Module 1

Part A

1)Derive the Haar transformation matrix for N = 8 and explain how it is constructed.
The Haar transformation is a wavelet transformation that decomposes a signal into a series of

increasingly fine-scale signals, known as wavelets. The Haar transformation matrix for N=8 is

given by:

H8 = [1 1 1 1 1 1 1 1
1 1 1 1 -1 -1 -1 -1
1 1 -1 -1 0 0 0 0
0 0 0 0 1 1 -1 -1
1 -1 0 0 1 -1 0 0
0 0 1 -1 0 0 1 -1
0 0 1 1 0 0 -1 -1

0 0 1 -1 0 0 -1 1]

To construct the Haar transformation matrix for N=8, we start by constructing the Haar

transformation matrix for N=4, which is given by:

H4 = [1 1 1 1
1 1 -1 -1
1 -1 0 0

0 0 1 -1]

We can then construct the Haar transformation matrix for N=8 by taking the Kronecker product

of H4 with itself:

H8 = H4 x H4 = [1 1 1 1 1 1 1 1
1 1 1 1 -1 -1 -1 -1
1 1 -1 -1 0 0 0 0

0 0 0 0 1 1 -1 -1]

We can then append two more rows to H8 to get the final Haar transformation matrix for N=8:

H8 = [1 1 1 1 1 1 1 1
1 1 1 1 -1 -1 -1 -1
1 1 -1 -1 0 0 0 0
0 0 0 0 1 1 -1 -1
1 -1 0 0 1 -1 0 0
0 0 1 -1 0 0 1 -1
0 0 1 1 0 0 -1 -1

0 0 1 -1 0 0 -1 1]

The Haar transformation matrix is constructed using a recursive approach, where the Haar

transformation matrix for N is obtained by taking the Kronecker product of the Haar

transformation matrix for N/2 with itself. This allows the Haar transformation to capture both the

coarse-scale and fine-scale features of the signal.

2)Determine the convolution and correlation between the following images.

Convolution and correlation are mathematical operations that involve using a kernel
or filter to process an input image. The output of the convolution or correlation
operation is a new image, which is obtained by sliding the kernel over the input
image and performing element-wise multiplications and summations at each
position.

Convolution is a linear operation that is used to apply a filter or kernel to an input

image in order to extract features or modify the image in some way.

Correlation is a similar operation to convolution, but the kernel or filter is flipped both
horizontally and vertically before it is applied to the input image.

In image processing, convolution and correlation are often used to process and
analyze images, extract features, and perform tasks such as smoothing, sharpening,
and edge detection. Convolution and correlation can be implemented using various
techniques, such as the fast Fourier transform (FFT) or discrete convolution,
depending on the specific requirements and constraints of the application

3)A common measure of transmission for digital data is baud rate defined by
number of bits transmitted per second. Find how many minutes it would take to
transmit a 2048*2048 images with 256 intensity levels using a 33.6k baud modem.

To determine how many minutes it would take to transmit a 2048x2048 image with 256 intensity

levels using a 33.6 kbps (kilobits per second) modem, we need to first calculate the total number

of bits required to represent the image. Since the image has 2048x2048 pixels and each pixel has

256 intensity levels, we can represent each pixel using 8 bits (2^8 = 256). This means that the

total number of bits required to represent the image is 2048x2048x8 = 4,294,967,296 bits.
Next, we can divide the total number of bits by the baud rate to determine the total time required

to transmit the image. Since the baud rate is 33.6 kbps, we can calculate the total time as:

$$\frac{4,294,967,296\text{ bits}}{33.6\text{ kbps}} = \frac{4,294,967,296\text{

bits}}{33,600\text{ bits/s}} = 127.7\text{ seconds}$$

To convert this time to minutes, we can divide the time in seconds by 60:

$$\frac{127.7\text{ seconds}}{60\text{ seconds/minute}} = 2.1\text{ minutes}$$

Therefore, it would take approximately 2.1 minutes to transmit a 2048x2048 image with 256

intensity levels using a 33.6 kbps modem.

