Basics of digital signal and image processing

DIGITAL IMAGE PROCESSING
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■ Text Books:
■ 1. Rafael C G., Woods R E. and Eddins S L, Digital Image Processing,
Prentice Hall, 3rd Edition, 2008.
■ Reference Books:
1. Fundamentals of Digital Image Processing-Anil K. Jain, 2nd Edition,
Prentice Hall of India.
2. S. Sridhar, Digital Image Processing, Oxford University Press, 2nd Ed,
2016.
3. Digital image processing, TMH, Jayaraman, S.; Esakkirajan, S. &
Veerakumar, T.,

DIGITAL IMAGE PROCESSING
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8 Hours
■ Module – 1
■ Introduction Fundamental Steps in Digital Image Processing,
■ Components of an Image Processing System,
■ Sampling and Quantization,
■ Representing Digital Images (Data structure),
■ Some Basic Relationships Between Pixels- Neighbors and Connectivity of pixels
in image,
■ Applications of Image Processing:
– Medical imaging,
– Robot vision,
– Character recognition,
– Remote Sensing

Image processing
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■ Signal processing is a discipline that deals with analysis and processing
of analog and digital signals and deals with storing , filtering , and
other operations on signals.
■ These signals include transmission signals , sound or voice signals ,
image signals , and other signals.
■ The field that deals with the type of signals, for which the input is an
image and the output is also an image is image processing.
■ As name suggests, it deals with the processing on images.

Image processing
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■ What is an Image ??
■ An image may be defined as a two-dimensional function, f(x,y),
– where x and y are spatial (plane) coordinates, and
– the amplitude of f at any pair of coordinates (x, y) is called the
intensity or gray level of the image at that point.
■ When x, y, and the intensity values of f are all finite, discrete
quantities, we call the image a digital image.
■ The field of digital image processing refers to processing digital
images by means of a digital computer.
■ A digital image is composed of a finite number of elements, each of
which has a particular location and value.
■ These elements are called picture elements, or pixels.

Image processing
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■ Images play the most important role in human perception.
■ However, we human beings, are limited to the visual band of the
electromagnetic (EM) spectrum, but imaging machines cover almost the
entire EM spectrum, ranging from gamma to radio waves.
■ They can operate on images generated by sources that humans are not
accustomed to associating with images.
■ These include ultrasound, electron microscopy, and computer-generated
images.
■ Thus, digital image processing covers a wide and varied field of applications

Image processing
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■ There is no fixed boundary regarding where image processing stops and
other related areas, such as image analysis and computer vision, start.
■ Sometimes a distinction is made by defining image processing as a
discipline in which both the input and output of a process are images.
■ But this will be a limiting and artificial boundary.
– Even the trivial task of computing the average intensity of an image
(which yields a single number) would not be considered an image
processing operation.
■ On the other hand, the fields such as computer vision whose ultimate goal is
to use computers to emulate human vision, including learning and being able
to make inferences and take actions based on visual inputs.

Image processing
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■ There are no clear-cut boundaries image processing and computer vision.
■ But we may consider three types of computerized processes low-, mid-, and
high-level processes.
■ Low-level processes involve primitive operations such as image
preprocessing to reduce noise, contrast enhancement, and image sharpening.
■ Here both inputs and outputs are images.
■ Mid-level processing on images involves tasks such as segmentation
(partitioning an image into regions or objects), description of those objects
to reduce them to a form suitable for computer processing, and classification
(recognition) of individual objects.
■ In this, inputs generally are images, but its outputs are attributes extracted
from those images (e.g., edges, contours, and the identity of individual
objects).

Image processing
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■ Higher-level processing involves “making sense” of an ensemble of
recognized objects, as in image analysis, and performing the cognitive
functions normally associated with vision.
■ Here we can see that a logical overlap between image processing and image
analysis exits and that is the area of recognition of individual regions or
objects in an image.
■ Thus, we can say digital image processing includes processes whose inputs
and outputs are images and, also processes that extract attributes from
images, up to and including the recognition of individual objects

Image processing
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■ To clarify these concepts, Lets see example of OCR.
■ This involves
■ The processes of acquiring an image of the area containing the text,
■ Preprocessing that image,
■ Extracting (segmenting) the individual characters,
■ Describing the characters in a form suitable for computer processing, and
■ Recognizing those individual characters are in the scope of DIP

Image processing
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■ Making sense of the content of the page may be seen as a part of image
analysis and even computer vision, depending on the level of complexity
implied by the statement “making sense.”

