Digital Image Fundamentals

Digital Image Fundamentals
Mr. A. B. Shinde

A. B. Shinde
Contents…
• Image Sensing and Acquisition
• Basic concept of sampling and quantization
• Representations of digital image
• Spatial and gray level resolution
• Zooming and shrinking of image,
• Basic relationship between pixels
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Illusions: Images
• Optical illusions:
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Image Sensing & Acquisition

A. B. Shinde
• Images are generated by the combination of an “illumination” source and
the reflection or absorption of energy from that source.
• Depending on the source:
illumination energy is
reflected from, or
transmitted through, objects.
• Example in the first category: Light reflected from a planar surface.
• Example in the second category: When X-rays pass through a patient’s
body for generating a diagnostic X-ray film.
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• Three principal sensor arrangements used to transform illumination
energy into digital images.
• Single Imaging Sensor
• Line Sensor
• Array Sensor
• Incoming energy is transformed into a voltage by the combination of
input electrical power and sensor material that is responsive to the
particular type of energy being detected.
• The output voltage waveform is the response of the sensor(s), and a
digital quantity is obtained from each sensor by digitizing its response.
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Single imaging sensor.
• Image Acquisition Using a
Single Sensor:
• Figure shows the components
of a single sensor.
• Most familiar sensor of this type
is the photodiode.
• To generate a 2-D image using
a single sensor, there has to be
relative displacements in both
the x- and y-directions.

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• Image Acquisition Using a
Single Sensor:
• A film negative is mounted onto a
drum whose mechanical rotation
provides displacement in one
dimension. The single sensor is
mounted on a lead screw that
provides motion in the
perpendicular direction.
• Because mechanical motion can
be controlled with high precision,
this method is an inexpensive
(but slow) way to obtain high-
resolution images.
Combining a single sensor with
motion to generate a 2-D image.

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Line sensor
 Image Acquisition Using Sensor Strips:
• A geometry that is used much more frequently than single sensors
consists of an in-line arrangement of sensors in the form of a sensor
strip, as shown in figure
• The strip provides imaging elements in one direction

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 Image Acquisition Using Sensor
Strips:
• Motion perpendicular to the strip provides
imaging in the other direction, as shown
in figure
• This is the type of arrangement used in
most flat bed scanners. Sensing devices
with 4000 or more in-line sensors are
possible.
• The imaging strip gives one line of an
image at a time, and the motion of the
strip completes the other dimension of a
two-dimensional image.
Image acquisition using
a linear sensor strip

A. B. Shinde
• Image Acquisition Using Sensor
Strips:
• Sensor strips mounted in a ring
configuration are used in medical and
industrial imaging to obtain cross-
sectional (“slice”) images of 3-D
objects as shown figure.
• A rotating X-ray source provides
illumination and the sensors opposite
the source collect the X-ray energy
that passes through the object.
• This technique is used in medical and
industrial computerized axial
tomography (CAT) imaging.
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Array sensor
 Image Acquisition Using Sensor Arrays:
• Figure shows individual sensors arranged in the
form of a 2-D array.
• Numerous electromagnetic and some ultrasonic
sensing devices are arranged in array format.
• This arrangement is found in digital cameras.
• The response of each sensor is proportional to
the integral of the light energy projected onto the
surface of the sensor.
• Since the sensor array is two-dimensional, its
key advantage is that a complete image can be
obtained by focusing the energy pattern onto the
surface of the array.

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 Simple Image Formation Model:
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Energy
(illumination)
source
Element of a Scene
Imaging
system
Projection of the scene
onto the image plane
Digitized image

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• Simple Image Formation Model:
• Images are two-dimensional functions of the form f(x, y).
• The value or amplitude of f at spatial coordinates is a positive scalar
quantity
• When an image is generated from a physical process, its intensity
values are proportional to energy radiated by a physical source.
• As a consequence, f(x, y) must be nonzero and finite; that is,
0 < f(x, y) < ꝏ
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Image Sampling & Quantization

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Continuous Image Digital Image

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• There are numerous ways to acquire images, but our objective is to
generate digital images from sensed data.
• The output of most sensors is a continuous voltage waveform whose
amplitude and spatial behavior are related to the physical phenomenon
being sensed.
• To create a digital image, we need to convert the continuous sensed
data into digital form.
• This involves two processes: sampling and quantization.
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Continuous
image.
Scan line
from A to B
Sampling and
quantization
Digital
scan line

