Digitized images and

Digitized Images and
its Properties

Ashish Khare
M. Sonka, V. Hlavac and R. Boyle, Image Processing,
Analysis and Machine Vision, PWS Publishing.

12/06/12 1

Image Functions (cont.)
 A signal is a function depending on some variable
with physical meaning.
 Signals can be
 one-dimensional (e.g., dependent on time),
 two-dimensional (e.g., images dependent on two
co-ordinates in a plane),
 three-dimensional (e.g., describing an object in
space),
 or higher-dimensional.

 A scalar function may be sufficient to describe a
monochromatic image, while vector functions are to
represent, for example, color images consisting of
12/06/12 component colors.
three 2

 The image can be modeled by a continuous
function of two or three variables.

 arguments are co-ordinates x, y in a plane,
while if images change in time a third variable
t might be added.

 The image function values correspond to the
brightness at image points.

 The function value can express other physical
quantities as well (temperature, pressure
distribution, distance from the observer, etc.).
12/06/12 3

 The brightness integrates different optical
quantities
- using brightness as a basic quantity allows us to
avoid the description of the very complicated process
of image formation.

 The image on the retina or on a TV camera
sensor is intrinsically 2D. We call such a 2D
image bearing information about brightness
points an intensity image.

 The 2D intensity image is the result of a
perspective projection of the 3D scene.
12/06/12 4

 When 3D objects are mapped into the
camera plane by perspective projection a lot
of information disappears as such a
transformation is not one-to-one.

 Recognizing or reconstructing objects in a 3D
scene from one image is an ill-posed
problem.

 Recovering information lost by perspective
projection is only one, mainly geometric,
problem of computer vision.
12/06/12 5

 The second problem is how to understand image
brightness?

 The only information available in an intensity
image is brightness, which is dependent on a
number of independent factors such as
 object surface reflectance properties (given by the
surface material, microstructure and marking),
 illumination properties,
 and object surface orientation with respect to a viewer
and light source.

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 Some scientific and technical disciplines work
with 2D images directly; for example,
 a character drawn on a sheet of paper,
 the image of a fingerprint, etc.

 Many basic and useful methods used in digital
image analysis do not depend on whether the
object was originally 2D or 3D.

 Related disciplines are photometry which is
concerned with brightness measurement, and
colorimetry which studies light reflectance or
emission depending on wavelength.
12/06/12 7

 A light source energy distribution C(x,y,t,λ)
depends in general on image co-ordinates (x, y),
time t, and wavelength λ.

 For the human eye and image sensors (e.g., TV
cameras) the brightness f depends on the
lightsource energy distribution C and the
spectral sensitivity of the sensor, S(λ)
(dependent on the wavelength).

 A monochromatic image f(x,y,t) provides the
brightness
12/06/12 distribution. 8

 In a color or multispectral image, the image is
represented by a real vector function f .

 Image processing often deals with static images,
in which time t is constant.
 A monochromatic static image is represented by
a continuous image function f(x,y) whose
arguments are two co-ordinates in the plane.
 Brightness values bounded by the minimum
(black) and the maximum (white) limits are gray
levels.
12/06/12 9

Quality of Images
 The quality of a digital image grows in proportion to the
spatial, spectral, radiometric, and time resolution.
 The spatial resolution is given by the proximity of
image samples in the image plane.
 The spectral resolution is given by the bandwidth of the
light frequencies captured by the sensor.
 The radiometric resolution corresponds to the number
of distinguishable gray levels.
 The time resolution is given by the interval between
time samples at which images are captured.
12/06/12 10

Images as a Stochastic Process
 Images f(x,y) can be treated as deterministic
functions or as realizations of stochastic
processes.

 Mathematical tools used in image description
have roots in linear system theory, integral
transformations, discrete mathematics and
the theory of stochastic processes.

12/06/12 11

Image Digitization
 An image captured by a sensor is expressed
as a continuous function f(x,y) of two co-
ordinates in the plane.

 Image digitization means that the function f(x,y)
is sampled into a matrix with M rows and N
columns.

 The image quantitation assigns to each
continuous sample an integer value.

 The continuous range of the image function
f(x,y)
12/06/12 is split into K intervals. 12

Image Digitization
 The finer the sampling (i.e., the larger M and
N) and quantitation (the larger K) the better the
approximation of the continuous image
function f(x,y).

 Two questions should be answered in
connection with image function sampling:
 First, the sampling period should be determined --
the distance between two neighboring sampling
points in the image
 Second, the geometric arrangement of sampling
points (sampling grid) should be set.
12/06/12 13

Sampling
 A continuous image function f(x,y) can be
sampled using a discrete grid of sampling
points in the plane.

 Two neighboring sampling points are
separated by distance ∆x along the x axis and
∆y along the y axis.

 ∆x and ∆y are called the sampling interval
and the matrix of samples constitutes the
discrete image.