Fundamental Steps in Digital Image Processing
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■ We saw two broad categories of image processing
– methods whose input and output are images, and
– methods whose inputs may be images but whose outputs are attributes
extracted from those images.
■ This organization is summarized in following Fig.

Image processing
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■ The diagram does not imply that every process is applied to an image.
■ It gives an idea of all the methodologies that can be applied to images for
different purposes and possibly with different objectives.
■ Let us briefly discuss the overview of each of these methods
■ Image acquisition
■ The first process in Fig.
■ We see more methods of image acquisition in subsequent sections
■ Acquisition could be as simple as being given an image that is already in
digital form.

Image processing
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■ Image enhancement:
■ The process of manipulating an image so that the result is more suitable than the
original for a specific application.
■ The word “specific” establishes at the outset that enhancement techniques are
problem oriented.
■ Thus, for example, a method that is quite useful for enhancing X-ray images may
not be the best approach for enhancing satellite images taken in the infrared band
of the electromagnetic spectrum.
■ This process enhances the quality.
■ As an output of image enhancement, certain parts of the image will be highlighted
while it will remove or blur the unnecessary parts of the image.
■ Image restoration:
■ This also deals with improving the appearance of an image but it is objective,
■ That is, restoration techniques mostly based on mathematical or probabilistic
models of image degradation.

Image processing
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■ But Enhancement, is based on human subjective preferences regarding what
constitutes a “good” enhancement result.
■ Color image processing
■ An area that has been gaining in importance because of the significant
increase in the use of digital images over the Internet.
■ Wavelets:
■ These are the foundation for representing images in various degrees of
resolution.
■ In particular, this is used for image data compression and for pyramidal
representation, in which images are subdivided successively into smaller
regions

Image processing
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■ Compression:
■ Deals with techniques for reducing the storage required to save an image, or
the bandwidth required to transmit it.
■ Although storage technology has improved over the years, the transmission
capacity might not have been improved.
■ Image compression is familiar to most users of computers in the form of
image file extensions, such as the jpg file extension used in the
■ JPEG (Joint Photographic Experts Group) image compression standard.

Image processing
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■ Morphological processing
■ deals with tools for extracting image components that are useful in the
representation and description of shape.
■ Segmentation procedures partition an image into its constituent parts or
objects.
■ Representation and description -usually follows the output of a segmentation
stage, which usually is raw pixel data, constituting either the boundary of a
region (i.e., the set of pixels separating one image region from another) or
all the points in the region itself.
■ In either case, the data has to be converted to a form suitable for computer
processing

Image processing
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■ Recognition is the process that assigns a label (e.g., “vehicle”) to an object
based on its descriptors.
■ This is the last stage of digital image processing where the methods for
recognition of individual objects is developed
■ Knowledge about a problem domain is coded into an image processing
system in the form of a knowledge database.
■ This knowledge may be as simple as detailing regions of an image where the
information of interest is known to be located, thus limiting the search that
has to be conducted in seeking that information.

Image processing
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■ :The knowledge base also can be quite complex, such as an interrelated list
of all major possible defects in a materials inspection problem or an image
database containing high-resolution satellite images of a region in
connection with change-detection applications.
■ In addition to guiding the operation of each processing module, the
knowledge base also controls the interaction between modules.
■ This distinction is made in Fig by means of use of double-headed arrows
between the processing modules and the knowledge base, as opposed to
single-headed arrows linking the processing modules.

Components of an Image Processing System
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■ Image sensing:
■ Two elements required for acquiring digital image:
– A physical device that is sensitive to the energy radiated by the object we
wish to image.
– The second, called a digitizer, is a device for converting the output of the
physical sensing device into digital form
■ E.g.:
■ In a digital video camera, the sensors produce an electrical output
proportional to light intensity.
■ The digitizer converts these outputs to digital data.