4)Compute the Haar transform of the N=2,and N=4 image

The Haar transform is a type of wavelet transform that decomposes an image into a
set of orthogonal basis functions, known as "wavelets", which can be used to
represent the image in a compact and efficient manner. The Haar transform is
particularly useful for image compression and analysis, and it has been used in a
variety of applications, including image recognition and facial detection.

To compute the Haar transform of an NxN image, we can follow these steps:

1. Divide the image into non-overlapping blocks of size NxN.

2. For each block, compute the average value of the pixels in the block, and
subtract this average value from the value of each pixel in the block. This
produces a set of NxN "difference" values for each block.
3. Repeat the process on the difference values, dividing them into blocks of size
N/2xN/2 and computing the average value and differences for each block.
4. Repeat the process until all blocks have been processed.

For example, to compute the Haar transform of a 2x2 image, we can follow these
steps:

1. Divide the image into two non-overlapping blocks of size 2x2.

2. For each block, compute the average value of the pixels in the block, and
subtract this average value from the value of each pixel in the block. This
produces two 2x2 "difference" matrices:
 $$\begin{bmatrix} a & b \ c & d \end{bmatrix} \rightarrow \begin{bmatrix}
a-\frac{a+b+c+d}{4} & b-\frac{a+b+c+d}{4} \ c-\frac{a+b+c+d}{4} &
d-\frac{a+b+c+d}{4} \end{bmatrix}$$
3. The Haar transform of the 2x2 image is then given by the concatenation of the
two difference matrices:
$$\begin{bmatrix} a-\frac{a+b+c+d}{4} & b-\frac{a+b+c+d}{4} \
c-\frac{a+b+c+d}{4} & d-\frac{a+b+c+d}{4} \end{bmatrix}$$

To compute the Haar transform of a 4x4 image, we can follow a similar process,
dividing the image into four non-overlapping blocks of size 2x2 and computing the
average and difference values for each block. The resulting Haar transform is then
given by the concatenation of the four difference matrices.

5)Obtain the slant transforms matrix For N=8.

The slant transform is a linear transform that maps an NxN image to a new image with the same

size, where the rows and columns of the new image are slanted versions of the rows and

columns of the original image. The slant transform can be represented by an NxN matrix, where

each element of the matrix corresponds to a position in the original image and is mapped to a

new position in the transformed image.

To obtain the slant transform matrix for N=8, we can follow these steps:

1. Create an NxN matrix with all elements equal to 0. This will be our slant transform matrix.
2. For each row i and column j in the original image, set the element at position (i,j) in the
slant transform matrix to 1.
3. Set the diagonal elements of the matrix to 0. These elements correspond to the pixels in
the original image that are not transformed.
4. For each row i and column j in the original image, set the element at position (i+1,j+1) in
the slant transform matrix to -1. This will cause the rows and columns of the transformed
image to be slanted versions of the rows and columns of the original image.

For example, the slant transform matrix for N=8 would be:

$$\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \ -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & -1 & 0 & 1 & 0

& 0 & 0 & 0 \ 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & -1 & 0 &

1 & 0 \ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \end{bmatrix}$$
I hope this helps! Let me know if you have any questions or if you would like further clarification.

10)Derive the DCT matrix for N = 4 and verify that it obeys the orthogonality property.

The discrete cosine transform (DCT) is a linear transform that maps an NxN image to a new

image with the same size, where the rows and columns of the new image are weighted sums of

the rows and columns of the original image. The DCT is often used for image compression and

analysis, and it can be represented by an NxN matrix, where each element of the matrix

corresponds to a position in the original image and is mapped to a new position in the

transformed image.

To derive the DCT matrix for N=4, we can follow these steps:

1. Create an NxN matrix with all elements equal to 0. This will be our DCT matrix.

2. For each row i and column j in the matrix, set the element at position (i,j) to:

$$DCT[i,j] = \cos\left(\frac{\pi(2i+1)j}{2N}\right)$$

3. Scale the matrix by a factor of $\sqrt{\frac{2}{N}}$ if $i=0$ and a factor of

$\sqrt{\frac{1}{N}}$ if $i>0$. This ensures that the DCT matrix is orthonormal, which

means that the rows and columns of the matrix are orthogonal (perpendicular) and have

unit length.