12/06/12 14

Sampling (cont.)
 The ideal sampling s(x,y) in the regular grid can be
represented using a collection of Dirac distributions

 The sampled image is the product of the continuous
image f(x,y) and the sampling function s(x,y)

12/06/12 15

Sampling (cont.)
 The collection of Dirac distributions can be
regarded as periodic with period x, y and
expanded into a Fourier series

where the coefficients of the Fourier expansion
can be calculated as

12/06/12 16

Sampling (cont.)
 In frequency domain,

12/06/12 17

Sampling (cont.)
 Thus the Fourier transform of the sampled
image is the sum of periodically repeated
Fourier transforms F(u,v) of the image.
 Periodic repetition of the Fourier transform
result F(u,v) may under certain conditions
cause distortion of the image which is called
aliasing; this happens when individual
digitized components F(u,v) overlap.
 There is no aliasing if the image function f(x,y)
has a band limited spectrum ......... its Fourier
transform F(u,v)=0 outside a certain interval of
frequencies |u| > U; |v| > V.
12/06/12 18

Sampling (cont.)
 From general sampling theory, overlapping of
the periodically repeated results of the Fourier
transform F(u,v) of an image with band limited
spectrum can be prevented if the sampling
interval is chosen according to

 This is the Shannon sampling theorem that has
a simple physical interpretation in image
analysis:
The sampling interval should be chosen in size
such that it is less than or equal to half of the
smallest interesting detail in the image.
12/06/12 19

Sampling (cont.)
 The sampling function is not the Dirac
distribution in real digitizers -- narrow impulses
with limited amplitude are used instead.

 As a result, in real image digitizers a sampling
interval about ten times smaller than that
indicated by the Shannon sampling theorem is
used - because the algorithms for image
reconstruction use only a step function.

12/06/12 20

Sampling (cont.)
 Original image of Golden Gate Bridge:

12/06/12 21

Sampling (cont.)
 Image resampled at half the original
resolution:
With image size adjusted

12/06/12 22

Sampling (cont.)
Without image size adjusted:

12/06/12 23

Sampling (cont.)
 Image resampled at twice the original resolution
With image size adjusted

12/06/12 24

Sampling (cont.)
Without image size adjusted

12/06/12 25

Sampling (cont.)
Image which has been downsampled at one-
tenth resolution, and upsampled at 10x
resolution

12/06/12 26

Sampling (cont.)
 A continuous image is digitized at sampling points.

 These sampling points are ordered in the plane and
their geometric relation is called the grid.

 Grids used in practice are mainly square or
hexagonal.

12/06/12 27

Sampling (cont.)
 One infinitely small sampling point in the grid
corresponds to one picture element (pixel) in
the digital image.

 The set of pixels together covers the entire
image.

 Pixels captured by a real digitization device
have finite sizes.

 The pixel is a unit which is not further divisible,
sometimes pixels are also called points.
12/06/12 28

Quantization
 A magnitude of the sampled image is
expressed as a digital value in image
processing.

 The transition between continuous values of
the image function (brightness) and its digital
equivalent is called quantitation.

 The number of quantitation levels should be
high enough for human perception of fine
shading details in the image.

12/06/12 29

Quantization (cont.)
 Most digital image processing devices use
quantitation into k equal intervals.

 If b bits are used ... the number of brightness
levels is k=2b.

 Eight bits per pixel are commonly used,
specialized measuring devices use twelve
and more bits per pixel.

12/06/12 30

Metric Properties of Digital Images
 Distance is an important example.

 The distance between (I,j) and (h,k)
 the Euclidean distance

 city block distance

 chessboard distance

12/06/12 31

(cont.)
 Pixel adjacency is another important concept in
digital images.

 4-neighborhood
 8-neighborhood

12/06/12 32

(cont.)
 It will become necessary to consider important
sets consisting of several adjacent pixels --
regions.

 Region is a contiguous set.

 Contiguity paradoxes of the square grid

12/06/12 33

(cont.)

12/06/12 34

(cont.)

12/06/12 35

(cont.)
 One possible solution to contiguity paradoxes
is to treat objects using 4-neighborhood and
background using 8-neighborhood (or vice
versa).

 A hexagonal grid solves many problems of
the square grids ... any point in the
hexagonal raster has the same distance to all
its six neighbors.

12/06/12 36

(cont.)
 Border R is the set of pixels within the region
that have one or more neighbors outside R ...
inner borders, outer borders exist.
 Edge is a local property of a pixel and its
immediate neighborhood --it is a vector given by
a magnitude and direction.
 The edge direction is perpendicular to the
gradient direction which points in the direction of
image function growth.
 Border and edge ... the border is a global
concept related to a region, while edge
expresses local properties of an image function.
12/06/12 37

(cont.)
 Crack edges ... four crack edges are attached
to each pixel, which are defined by its relation
to its 4-neighbors. The direction of the crack
edge is that of increasing brightness, and is a
multiple of 90 degrees, while its magnitude is
the absolute difference between the brightness
of the relevant pair of pixels.

12/06/12 38

Topological Properties of Images
 Topological properties of images are invariant to
rubber sheet transformations.

 Stretching does not change contiguity of the
object parts and does not change the number of
holes in regions.

 One such image property is the Euler--Poincare
characteristic defined as the difference between
the number of regions and the number of holes
in them.

12/06/12 39

Topological Properties of Images
(cont.)
 Convex hull is used to describe topological
properties of objects.
 The convex hull is the smallest region which
contains the object, such that any two points
of the region can be connected by a straight
line, all points of which belong to the region.

12/06/12 40

Histograms
 Brightness histogram provides the frequency
of the brightness value z in the image.

12/06/12 41

Noise in Image
 Images are often degraded by random noise.

 Noise can occur during image capture,
transmission or processing, and may be
dependent on or independent of image
content.

12/06/12 42

Noise in Image (cont.)
 White noise - constant power spectrum (its
intensity does not decrease with increasing
frequency).
 Gaussian noise is a very good approximation of
noise that occurs in many practical cases.

 During image transmission, noise which is usually
independent
12/06/12 of the image signal occurs.
43

 Noise may be
 additive noise

 multiplicative noise

 impulse noise (saturated = salt and pepper noise)
12/06/12 44


Original Image Noise Noisy Image

12/06/12 45

Digitized images and

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