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■ Specialized image processing hardware:
■ Consists of the digitizer plus hardware that performs other primitive
operations, such as anALU, that performs arithmetic and logical operations
in parallel on entire images.
■ This type of hardware is also called a front-end subsystem, and its most
distinguishing characteristic is speed.
■ This unit performs functions that require fast data throughputs that the
typical main computer cannot handle.
■ The computer
■ This could be is a general-purpose computer ranging from a PC to a
supercomputer.
■ For any general-purpose image processing systems, any well-equipped PC-
type machine is suitable for off-line image processing tasks.

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■ Image processing Software:
■ consists of specialized modules that perform specific tasks.
■ A well-designed package facilitates the user to write minimum code that,
utilizes the available specialized modules.
■ More sophisticated software packages allow the integration of those
modules and general-purpose software commands from at least one
computer language.

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■ Mass storage:
■ Is a must in image processing applications.
■ W.k.t. Storage is measured in bytes, Kbytes, Mbytes, Gbytes and Tbytes
■ An image of size 1024 X 1024 pixels, in which the intensity of each pixel is
an 8-bit quantity, requires 1MB of storage space if the image is not
compressed.
■ When dealing with large number of images, providing adequate storage in
an image processing system can be a challenge.
■ Digital storage for image processing applications are of three categories:
– Short-term storage for use during processing,
– On-line storage for relatively fast recall, and
– Archival storage, characterized by infrequent access.

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■ One method of providing Short-term storage is computer memory.
■ Another is by specialized boards, called frame buffers, that store one or
more images and can be accessed rapidly, usually at video rates (e.g., at 30
complete images per second).
■ Frame buffers are located in the specialized image processing hardware unit
■ On-line storage generally takes the form of magnetic disks or optical-media
storage.
■ Finally, archival storage is characterized by massive storage requirements
but infrequent need for access.
■ Magnetic tapes and optical disks housed in “jukeboxes” are the usual media
for archival applications.

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■ Image displays used today are mainly color TV monitors.
■ Monitors are driven by the outputs of image and graphics display cards that
are an integral part of the computer system.
■ In some cases, it is necessary to have stereo displays, and these are
implemented in the form of headgear containing two small displays
embedded in goggles worn by the user.
■ Hardcopy devices are used for recording images
■ E.g.: laser printers, film cameras, heat-sensitive devices, inkjet units, and
digital units, such as optical and CDROM disks.

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■ Networking is a default function in any computer system in use today.
■ Because of the large amount of data inherent in image processing
applications, main issue is the bandwidth for image transmission
■ In dedicated networks, this is not a problem, but communications with
remote sites via the Internet are not always as efficient.
■ Fortunately, this situation is improving quickly as a result of optical fiber
and other broadband technologies

Sampling and Quantization
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■ Before proceeding with this step we must have acquired the image of
interest using any of the available methods
■ 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.

■ Basic Concepts
■ An image may be continuous with respect to the x- and y-coordinates,
■ and also in amplitude
■ Let us say figure below shows a continuous image f that we want to convert
to digital form
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■ 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.
■ The plot of amplitude (intensity level) values of the continuous image f
along the line segmentAB is shown below.
■ The random variations are due to image noise
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■ To sample this function, we take equally spaced samples along lineAB, as shown
in Fig. below.
■ The samples are shown as small white squares superimposed on the function.
■ The spatial location of each sample is indicated by a vertical tick mark in the
bottom part of the figure.
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■ The set of these discrete locations gives the sampled function.
■ The values of the samples still span (vertically) a continuous range of
intensity values.
■ In order to form a digital function, the intensity values also must be
converted (quantized) into discrete quantities.
■ The right side of Fig. shows the intensity scale divided into eight discrete
intervals, ranging from black to white.
■ The vertical tick marks indicate the specific value assigned to each of the
eight intensity intervals.
■ 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 Fig.
■ Starting at the top of the image and carrying out this procedure line by line
produces a two-dimensional digital image.
■ It is implied in Fig. that, in addition to the number of discrete levels used,
the accuracy achieved in quantization is highly dependent on the noise
content of the sampled signal
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• This is used to convert the acquired image into digital form.
• Before applying this step we must have acquired the image of interest
using any of the available methods
• 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.

