning. But my worthy landlord possessed all the virtues.
He had in the early part of his life resided much at Venice, and
brought from thence the economy and frugality which
distinguish them in their private families, their temperance, their
inviolable secrecy of public and private affairs, and a certain
steadiness and serenity to which the English are supposed to be
utter strangers. His long residence there, made him well known
to the duchess de Salis, whose distant relation he had married.
This lady had resided some years at Rotterdam with her family.
She was only daughter to the Count de Trevier, was heiress to a
large fortune, and possessed exquisite beauty, good-sense, and
every accomplishment that was likely to preserve and to
improve the authority beauty gives to make it indefectible and
interminable. But the duke, her husband, unfortunately was
soon satiated with the regularity of her virtues: His affections
could not long be preserved by a woman of her amiable
undisguised character. When custom had taken off the edge
from his passion, he endeavoured to rouse his torpid mind by a
change of object. That vivacity which the tender passions impart
to pleasure, was a powerful incentive for him to indulge them.
His heart found fresh delight in gallantry, to which he was
naturally prone: a dangerous delight, which, habituating the
mind to the most lively transports, gives it a distaste to all
moderate and temperate enjoyments: from thence forward the
innocent and tranquil joys which nature offers, lose all their
relish. His sophisticated mind made him blind to the merit of his
wife, who loved him tenderly.—She felt most severely his
neglect, and contracted insensibly a settled melancholy, which

served the more effectually to alienate his affections from her.
She became miserable:—and no temper can be so invincibly
good as to hold out against the siege of constant slights and
neglects. Misfortunes she had strength of mind to support, and
death she could have encountered with greater resolution than
the displeasure and peevishness of the man she loved.
Wherever there is love, there is a degree of fear—we are
naturally afraid of offending, or of doing any thing which may
lessen us in the esteem of an object that is dear to us: and if
we are conscious of any act by which we may have incurred
displeasure, we are impatient and miserable, till, by intreaties
and tokens of submission, we have expiated the offence and are
restored to favor.
By the duchess's earnest solicitude to please, she destroyed her
own purpose, and her obedience, like water flung upon a raging
fire, only inflamed her husband's follies; and therefore, when he
was in an ill humour, the duke vented his range on her. He did
not care how often he quarrelled with, or, to speak more
properly, how often he insulted her; for that could not be called
a quarrel wherein she acted no part but that of suffering. But
though his displeasure was grievous to her, yet she could bear it
better than his indifference—for resentment argues some
degree of regard. But whilst she was breaking her heart for him,
he passed his time in gallantry—though his affections were
always the satire of a woman's virtue—the ruin of a woman's
reputation.
A favourite mistress, by pursuing a different plan from that of
the duchess, secured his affections. She kept alive his ardour by
her caprices. Affectation always exceeds the reality. But is not
the extravagance of some men's fancy to be pitied, who lodge
all their passions in a mistress, a dog, or a horse, which but in
general do them no service but what they are prompted to
through necessity or instinct? Art and cunning are unknown to a
woman of virtue, whose conduct is determined by her
principles, whose anxiety alone is excited by affection.

After five years, in which the duchess had a son and a daughter,
and in which she had experienced many of the vexations, but
few of the satisfactions of a married state; the duke left her, and
resided entirely in Paris with his mistress. She retired to the
country, to a family-seat of her father's, and devoted her time
entirely to the education of her children, and that of a young
lady (of great beauty and fortune) whose mother with her last
breath bequeathed to her care.
She from time to time wrote the duke letters, expressing great
resignation, and such a tenderness for him as she thought
might have power to touch his heart. I am obedient to your
wishes, said she, I will not urge, with one unwelcome word,
this unkindness—I'll conceal it—If your heart has made a choice
more worthy, I forgive it—pursue your pleasures—drive without
a rein your passions—I am the mistress of my own mind, that
shall not mutiny—If I retrieve you, I shall be thankful—If not,
you are and must be still my lord.
To letters such as these she never received any answer! as the
charms of a woman's eloquence never have any force, when
those of her person are expired (in the eyes of her lover I
mean): it might be perhaps as easy to persuade a man to
dance, who had lost the use of his limbs.
I shall pass over the first ten years of her retirement, as they
furnish nothing more than the unwearied attentions she took in
employing every means for the instruction of her son, daughter,
and ward. I shall only observe, that the regularity of her
conduct gained her the esteem of every one. She was a friend
to virtue under any denomination, and an enemy to vice under
any colour. She established an institution for the provision of the
infirm and destitute. This was constructed on that wise and
excellent plan, that excludes the undeserving from participating
in the charity, and extends only to those who, from their real
necessities, are proper objects of benevolence.—At that period
she was advised to take her son to the capital. But she wisely