The resulting DCT matrix for N=4 is:

$$\begin{bmatrix} \sqrt{\frac{2}{4}} & \sqrt{\frac{1}{4}} & \sqrt{\frac{1}{4}} & \sqrt{\frac{1}{4}} \

\sqrt{\frac{1}{4}} & \sqrt{\frac{2}{4}} & \sqrt{\frac{1}{4}} & -\sqrt{\frac{1}{4}} \ \sqrt{\frac{1}{4}} &

\sqrt{\frac{1}{4}} & -\sqrt{\frac{2}{4}} & \sqrt{\frac{1}{4}} \ \sqrt{\frac{1}{4}} & -\sqrt{\frac{1}{4}}

& \sqrt{\frac{1}{4}} & -\sqrt{\frac{2}{4}} \end{bmatrix}$$

To verify that this matrix obeys the orthogonality property, we can take the dot product of each

pair of rows and columns and verify that the result is 0 for all pairs except for the pairs that

consist of the same row or column. For example, the dot product of the first and second rows is:
$$\begin{bmatrix} \sqrt{\frac{2}{4}} & \sqrt{\frac{1}{4}} & \sqrt{\frac{1}{4}} & \sqrt{\frac{1}{4}}

\end{bmatrix} \cdot \begin{bmatrix} \sqrt{\frac{1}{4}} & \sqrt{\frac{2}{4}} & \sqrt{\frac{1}{4}} &

-\sqrt{\frac{1}{4}} \end{bmatrix} = \frac{2}{4} + \frac{1}{4} + \frac{1}{4} - \frac{1}{4} = 0$$

Similarly, the dot product of the first and third rows is:

$$\begin{bmatrix} \sqrt{\frac{2}{4}} & \sqrt{\frac{1}{4}}

Part B

1)Explain any four basic relationships between pixels.

In digital image processing, the relationship between pixels is an important concept

that plays a key role in various image analysis and manipulation tasks. Here are four

basic relationships between pixels that are commonly used in image processing:

1. Adjacency: Pixels that are adjacent to each other in an image are said to be

adjacent pixels. Adjacency can be defined in various ways, depending on the

specific application. For example, pixels may be considered adjacent if they

share an edge or if they are located within a certain distance of each other.

Adjacency is often used to define neighborhoods or regions in an image,

which can be useful for tasks such as smoothing, edge detection, and image

segmentation.

2. Distance: The distance between two pixels in an image is a measure of the

separation between them. Distance can be calculated using various metrics,

such as the Euclidean distance, Manhattan distance, or other measures.

Distance is often used to measure the similarity or dissimilarity between

pixels, which can be useful for tasks such as image registration, pattern

matching, and clustering.

3. Orientation: The orientation of a pixel in an image is a measure of its direction

or angle relative to a reference direction or point. Orientation is often used to

 detect lines, edges, and other features in an image, and it can be calculated

using various techniques, such as gradient-based methods or the Hough

transform.

4. Intensity: The intensity of a pixel in an image is a measure of its brightness or

color. Intensity is often used to represent the grayscale or color value of a

pixel, and it can be quantified using various scales, such as the 8-bit or 16-bit

scales. Intensity is often used to represent the visual appearance of an image

and to perform tasks such as contrast enhancement and color correction.

2)Demonstrate the components of digital image processing system and explain each block.

A digital image processing system is a system that processes digital images for

various purposes, such as visualization, analysis, manipulation, and communication.

The components of a digital image processing system can be broadly grouped into

the following categories:

1. Input device: This is a device that captures or receives digital images, such as

a digital camera, scanner, or video capture device. The input device may also

include additional hardware or software components, such as lenses, sensors,

or image processing algorithms, that are used to acquire, pre-process, or

enhance the images.

2. Storage: This is a component that stores digital images in a digital format,

such as a hard drive, solid-state drive, or cloud storage. The storage

component may also include additional hardware or software components,

such as file systems, backup systems, or metadata management systems,

that are used to organize, protect, or retrieve the images.