• Basic Concepts
• An image may be continuous with respect to the x- and y-coordinates, and also in
amplitude
• Let us say figure below shows a continuous image f that we want to convert to
digital form
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• 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.
• The plot of amplitude (intensity level) values of the continuous image f along the
line segmentAB is shown below.
• The random variations are due to image noise
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• To sample this function, we take equally spaced samples along lineAB, as shown
in Fig. below.
• The samples are shown as small white squares superimposed on the function.
• The spatial location of each sample is indicated by a vertical tick mark in the
bottom part of the figure.
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• The set of these discrete locations gives the sampled function.
• The values of the samples still span (vertically) a continuous range of intensity
values.
• In order to form a digital function, the intensity values also must be converted
(quantized) into discrete quantities.
• The right side of Fig. shows the intensity scale divided into eight discrete
intervals, ranging from black to white.
• The vertical tick marks indicate the specific value assigned to each of the eight
intensity intervals.
• 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
Fig.
• Starting at the top of the image and carrying out this procedure line by line
produces a two-dimensional digital image.
• It is implied in Fig. that, in addition to the number of discrete levels used, the
accuracy achieved in quantization is highly dependent on the noise content of the
sampled signal
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Representing Digital Images
• 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, f(x, y), containing
M rows and N columns, where (x, y) are discrete coordinates.
• We use integer values for these discrete coordinates: x = 0, 1, 2, ..M-1 and y = 0,
1,.. N-1.
• Thus, 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).
• Notation (0, 1) implies the second sample along the first row.
• In general, the value of the image at any coordinates (x, y) is denoted f(x,y), where
x and y are integers.

• Three basic ways to represent f(x, y):
• First method is a plot of the function f(x, y), with two axes determining spatial
location and the third axis being the values of f (intensities) as a function of the
two spatial variables x and y as shown below
• 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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• Second method is as shown below
• This is much more common.
• 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.
• Amonitor or printer simply converts these three values to black, gray, or white,
respectively
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• The third representation is 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 360,000 numbers.
• The last two representations are the most useful.
• Image displays allow us to view results at a glance and Numerical arrays are used
for processing and algorithm development.
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• The third representation is 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 360,000 numbers.
• The last two representations are the most useful.
• Image displays allow us to view results at a glance and Numerical arrays are used
for processing and algorithm development.
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• We can write the representation of an M X N numerical array in equation form as
• Both sides of this equation are equivalent ways of expressing a digital image
• The right side is a matrix of real numbers.
• Each element of this matrix is called an image element, picture element, pixel, or
pel.
• We stick on to the terms image and pixel to denote a digital image and its elements
respectively.
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• It is also advantageous to use traditional matrix notation to denote a digital image
and its elements as shown below
• Here any aij = f(x=i, y=j) = f(i,j)
• We can represent an image as a vector, v.
• For example, a column vector of size MN X 1 is formed by letting the first M
elements of v be the first column ofA, the next M elements be the second column,
and so on.
• Alternatively, we can use the rows instead of the columns ofA to form such a
vector.
• Either representation is valid, as long as we are consistent.
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• In the figure below we see that the origin of a digital image is at the top
left, with the positive x-axis extending downward and the positive y-
axis extending to the right.
• This is a conventional representation based on the fact that many image
displays (e.g., TV monitors) sweep an image starting at the top left and
moving to the right one row at a time.
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• Mathematically, the first element of a matrix is by convention at the top
left of the array, so choosing the origin of at that point makes sense.
• We henceforth show the axes pointing downward and to the right,
instead of to the right and up.
• The digitization process requires that decisions be made regarding the
values for M, N, and for the number, L, of discrete intensity levels.
• There are no restrictions placed on M and N, other than they have to be
positive integers.
• Due to storage and quantizing hardware considerations, the number of
intensity levels typically is an integer power of 2, L = 2k

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• We assume that the discrete levels are equally spaced and that they are
integers in the interval [0,L-1]
• Sometimes, the range of values spanned by the gray scale is referred to
informally as the dynamic range.
• This is a term used in different ways in different fields.
• We define the dynamic range of an imaging system to be the ratio of the
maximum measurable intensity to the minimum detectable intensity
level in the system.
• As a rule, the upper limit is determined by saturation and the lower limit
by noise

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• Saturation is the highest value beyond which all intensity levels are
clipped
• Noise in this case appears as a grainy texture pattern.
• Noise, especially in the darker regions of an image (e.g., the stem of the
rose) masks the lowest detectable true intensity level

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• Dynamic range of an imaging system is defined as the ratio of the
maximum measurable intensity to the minimum detectable intensity
level in the system.
• As a rule, the upper limit is determined by saturation and the lower limit
by noise
• Dynamic range establishes the lowest and highest intensity levels that a
system can represent and, consequently, that an image can have.
• Image contrast is closely associated with this, which is defined as the
difference in intensity between the highest and lowest intensity levels in
an image.