considered that the education which commonly attends high
birth or great fortune, very often corrupts or sophisticates
nature; whilst in those of the middle state she remains unmixed
and unaltered. I have somewhere read; Jamais les grandes
passions et les grandes vertus ne sont nées, ne se sont
nourries que dans le silence la retrait. L'homme en societé
perd tous ses traits distinctifs: ce n'est plus qu' une froide copie
de ce qui l'environne. Voilà pour quelle raison on nous accuse
de manquer de caractere: nous ne vivons pas assez avec nous-
mêmes, nous empruntons trop des autres.
The duchess procured for her son's tutor, a very respectable
man, who was at the utmost pains in forming his morals, and
improving his understanding; while so many of the degenerate
nobility in great cities are trifling away their time and their
fortunes, in idle dissipations, in sensual enjoyments, or irrational
diversions, and making mere amusement the great business of
their lives. Happiness and merit are the result, not so much of
truth and knowledge, as of attaining integrity and moderation.
Many ridiculed the duchess's plan of education, of debarring
herself from those pleasures and enjoyments her youth, rank,
and beauty so well intitled her to: But she often observed it
would be the height of imbecility to judge of her felicity by the
imagination of others; considering nothing under the title of
happiness, but what she wished to be in the possession of, or
what was the result of her own voluntary choice. Women of the
world counteract their intention, in so assiduously courting
pleasure, as it only makes it fly further from them. They will not
understand, that pleasure is to be purchased, and that industry
is the price of it; to reject the one, is to renounce the other.
They are to learn that pleasure, which they idolize, must now
and then be quitted in order to be regained. They have tried in
vain to perpetuate it, by attempting variety and refinement.
Their fertile invention has multiplied the objects of amusement,
and created new ones every day, without making any real
acquisition. All these fantastic pleasures, which are founded on

variety, make no lasting impressions on the mind; they only
serve to prove the impossibility of permanent happiness, of
which some women entertain chimerical expectations: but the
duchess was too rational to make amusement her principal
object. A woman that is hurried away by a fondness for it, is,
generally speaking, a very useless member of the community: A
party of pleasure will make her forget every connection: and she
is often sick without knowing where her complaint lies, because
she has nothing to do, and is tired of being well.
The duchess had loved her husband passionately. If any person
had a desire of ingratiating themselves with her, they had only
to begin by him: To praise, to please, or admire him, opened to
them a reception in her heart. But our best virtues, when
pushed to a certain degree, are on the point of becoming vices:
She soon found she was to blame, in dedicating herself too
fondly even to this beloved object. She exhausted her whole
sensibility on him, and in proportion to the strength of her
attachment, was the mortification she endured in being
abandoned by him. But had not even this been her fate, the
extravagant excesses of passion are but too generally followed
by an intolerable langour. The woman who wishes to preserve
her husband's affection, should be careful to conceal from him
the extent of hers: there should be always something left for
him to expect. Fancy governs mankind: and when the
imagination is cloyed, reason is a slave to caprice.
Women do not want judgment to determine, penetration to
foresee, nor resolution to execute; and Providence has not given
them beauty to create love, without understanding to preserve
it. The pleasures of which they are susceptible, are proportioned
to the capacity and just extent of their feelings. They are not
made for those raptures which transport them beyond
themselves: these are a kind of convulsions, which can never
last. But there are infinite numbers of pleasures, which, though
they make slighter impressions, are nevertheless more valuable.
These are renewed every day under different forms, and instead