3. Processor: This is a component that performs various operations on digital

images, such as image analysis, image manipulation, or image synthesis. The

 processor may be a general-purpose computer or a specialized device, such

as a graphics processing unit (GPU) or a digital signal processor (DSP), and it

may use various algorithms, software libraries, or programming languages to

perform the operations.

4. Output device: This is a device that displays or communicates digital images,

such as a monitor, printer, or network interface. The output device may also

include additional hardware or software components, such as display drivers,

protocols, or image formats, that are used to prepare the images for display or

transmission.

5. User interface: This is a component that allows the user to interact with the

digital image processing system, such as through a graphical user interface

(GUI), a command-line interface (CLI), or a programming interface. The user

interface may include various input and output devices, such as a keyboard,

mouse, or touch screen, and it may provide various features and tools, such

as menus, buttons, or sliders, to control the system and access its

functionality.

3)Define digital image. Discuss how digital images are represented with neat diagrams.

A digital image is a representation of a visual image in a digital format, such as a

computer file or a stream of data. Digital images can be created by capturing

real-world scenes or objects using devices such as cameras or scanners, or they can

be synthesized using software or algorithms. Digital images are often used for

various purposes, such as visualization, analysis, manipulation, communication, and

storage.

There are several ways to represent digital images, and the specific representation

used depends on the requirements and constraints of the application. Some

common ways to represent digital images are:

 1. Pixel array: This is a common representation for digital images, where each

pixel in the image is represented by a numerical value or a set of values that

correspond to its color or intensity. Pixel arrays are often organized as 2D

grids of rows and columns, where each row and column corresponds to a

position in the image. The values of the pixels can be encoded using various

scales, such as the 8-bit or 16-bit scales, and they can be stored in various

formats, such as binary, ASCII, or compressed.

2)Vector graphics: This is a representation for digital images that uses geometric

primitives, such as points, lines, curves, and shapes, to describe the image. Vector

graphics are often used for images that have smooth, continuous, or scalable

elements, and they can be stored in various formats, such as PostScript, SVG, or PDF.

3)Bitmap graphics: This is a representation for digital images that uses a pattern of

bits or pixels to represent the image. Bitmap graphics are often used for images that

have discrete, pixelated, or binary elements, and they can be stored in various
4)Discuss sampling and quantization With necessary diagrams.

Sampling and quantization are two important concepts in digital image processing

that are used to convert continuous signals, such as images, into digital form.

Sampling is the process of selecting a set of samples from a continuous signal at

discrete intervals of time or space. Sampling is used to represent the continuous

signal by a finite set of points, which can be processed and manipulated using digital

techniques. Sampling is often performed by an analog-to-digital converter (ADC),

which converts the continuous signal into a sequence of discrete values.

Quantization is the process of mapping a set of continuous values to a set of

discrete values, using a finite number of levels or bins. Quantization is used to

represent the samples of the continuous signal by a finite set of integers or symbols,

which can be encoded and stored using a limited number of bits. Quantization is

often performed by a digital-to-analog converter (DAC), which converts the discrete

values into a continuous signal.

Sampling and quantization are often used together in digital image processing to

represent images by a finite set of pixels, where each pixel is represented by a finite

set of bits

5)Illusttrate the effect of increasing sampling frequency and quantization levels on

image with an example

Sampling frequency and quantization levels are two important parameters that

affect the quality and accuracy of digital images.

Increasing the sampling frequency means increasing the number of samples per unit

time or space, which can improve the resolution and detail of the image. However,

increasing the sampling frequency also increases the amount of data required to

represent the image, which can lead to larger file sizes and longer processing times.
Increasing the quantization levels means increasing the number of levels or bins

used to represent the samples, which can improve the precision and accuracy of the

image. However, increasing the quantization levels also increases the number of bits

required to represent the samples, which can lead to larger file sizes and longer

processing times.

6)List and explain applications of image processing

Image processing is a field of digital signal processing that involves the

manipulation and analysis of images for various purposes. Image processing has a

wide range of applications in various fields, such as medicine, engineering, science,

art, and entertainment. Some examples of applications of image processing are:

1. Medical imaging: Image processing is used in medical imaging to visualize

and analyze medical images, such as X-rays, CT scans, MRI scans, and

ultrasound images. Image processing can be used to improve the quality,

contrast, resolution, and accuracy of medical images, and to extract

information and features that are relevant to diagnosis, treatment, and

research.