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• The number of bits required to store a digitized image is given by
• b = M *N * k
• When M = N this equation becomes
• b = N2*k
• Following table shows the number of bits required to store square
images with various values of N and k.

• An image of 2k intensity levels, is referred as a “k-bit image.”
• E.g.: an image with 256 possible discrete intensity values is called an 8-
bit image.
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• We can see that, 8-bit images of size 1024X1024 and higher are demanding for
significant storage requirements
• Spatial resolution and Intensity resolution:
• We know that, pixel is the smallest element of an image and it can store a value
proportional to the light intensity at that particular location.
• Resolution is a common term used with images
• The resolution can be defined in many ways.
• Pixel resolution,
• Spatial resolution,
• Temporal resolution,
• Spectral resolution.
• Let us see pixel resolution first

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• You might have seen that in PC settings, monitor resolution of 800 x 600, 640 x 480 etc
• In pixel resolution, the term resolution refers to the total number of count of pixels in an
digital image.
• If an image has M rows and N columns, then its resolution can be defined as M X N.
• If we define resolution as the total number of pixels, then pixel resolution can be defined
with set of two numbers.
• The first number, the width of the picture, or the pixels across columns, and the second
number is height of the picture, or the pixels across its width.
• We can say that the higher is the pixel resolution, the higher is the quality of the image.
• We can define pixel resolution of an image as 4500 X 5500

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Spatial and Intensity resolution
• Spatial resolution
• Spatial resolution can be defined as the smallest observable detail in an image.
• Can be defined in several ways
• Dots (pixels) per unit distance,
• Line pairs per unit distance
• spatial resolution means that we cannot compare two different types of images to
see that which one is clear or which one is not.
• If we have to compare the two images, to see which one is more clear or which
has more spatial resolution, we have to compare two images of the same size.

• Both the pictures has same dimensions which are of 227 X 222.
• Now when we compare them, we see that the picture on the left side has
more spatial resolution or it is more clear then the picture on the right side
which is a blurred image.
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• Measuring spatial resolution
• Since the spatial resolution refers to clarity, so for different devices, different
measure has been made to measure it.
• Dots per inch (DPI) - usually used in monitors.
• Lines per inch (LPI) - usually used in laser printers.
• Pixels per inch (PPI) is measure for different devices such as tablets , Mobile
phones e.t.c.
• Let us see the effects of reducing spatial resolution in an image.
• The images in Figs. (a) through (d) are shown in 1250, 300, 150, and 72 dpi,
respectively.

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• As spatial resolution reduces, image size also reduces.
• Image (a) was of size 3692 X 2812 pixels while image (d) was if 213 X 162
• To compare the images, smaller images are brought back to the original size by
zooming them.
• Images (a) and (b) there are no much differences
• But (c) and (d) show significant degradation

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• Intensity resolution:
• Refers to the smallest observable change in the intensity level
• It is defined as the number of bits used to quantize the intensity
• We have seen that, the number of intensity levels choses is usually power of 2
• Most commonly used number is 8 bits
• E.g.: an image with intensity quantized to 256 levels is said to have 8-bit intensity
resolution
• Effects of varying the number of intensity levels in a digital image.
• Consider a CT scan image of size 452 X 374 displayed with k=8, (256 intensity
levels).