of excluding each other, unite together in happy concert,
producing that temperate glow of mind which preserves it
vigorous, and keeps it in a delightful equanimity. How much are
those of the fair-sex to be pitied who are insensible to such
attainments, and who look upon life as gloomy, which is exempt
from the agitation of unruly passions! As such prepossessions
deprive them of pleasures which are much preferable to those
which arise from dangerous attachments, the duchess knew
how to make choice of her amusements, and improved her
understanding at the same time that she gratified her feelings.
Life to those who know how to make a proper use of it, is
strewed with delights of every kind, which, in their turn, flatter
the senses and the mind; but the latter is never so agreeably
engaged as in the conversation of intelligent persons, who are
capable of conveying both instruction and entertainment. The
duchess preferred the conversation of such, to men of the
world; being sensible she had every thing to gain on one side,
and every thing to lose on the other.
The Baron de Luce resided in the same part of the country. He
was a man of great gallantry, wit, and humour. He judged it
impossible that a woman in the bloom of beauty, possessed of
the united advantages resulting from rank, riches, and youth,
should retire to an obscure part of the world, and sequester
herself from (what he judged) the pleasures of life, without
being compelled by her husband or prompted by some secret
inclination which she wished to conceal. Determined to unravel
this mystery, and to amuse himself during the time he staid in
the neighbourhood, he tried to insinuate himself into her good
opinion—but without giving any offence she avoided entering
into his plans. He still persisted in his intentions, judging, as he
wrote well, the duchess would be glad to enter into a
correspondence; but he found nothing in the reception she gave
him that was for his purpose, to embellish the history of his
amours. But what he undertook at first from vanity, became at
last sufficient punishment for him. The more he saw of her

conduct the more his respect increased, but which instead of
making him relinquish his intentions (from a conviction of the
inefficacy of the pursuit) made him persist in them, as he then
felt the passion which at first he feigned.
The duchess knew the predicament on which she stood; but as
the hatred of men of a certain character is less pernicious than
their love, she gave orders never to admit him into her
presence. The good or bad reputation of women depends not so
much upon the propriety of their own conduct, as it does upon
a lucky or unlucky combination of circumstances in certain
situations. Some men calumniate them for no other reason, but
because they are in love with them. They revenge themselves
upon them for the want of that merit which renders them
despicable in their eyes. This was the case with the Baron; he
insinuated there were reasons which he knew that rendered it
highly proper for the duchess to live in the manner she did,
speaking in a style which conveyed more than met the ear! The
people he addressed greedily listened to what seemed to bring
the duchess more on a footing with themselves; a thousand
stories were circulated to her prejudice (though innocence
itself): Thus if there be but the least foundation for slander,
some people believe themselves fully authorized to publish
whatever malice dares invent. But there are no enemies more
dangerous to the reputation of women, than lovers that cannot
gain the reciprocal affection of their mistresses. These reports
were confirmed from another cause—A lady of fortune in the
neighbourhood became much attached to a man who resided
with the duchess as her son's tutor; he was ingenuous, sensible,
and much respected. She offered him her hand, and as she
possessed a handsome fortune could not conceive how he could
decline that happiness. As he was constantly at home,
agreeable to the stories that had been circulated, she concluded
at once (and then affirmed) he was a favourite of the duchess.
Self-love is of the nature of the polypus; though you sever her
branches or arms, and even divide her trunk, yet she finds

means to reproduce herself. In consequence of the information
the duke received from this lady, who wrote to him in the
character of an anonymous friend, he left Paris and his mistress
abruptly; and, to the great surprise of his wife, came to—. He
accosted her in a distant, but respectful manner.—Nothing gives
so sharp a point to one's aversion as good-breeding—The
duchess, unconscious of having given him any occasion of
offence, was highly delighted at his return, flattering herself
with a return of his affection. And as she considered him the
aggressor, received him graciously, insisting that no mention
should be made of past transactions; assuring him that she still
retained the same love for him, and as she regarded him as the
first of human beings, had perhaps been too sanguine in
expecting his constancy, as so many temptations must occur
from his superiority to the rest of mankind. She thought he was
but too amiable—that his very vices had charms beyond other
men's virtues. Adding that (grievous as his neglect had been to
her) yet she had never done anything that could reflect upon his
honor! He heard her in a sullen humour; his inclinations were
revived by remarking, that time, instead of diminishing, had
added to her charms: this increased his resentment, and he
answered, that the worst a bad woman can do, is to make
herself ridiculous; it is on herself only that she can entail infamy
—but men of honor have a degree of it to maintain, superior to
that which is in a woman's keeping. Had she had a mind to
retaliate, she might easily have said, that a man of honor and
virtue which, in themselves indeed, are always inseparably
connected, are but too often separated in the absurd and
extravagant opinions of mankind. For what a strange perversion
of reason is it, to call a person a man of honor who has scarcely
a grain of virtue! She only observed, we are indeed civilized into
brutes; and a false idea of honor has almost reduced us into
Hob's first state of nature, by making us barbarous. Honor now
is no more than an imaginary being, worshipped by men of the
world, to which they frequently offer human sacrifices. He told