2. Computer vision: Image processing is used in computer vision to enable

computers to perceive, understand, and interpret the visual world. Computer

vision involves tasks such as image recognition, object detection, tracking,

segmentation, and 3D reconstruction, and it has applications in areas such as

robotics, surveillance, transportation, and augmented reality.

3. Remote sensing: Image processing is used in remote sensing to extract

information from images and data acquired by sensors that are mounted on
 satellites, aircraft, drones, or other platforms. Remote sensing can be used to

monitor and map the Earth's surface, atmosphere, oceans, and other features,

and it has applications in areas such as agriculture, forestry, meteorology, and

environmental management.

4. Graphics and multimedia: Image processing is used in graphics and

multimedia to create, edit, and manipulate images, videos, and other media

for various purposes. Image processing can be used to enhance the

appearance, quality, and effect of media, and to generate special effects,

animations, and interactive content.

5. Industrial inspection: Image processing is used in industrial inspection to

automate and improve the efficiency of inspection tasks, such as defect

detection, measurement, and classification. Image processing can be used to

analyze images of products, parts, or materials, and to extract information

and features that are relevant to quality, performance, and safety.

7)What is spatial resolutions? Discuss the effect on the image by reducing it.

Spatial resolution is a measure of the detail and clarity of an image, and it is defined

as the minimum distance between two points in the image that can be distinguished

as separate entities. Spatial resolution is often expressed in terms of pixels per inch

(PPI) or pixels per millimeter (PPM), and it is an important factor that determines the

quality and usefulness of an image for various tasks.

Reducing the spatial resolution of an image means reducing the number of pixels or

the size of the pixels, which can lower the detail and clarity of the image. Reducing

the spatial resolution can cause the image to become blurry, pixelated, or distorted,

and it can make it difficult to see small or subtle features in the image. Reducing the

spatial resolution can also reduce the file size of the image, which can be useful for
storage or transmission purposes, but it can also reduce the information content and

the potential uses of the image.

Here is an example that illustrates the effect of reducing the spatial resolution of an

image:

In this example, the original image (left) has a high spatial resolution, with clear and

detailed features. When the spatial resolution is reduced (middle and right), the

image becomes progressively blurrier and pixelated, and the features become less

distinct and more difficult to discern.

8)Explain the concept of nonuniform sampling and quantization with examples.

Nonuniform sampling and quantization are variations of the traditional sampling and

quantization techniques that are used in digital image processing. Nonuniform

sampling and quantization involve the use of irregular or variable intervals, levels, or

distributions to sample or quantize the image, rather than the use of uniform or

constant intervals, levels, or distributions. Nonuniform sampling and quantization

can be used to adapt the representation of the image to the characteristics or

patterns of the image, or to the requirements or constraints of the application.

Some examples of nonuniform sampling and quantization are:

1. Nonuniform sampling: Nonuniform sampling involves the use of nonuniform

intervals to sample the image, such as variable intervals that are based on the

intensity, gradient, or frequency of the image. Nonuniform sampling can be

used to improve the efficiency or accuracy of the representation of the image,

or to reduce the data rate or complexity of the image. For example,

nonuniform sampling can be used to sample the low-frequency or smooth

 parts of the image more densely than the high-frequency or detailed parts of

the image, or vice versa.

2. Nonuniform quantization: Nonuniform quantization involves the use of

nonuniform levels or bins to quantize the image, such as variable levels or

bins that are based on the intensity, gradient, or probability of the image.

Nonuniform quantization can be used to improve the quality or efficiency of

the representation of the image, or to reduce the data rate or complexity of the

image. For example, nonuniform quantization can be used to assign more

levels or bins to the high-intensity or high-probability parts of the image, and

fewer levels or bins to the low-intensity or low-probability parts of the image,

or vice versa.