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• Following images are obtained by reducing the number of bits from k=7 to k=1
while keeping the image size constant at 452 X 374 pixels.
• The 256-, 128-, and 64-level images are visually identical for all practical
purposes
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• However, the last figure with 32- level has an imperceptible set of very
fine ridge-like structures in areas of constant or nearly constant intensity
(particularly in the skull).
• This effect, caused by the use of an insufficient number of intensity
levels in smooth areas of a digital image, is called false contouring,
• False contouring generally is quite visible in images displayed using 16
or less uniformly spaced intensity levels, as seen in the following
images

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The results in previous two examples illustrate the effects produced on image
quality by varying N and k independently.
However, these results only partially answer the question of how varying N and k
affects images
Because we have not considered yet any relationships that might exist between
these two parameters.
An early study by Huang [1965] attempted to quantify experimentally the effects
on image quality produced by varying N and k simultaneously.
The experiment consisted of a set of subjective tests on images which have very
little detail, moderate detail and large amount of details

Sets of these three types of images were generated by varying N and k, and observers
were then asked to rank them according to their subjective quality.
Results were summarized in the form of so-called isopreference curves in the Nk-plane
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Each point in the Nk-plane represents an image
having values of N and k equal to the coordinates of
that point.
Points lying on an isopreference curve correspond to
images of equal subjective quality.
It was found that the isopreference curves tended to
shift right and upward, but their shapes in each of the
three image categories were similar to those in Fig
Shift up and right in the curves simply means larger
values for N and k, which implies better picture
quality.
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We can see that isopreference curves tend to become more vertical as
the detail in the image increases.
This result suggests that for images with a large amount of detail only a
few intensity levels may be needed.
❖E.g.: The isopreference curve corresponding to the crowd is nearly
vertical.
This means, for a fixed value of N, the perceived quality for this type
of image is nearly independent of the number of intensity levels used

Some Basic Relationships Between Pixels
❖ Here we study several important relationships between pixels in a digital image.
❖ We denoted an image as f(x, y).
❖ Hereafter to refer to a particular pixel, we use lowercase letters, such as p and q.
❖ Neighbors of a Pixel
❖ A pixel p at coordinates (x, y) has four horizontal and vertical neighbors as shown
❖ Their coordinates are given by (x + 1, y), (x - 1, y), (x, y + 1), (x, y - 1)
❖ This set of pixels, called the 4-neighbors of p, is denoted by N4(p)
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❖ Each pixel is a unit distance from (x, y),
❖Some of the neighbor locations of p lie outside the digital image if is on the border of
the image
❖The four diagonal neighbors of p have coordinates (x+1, y+1), (x+1, y-1), (x-1, y+1),
(x-1, y-1)
❖ These are denoted by ND(p)
❖These points, together with the 4-neighbors, are called the 8-neighbors of p, denoted by
N8(p).
❖Some of the neighbor locations in ND(p) and N8(p) fall outside the image if is on the
border of the image.

• Neighbours of a pixel.
• N4 - 4-neighbors
• ND - diagonal neighbours
• N8 - 8-neighbors (N4 U ND)
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Adjacency, Connectivity, Regions, and Boundaries
Adjacency
Two pixels are connected if they are neighbors and their gray levels satisfy some
specified criterion of similarity.
For example, in a binary image two pixels are connected if they are 4-neighbors and
have same value (0/1).
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.
We consider three types of adjacency:

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❖ 4-adjacency.
❖ Two pixels p and q with values from V are 4-adjacent if q is in the set N4(p)
❖ 8-adjacency.
❖ Two pixels p and q with values from V are 8-adjacent if q is in the set N8(p)
❖ 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.

to denote set intersection and union, respectively.
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❖ We use the symbols ∩ and
❖ For any given setsAand B,
❖ their intersection is the set of elements that are members of bothAand B.
❖The union of these two sets is the set of elements that are members of A, of B, or of
both.

• Connectivity
• Consider a set of gray scale values V = { 0, 1, 2}
• Let p be a pixel such that f(p) which has intensity level
• 4-connectivity:
• Consider a pixel q such that q is N4-neighbor of p and intensity of q is
f (q) V
• Then we say q is having 4 connectivity with p
• 8-connectivity:
• Consider a pixel q such that q is N8-neighbor of p and intensity of q is
f (q) V
• Then q is having 8-connectivity with p
f ( p) V
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• m-connectivity (mixed connectivity):
• Consider two pixels p and q such that, intensity of p and q are given by
f ( p) V f (q) V
• Pixels p and q are said to be having m-connectivity, if
• Examples
q N4 ( p)
i. or
ii. The set N4 ( p) N4 (q) is empty
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• Path
•A (digital) path (or curve) from pixel p with coordinates (x0, y0) to pixel q with
coordinates (xn, yn) is a sequence of distinct pixels with coordinates (x0, y0), (x1, y1),
…, (xn, yn)
• where (xi, yi) and (xi-1, yi-1) are adjacent for 1 ≤ i ≤ n.
• Here n is the length of the path.
• If (x0, y0) = (xn, yn), the path is closed path.
• We can define 4-, 8-, and m-paths based on the type of adjacency used.