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 Basic Concepts in Sampling and
Quantization:
• Figure shows a continuous image f that we want
to convert to digital form.
• An image may be continuous with respect to the
x- and y-coordinates, and also in amplitude.
• To convert it to digital form, we have to sample
the function in both coordinates and in amplitude.
• Digitizing the coordinate values is called
sampling.
• Digitizing the amplitude values is called
quantization.
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Continuous
image

A. B. Shinde
Quantization:
• The one-dimensional function in figure is a
plot of amplitude (intensity level) values of
the continuous image along the line
segment AB.
• The random variations are due to image
noise
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A scan line from
A to B in the
continuous image

A. B. Shinde
Quantization:
• The sample points are indicated by a
vertical tick marks at the bottom the
figure.
• The samples are shown as small white
squares superimposed on the function.
• The set of these discrete locations gives
the sampled function.
• To form a digital function, the intensity
values must be converted (quantized)
into discrete quantities.
• The right side of figure shows the
intensity scale divided into eight discrete
intervals, ranging from black to white.
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Sampling and
quantization

A. B. Shinde
Quantization:
• The continuous intensity levels are
quantized by assigning one of the eight
values to each sample.
• The assignment is made depending on
the vertical proximity of a sample to a
vertical tick mark.
• The digital samples resulting from both
sampling and quantization are shown in
figure.
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Digital
scan line

A. B. Shinde
• First figure shows a continuous image projected onto the plane of an
array sensor.
• Second figure shows the image after sampling and quantization.
• The quality of a digital image is determined to a large degree by the
number of samples and discrete intensity levels used in sampling and
quantization.
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Continuous image projected
onto a sensor array
Result of image
sampling and quantization

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Representing Digital Image

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• Let f(s, t) represent a continuous image function of two continuous
variables, s and t.
• We convert this function into a digital image by sampling and
quantization.
• Suppose that we sample the continuous image into a 2-D array, (x, y),
containing M rows and N columns, where (x, y) are discrete coordinates:
x = 0, 1, 2, ….. ,M - 1 and y = 0, 1, 2, ….. , N - 1.
• For example, the value of the digital image at the origin is f(0, 0), and the
next coordinate value along the first row is f(0, 1).
• In general, the value of the image at any coordinates (x, y) is denoted
f(x, y), where x and y are integers.
• The section of the real plane spanned by the coordinates of an image is
called the spatial domain, with x and y being referred to as spatial
variables or spatial coordinates.
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x
y
Origin
(0,0)
Pixel

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• A digital image is composed
of M rows and N columns of
pixels each storing a value
• Pixel values are most
often grey levels in the
range 0-255 (black – white)
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Image plotted
as a surface
Image
displayed as a
visual intensity
array
Image shown as a 2-D
numerical array
(0 – Black,
0.5 – Gray,
1 - white )

A. B. Shinde
 Figure shown is a plot of the
function, with two axes determining
spatial location and the third axis
being the values of f (intensities).
 This representation is useful when
working with gray-scale sets
whose elements are expressed as
triplets of the form (x, y, z) , where
x and y are spatial coordinates and
z is the value of f at coordinates
(x, y).
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• It shows f(x, y), as it would appear on a
monitor or photograph.
• Here, the intensity of each point is
proportional to the value of f at that
point.
• In this figure, there are only three
equally spaced intensity values. If the
intensity is normalized to the interval
[0, 1], then each point in the image has
the value 0, 0.5, or 1.
• A monitor or printer simply converts
these three values to black, gray, or
white, respectively.
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• This representation is simply to
display the numerical values of
f(x, y) as an array (matrix).
• In this example, f is of size
600 x 600, elements or 3,60,000
numbers.
• When developing algorithms, this
representation is quite useful when
only parts of the image are printed
and analyzed as numerical values.
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• In equation form, we write the representation of an numerical array as:
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• Sometimes, it is advantageous to use a more traditional matrix notation
to denote a digital image and its elements:
Each element of this matrix is called an image element, picture
element, pixel, or pel.

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Spatial & Gray Level Resolution

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Spatial and Gray Level Resolution
• Spatial resolution:
• It is a measure of the smallest discernible detail in an image.
• Spatial resolution can be:
– line pairs per unit distance, and
– dots (pixels) per unit distance
• Dots per unit distance is a measure of image resolution used commonly
in the printing and publishing industry, expressed as dots per inch (dpi).
• For example:
– Newspapers are printed with a resolution of 75 dpi,
– Magazines at 133 dpi,
– Glossy brochures at 175 dpi, and
– Book page is printed at 2400 dpi.
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1250 dpi, 300 dpi, 150 dpi, 72 dpi,
 Effects of Reducing Spatial Resolution

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Vision specialists will often talk about pixel size