her she needed not be troubled for her minion: and retiring to
rest, left her quite at a loss to account for his conduct.
It is not sufficient we know our own innocence; it is necessary,
for a woman's happiness, not to be suspected.
For unfortunately after she has been once censured (however
falsely) she must expect the envenomed shafts of malice ever
ready to be let fly at her, and that in the transaction of any
affairs that will admit of two interpretations (to avoid the worst,
and enjoy an unblemished reputation). It is not enough to
govern herself with propriety, there must be nothing that will
carry two interpretations in the accidents of her life: A woman
must therefore be necessarily always guilty, when innocence has
need of many justifications. Happy are those who are not
exposed to such inconveniences!
The Duke most injudiciously next morning publicly dismissed the
object of his jealousy, and, by his want of prudence, confirmed
every thing that had been falsely alledged against his innocent
wife, who continued ignorant of it for some months.
When acquainted with it—The less ground she saw for the
reports against her honor, the more courage and greater
resolution she had to condemn them. She thought herself
unfortunate to have lost the merit of her innocence by
scandalous reports which she thanked Heaven she had not
incurred by her guilt: and was so far from slighting the
probabilities that might confirm opinions founded against her,
that she by no means thought herself in the same situation with
others, who had never been contemned, and that consequently
she was not at liberty to act on some occasions as they might
do.
How many women err from the obstinacy of people in defaming
them—they give up the point, despairing of success in
conciliating the esteem of a world who never retract censure—It
is not with detraction as it is with other things that displease by

repetition: Stories that have been told a thousand times, are still
new when revived to the prejudice of another. The duchess bore
all these calumnies with patience, which was never yet a solitary
virtue: like an angler she endeavoured to humour the duke's
waywardness, flattering herself that her study to please would
conquer his disagreeable temper; and that if she could not
become a pleasing wife, she might at least be thought an
agreeable companion, a serviceable friend. Hope was the only
blessing left us, when Pandora's fatal box let out all the
numberless evils which infest these sublunary regions. But she
was at last obliged to resign all ideas of submitting longer to his
caprices. He became jealous even of his menial servants; and
she could speak to no man without incurring his suspicions;
which produced to her the most mortifying scenes. Like that
conqueror of China, who forced his subjects into a general
revolt, because he wanted to oblige them to cut their hair and
their nails, he reduced her to form the resolution of leaving him,
because (as he represented it) he had dismissed a servant. But
it was in reality his temper and abuse that occasioned it—and
when she was under the necessity of taking that step, she
rather let the world judge amiss of her, than justify herself at
her husband's expense. No condescensions on her part could
affect him, as daily experience convinced her, that from a
consciousness of the part he himself had acted, he could never
love her. Are there not many occasions in life in which it would
be reasonable to say, I conjure you to forget and forgive the
injury you have done me?
They at last parted amicably: she came to Rotterdam with her
family, and there I contracted an intimacy with her son, who
was an amiable young man about my own age. There I first
beheld the lovely Adelaude, Countess de Sons, the duchess's
ward: the first time I saw her, and the charming Julia, I know I
had a heart; until then I was insensible—These young ladies
were instructed in all the arts of Minerva; Julia was skilled in
music; but the countess's voice was, accompanied with the lyre,