• .
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• .
8
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• .
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• Examples on paths.
• Example # 1: Consider the image segment shown in figure. Compute length of the
shortest-4, shortest-8 & shortest-m paths between pixels p & q where, V = {1, 2}.
4 2
3 3
2 3
p 2 1
3
1
2
2
2 q
3
2
3

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•Examples on paths.
•Example # 1: Consider the image segment shown in figure. Compute length of the
shortest-4, shortest-8 & shortest-m paths between pixels p & q where, V = {1, 2}.
4 2
3 3
2 3
p 2 1
3
1
2
2
2 q
3
2
3

•Let us start finding the path between p and q as shown below.
•So shortest -4 path does not exist
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•Shortest-8 path.
•Thus Shortest-8 path = 4
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•Shortest-m path
•Shortest m-path =5
4
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•Connected in S
Let S represent a subset of pixels in an image.
Two pixels p with coordinates (x0, y0) and q with coordinates (xn, yn)
are said to be connected in S if there exists a path
(x0, y0), (x1, y1), …, (xn, yn)
where
Let S represent a subset of pixels in an image
For every pixel p in S, the set of pixels in S that are connected to p is called a
connected component of S.
If S has only one connected component, then S is called Connected Set.
i,0  i  n,(xi , yi ) S

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Region of an image
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 to be adjacent are said to be disjoint.
We consider 4- and 8- adjacency when referring to regions.

•In the above image, two regions are adjacent if 8 adjacency is used.
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•Foreground and background of image
•Suppose that an image contains K disjoint regions, Rk, k= 1,2,3..K with
none of them touching the image border.
denote its
•Let Ru denote the union of all the K regions, and let (Ru)c
complement
•The complement of a set S is the set of points that are not in S.
•We call all the points in Ru the foreground, and all the points in (Ru)c the
background of the image.

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•The boundary (border or contour) of a region R
•The boundary of a region R is the set of pixels in the region that have one or more
neighbors that are not in R.
•If R happens to be an entire image, then its boundary is defined as the set of pixels
in the first and last rows and columns in the image.
•This extra definition is required because an image has no neighbors beyond its
borders
•Normally, when we refer to a region, we are referring to subset of an image, and
any pixels in the boundary of the region that happen to coincide with the border of
the image are included implicitly as part of the region boundary.

•Distance Measures
•Consider three pixels p, q, and z, with coordinates (x, y), (s, t), and (v, w), respectively
•The distance between any two points can be calculated as distance metric or distance
measure
•But it must satisfy the following properties
•The Euclidean distance between p and q is defined as
• De (p,q) = [(x – s)2 + (y - t)2]1/2
p (x,y)
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q (s,t)

•City-block distance
•The D4 distance (also called city-block distance) between p and q is defined as:
D4 (p,q) = | x – s | + | y – t |
• Example:
•The pixels with distance D4 ≤ 2 from (x,y) form the following contours of constant
distance.
•The pixels with D4 = 1 are the 4-neighbors of (x,y)
p (x,y)
q (s,t)
D4
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•Pixels having a D4 distance from (x,y), less than or equal to some value r form a Diamond
centered at (x,y)
•In case of Euclidean Distance, 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)

•Distance Measures.
•Chessboard Distance (D8 distance):
•The D8 distance between p,q is defined as D8(p, q) = max(|x-s|, |y-t|)
•In this case, pixels having D8 distance from (x, y) less than or equal to
some value r form a square centered at ( x, y).
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•Applications of Image Processing:
• Medical imaging,
• Robot vision,
• Character recognition,
• Remote Sensing

Basics of digital signal and image processing

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