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Graphic designers will talk about dots per inch (DPI)
1024 * 1024 512 * 512 256 * 256
128 * 128 64 * 64 32 * 32

A. B. Shinde
• Intensity resolution:
• It refers to the smallest discernible change in intensity level.
• The number of intensity levels usually is an integer power of two
• The most common number is 8 bits, with 16 bits being used in some
applications.
• Intensity quantization using 32 bits is rare.
• For example, it is common to say that an image whose intensity is
quantized into 256 levels has 8 bits of intensity resolution.
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Number of Bits
Number of
Intensity Levels
Examples
1 2 0, 1
2 4 00, 01, 10, 11
4 16 0000, 0101, 1111
8 256 00110011, 01010101
16 65,536 1010101010101010
• The more intensity levels used, the finer the level of detail discernable
in an image
• Intensity level resolution is usually given in terms of the number of bits
used to store each intensity level

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 Effects of Reducing Intensity Resolution
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256-level 128-level 64-level 32-level
16-level 8-level 4-level 2-level

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 Effects of Reducing Intensity Resolution
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24 - bits 8 - bits 4 - bits 1 - bit

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Image Interpolation
(Zooming & Shrinking)

A. B. Shinde
Image Interpolation
• Interpolation is a basic tool used extensively in tasks such as zooming,
shrinking, rotating, and geometric corrections.
• Image shrinking and zooming, are image resampling methods.
• Interpolation is the process of using known data to estimate values at
unknown locations.
• Suppose that an image of size 500 x 500 pixels has to be enlarged 1.5
times to 750 x 750 pixels.
• Zooming needs to create an imaginary 750 x 750 grid with the same
pixel spacing as the original, and then shrink it so that it fits exactly over
the original image.
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Image Interpolation
• To perform intensity-level assignment for any point in the overlay, we
look for its closest pixel in the original image and assign the intensity of
that pixel to the new pixel in the 750 x 750 grid.
• After assigning intensities to all the points in the overlay grid, we expand
it to the original specified size to obtain the zoomed image.
• This method is called as nearest neighbor interpolation because it
assigns to each new location the intensity of its nearest neighbor in the
original Image.
• This approach is simple but, it has the tendency to produce undesirable
artifacts, such as severe distortion of straight edges.
• For this reason, it is used infrequently in practice.
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Image Interpolation
• Another approach is bilinear interpolation, in which we use the four
nearest neighbors to estimate the intensity at a given location.
• Let (x, y) denote the coordinates of the location to which we want to
assign an intensity value and let v(x, y) denote that intensity value.
• For bilinear interpolation, the assigned value is obtained using the
equation
v(x, y) = ax + by + cxy + d
• where the four coefficients are determined from the four equations in
four unknowns that can be written using the four nearest neighbors of
point (x, y).
• Bilinear interpolation gives much better results than nearest neighbor
interpolation, with a increase in computational burden.
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Image Interpolation
• The next level of complexity is bicubic interpolation, which involves the
sixteen nearest neighbors of a point. The intensity value assigned to
point is obtained using the equation
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• Where the sixteen coefficients are determined from the sixteen
equations in sixteen unknowns that can be written using the sixteen
nearest neighbors of point (x, y).
• Generally, bicubic interpolation does a better job of preserving fine
details than bilinear interpolation technique.
• Bicubic interpolation is the standard used in commercial image editing
programs, such as Adobe Photoshop and Corel Photopaint.

A. B. Shinde
Image Interpolation
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Original Image
(3692 x 2812
pixels)
Image reduced
to 72 dpi and
zoomed back to
its original size,
using nearest
neighbor
interpolation
Image shrunk
and zoomed
using bilinear
interpolation
Image shrunk
and zoomed
using bicubic
interpolation
It is possible to use more neighbors in interpolation, and there are more
complex techniques, such as using splines and wavelets, that in some
instances can yield better results.

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Basic Relationship Between
Pixels

A. B. Shinde
Basic Relationship Between Pixels
 As we know, an image is denoted by f(x, y)
 Neighbors of a Pixel:
• A pixel p at coordinates (x, y) has four horizontal and vertical neighbors
whose coordinates are given by
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This set of pixels, called the 4-neighbors of p, is denoted by N4(p).
Each pixel is a unit distance from (x, y), and some of the neighbor
locations of p lie outside the digital image if (x, y) is on the border of the
image.

A. B. Shinde
 Neighbors of a Pixel:
 The four diagonal neighbors of p have coordinates
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and are denoted by ND(p).
 These points, together with the 4-neighbors, are called the 8-neighbors
of p, denoted by N8(p).
 As before, some of the neighbor locations in ND(p) and N8(p) fall outside
the image if (x, y) is on the border of the image.