more moving than that of Orpheus. Her hair hung waving in the
wind without any ornament, which the duchess had taught her
to despise: her motions were all perfectly easy, her smiles
enchanting! Without dress she had beauty, unconscious of any,
and thus were heightened all her charms.
The marquis enquired what I thought of his sister, and her fair
friend? I answered, They were charming, and asked if it was
possible he had resisted the charms of the beautiful countess?
He replied, I will own to you, my dear friend, I have not:
Adelaude is formed for love; my heart is naturally susceptible;
she has been my constant companion: he must be something
more, or something less than a man (a god or a devil) who hath
escaped, or who can resist love's empire.—The gods of the
heathens could not; Jupiter, Mars, Mercury, Apollo, their amours
are as famous as their names: so that sturdiness in human
nature, where it is found, which can resist, argues plainly how
much the devil is wrought up in the composition. But if my
sensibility had not been so great, yet so many opportunities she
has had to engage my affections, could not fail of rivetting me
her's for ever, You are beloved then said I hastily. Yes,
replied he, Adelaude calls me her dearest brother; but
entertains no ideas beyond that relation; and I am fearful of
letting her know the extent of my sentiments, lest it should
render her constrained in her manner to me; and the charming
naiveté of her behaviour forms the charms of my life! The
marks of that innocent affection, which first attached me to her,
have hitherto been looked upon as a childish play: and as no
one has troubled their head about the consequences of it, I
have taken care to profit by the liberty allowed me.—You make
me no answer!—Wherefore this gloomy silence, your dejected
air, and languishing looks? I pretended an indisposition, and
left him under the greatest oppressure of spirits; I loved, I
adored the charming Countess! judge then of the horror of my
situation.—

How many sacrifices could I not willingly have made to
friendship! My passion I thought was indeed the only one I
could not make: how was it possible I should? but convinced of
the happiness of my rival, what did I not suffer? I saw a pair of
happy lovers, suited to each other; I thought it would be safe to
alienate her affections; and considered myself only in the light
of a dependent on your bounty: in such a situation, had my
friend been uninterested, could I hazard addressing a young
lady of the countess's rank and fortune? I became melancholy
and distrait. Many people, and particularly those who have no
idea of that delicacy of passion peculiar to susceptible minds,
looked on me as a particular kind of a young man. To please
such persons, I must have devoted my time to them: you will
easily conceive then, I could well enough bear the want of their
good opinion. Such become the artificers of their own
misfortunes, by the false idea they form of pleasure, and they
philtre (if I may use the term) their own sorrows.
It was what is called pleasure, that sunk into ruin the ancient
states of Greece; that destroyed the Romans, that overturns
cities; that corrupts courts; that exhausts the fortunes of the
great; that consumes youth; that has a retinue composed of
satiety, indigence, sickness, and death. But my passion, as
much as a dislike to their manner of life, secured me from their
dissipations. The constant endeavours I used to suppress an
inclination I could not overcome, had a fatal effect on my
constitution—I was threatened with a consumption!—This I
carefully concealed, lest your kindness should have urged my
removal from a place, which I could not determine to quit:
though I carefully avoided the sight of those who were
interesting to me in it.
At this time the marquis received a peremptory command to
rejoin his father. He came to me in the greatest distress: How,
said he, can I resolve to leave the countess?—She is now
beautiful as an angel, exclusive of her immense fortune; to
remain single cannot possibly be long in her power, for her

beauty must necessarily strike every eye, and charm every
heart. But I will go and unburthen myself to my father; her
riches and rank will insure his approbation. You, my friend,
alone are acquainted with the secret of my heart. See the lovely
Adelaude often; to you I confide the secrets of my soul.
Farewel.
The marquis set out, and soon informed me that his father
would not yet hear of his marriage, and had insisted on his
immediately joining a regiment in which he had procured him a
command: It was in time of war; his honor at stake, and love
was subordinate to his glory. The susceptible mind is capable of
enjoying a thousand exquisite delights to which those are
strangers, whose pleasures are less refined; but what chagrin,
what regret, what pain does not so delicate a passion bring on
the heart that entertains it? Quand on est né trop tendre, on ne
doit pas aimer, says some French author. But the sufferings of
my friend could not equal mine; the object of my passion being
daily before my eyes heightened my inquietude. The general
characters of men, I am apt to believe, are determined by their
natural constitutions, as their particular actions are by their
immediate objects. The innocent marks of partiality she honored
me with, made me in constant fears of acting dishonorably to
the marquis. The duchess fell soon after into a languishing
illness, which in a short time put a period to her life: The duke
came, but too late, to receive her last breath. He at first
appeared inconsolable for her death; but his grief insensibly
decreased, and softened into that mournful and tender regard,
which a sense of her merit, and his own unkindness to her,
could not fail of exacting from him. Disgusted at an union,
which had caused him (from his own errors) so much
uneasiness, he formed a resolution carefully to avoid entering
again into a similar engagement. But he saw every day before
him the lovely Adelaude: he loved her; it was perhaps
impossible for him to do otherwise. He declared his passion; but

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