A. B. Shinde
 Adjacency, Connectivity, Regions, and Boundaries:
• Let V be the set of intensity values used to define adjacency.
• In a binary image, V = {1} if we are referring to adjacency of pixels with
value 1.
• In a gray-scale image, the idea is the same, but set V typically contains
more elements.
• For example,
• In the adjacency of pixels with a range of possible intensity values 0 to
255, set V could be any subset of these 256 values.
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 We consider three types of adjacency:
(a) 4-adjacency: Two pixels p and q with values from V are 4-adjacent if q is
in the set N4(p).
(b) 8-adjacency: Two pixels p and q with values from V are 8-adjacent if q is
in the set N8(p).
(c) m-adjacency (mixed adjacency): Two pixels p and q with values from V
are m-adjacent if
(i) q is in N4(p), or
(ii) q is in ND(p) and the set N4(p) ∩ N4(q) has no pixels whose
values are from V.
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• Adjacency, Connectivity, Regions, and Boundaries:
• Mixed adjacency is a modification of 8-adjacency. It is introduced to
eliminate the ambiguities that often arise when 8-adjacency is used.
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A. B. Shinde
Consider the pixel arrangement shown in figure
for V = {1}
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The three pixels at the top of figure show multiple
(ambiguous) 8-adjacency, as indicated by the dashed
lines.
Above ambiguity is removed by using m-adjacency, as
shown in figure.

A. B. Shinde
• A (digital) path (or curve) from pixel p with coordinates (x, y) to pixel q
with coordinates (s, t) is sequence of distinct pixels with coordinates.
(x0, y0), (x1, y1), …… , (xn, yn)
where,
(x0, y0) = (x, y) (xn, yn) = (s, t)
and pixels
(xi, yi) and (xi-1, yi-1) are adjacent for 1 ≤ i ≤ n.
• In this case, n is the length of the path. If (x0, y0) = (xn, yn), the path is a
closed path. We can define 4, 8 or m-paths depending on the type of
adjacency specified.
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The paths shown in figure between the top right and
bottom right points are 8-paths.
The paths shown in figure between the top right and
bottom right points are m-paths.

A. B. Shinde
• Let S represent a subset of pixels in an image.
• Two pixels p and q are said to be connected in S if there exists a path
between them consisting entirely of pixels in S.
• For any pixel p in S, the set of pixels that are connected to it in S is
called a connected component of S.
• If it only has one connected component, then set S is called a connected
set.
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• Let R be a subset of pixels in an image, We call R a region of the image
if R is a connected set.
• Two regions, Ri and Rj are said to be adjacent if their union forms a
connected set.
• Regions that are not adjacent are said to be disjoint.
• We consider 4- and 8-adjacency when referring to regions.
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The two regions (of 1s) in figure are adjacent only if 8-
adjacency is used
(according to the earlier definition, a 4-path between the
two regions does not exist, so their union is not a
connected set).
The circled point is part of the boundary of the 1-valued
pixels only if 8-adjacency between the region and
background is used
The inner boundary of the 1-valued region does not
form a closed path, but its outer boundary does.

A. B. Shinde
 Distance Measures:
• For pixels p, q, and z, with coordinates (x, y), (s, t), and (v, w),
respectively, D is a distance function or metric if
• D(p, q) ≥ 0 (D(p, q) = 0 iff p = q),
• D(p, q) = D(q, p), and
• D(p, z) ≤ D(p, q) + D(q, z).
• The Euclidean distance between p and q is defined as
De(p, q) = [(x - s)2 + (y - t)2 ]1/2
• The pixels having a distance less than or equal to some value r from
(x, y) are the points contained in a disk of radius r centered at (x, y).
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 The distance (called the city-block distance) between p and q is defined
as
D4(p, q) = | x - s | + | y - t |
• The pixels having a distance from (x, y) less than or equal to some value
r form a diamond centered at (x, y).
• For example, the pixels with D4 distance ≤ 2 from (x, y) (the center point)
form the following contours of constant distance:
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The pixels with D4 = 1 are the 4-neighbors of (x, y).

A. B. Shinde
 The distance (called the chessboard distance) between p and q is
defined as
D8(p, q) = max( | x - s | , | y - t | )
• Here, the pixels with D8 distance from (x, y) less than or equal to some
value r form a square centered at (x, y).
• For example, the pixels with D8 distance ≤ 2 from (x, y) (the center point)
form the following contours of constant distance:
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The pixels with D8 = 1 are the 8-neighbors of (x, y).

This presentation is published only for educational purpose
abshinde.eln@gmail.com

Digital Image Fundamentals

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