9)Discuss the most commonly used distance measures in image processing

Distance measures are mathematical functions that are used to quantify the

separation or similarity between two or more points, vectors, or sets in an image or a

space. Distance measures are often used in image processing to compare, classify,

cluster, or match images, or to measure the dissimilarity, similarity, or error between

images.

Here are some examples of distance measures that are commonly used in image

processing:

1. Euclidean distance: This is a measure of the straight-line distance between

two points in a space, and it is defined as the square root of the sum of the

squares of the differences between the coordinates of the points. Euclidean

distance is often used to measure the distance between pixels, patches, or

features in an image, and it is a popular choice for tasks such as image

registration, pattern matching, and clustering.

 2. Manhattan distance: This is a measure of the distance between two points in

a space that is based on the absolute differences between the coordinates of

the points, and it is defined as the sum of the absolute differences. Manhattan

distance is often used to measure the distance between pixels, patches, or

features in an image, and it is a popular choice for tasks such as image

segmentation and image recognition.

3. Cosine similarity: This is a measure of the similarity between two vectors in a

space that is based on the angle between the vectors, and it is defined as the

dot product of the vectors divided by the product of their magnitudes. Cosine

similarity is often used to measure the similarity between vectors, such as

image patches or features, and it is a popular choice for tasks such as image

retrieval and text analysis.

4. Earth Mover's Distance (EMD): This is a measure of the distance between two

sets of points or distributions in a space that is based on the amount of work

required to transform one set into the other, and it is defined as the minimum

cost of transport between the sets. EMD is often used to measure the

distance between sets, such as image histograms or distributions, and it is a

popular choice for tasks such as image classification and image retrieval.

10)The image refers to a two dimensional light intensity function. Discuss in detail.

An image can be represented as a two-dimensional light intensity function, also

known as an image function or a continuous image. A two-dimensional light intensity

function is a mathematical function that maps a set of two-dimensional coordinates

to a set of intensity values, and it describes the intensity or brightness of the image

at each point in the image.

A two-dimensional light intensity function is typically defined as a function of two

variables, such as x and y, which correspond to the horizontal and vertical

coordinates of the image, respectively. The intensity values of the function can be

continuous or discrete, and they can be encoded using various scales, such as the

8-bit or 16-bit scales. The intensity values of the function can represent the

luminance, chrominance, or other characteristics of the image, and they can be used

to display, analyze, or manipulate the image.

A two-dimensional light intensity function can be represented graphically as a

surface or a contour plot, where the intensity values are plotted as a function of the

coordinates. A two-dimensional light intensity function can also be represented

numerically as a matrix or a vector, where the intensity values are stored as elements

or coefficients. A two-dimensional light intensity function can be manipulated using

various image processing techniques, such as filtering, enhancement, restoration, or

transformation, and it can be displayed or communicated using various image

formats, such as JPEG, PNG, or GIF.

11)Discuss the image acquisition using a single sensor, sensor strips and sensor

arrays.
Image acquisition is the process of capturing, collecting, or generating images using various

devices or methods. Image acquisition involves the use of sensors, which are devices that are

used to detect and measure various physical quantities, such as light, sound, temperature, or

pressure. Sensors can be used to acquire images of various scenes, objects, or phenomena, and

they can be used for various purposes, such as visualization, analysis, manipulation,

communication, or storage.

There are several ways to acquire images using sensors, and the specific method used depends

on the requirements and constraints of the application. Some examples of image acquisition

using sensors are:

1. Single sensor: Image acquisition using a single sensor involves the use of a single sensor

to capture or generate an image of a scene or an object. Single-sensor image acquisition

 can be used to acquire monochrome or color images, and it can be performed using

various types of sensors, such as cameras, scanners, or sensors that are specialized for

specific wavelengths or ranges of the electromagnetic spectrum. Single-sensor image

acquisition can be affected by various factors, such as the size, resolution, sensitivity,

and dynamic range of the sensor, and the illumination, exposure, focus, and alignment of

the scene or the object.

2. Sensor strips: Image acquisition using sensor strips involves the use of a linear or

curvilinear array of sensors to capture or generate an image of a scene or an object.

Sensor-strip image acquisition can be used to acquire images with a wider field of view, a

higher resolution, or a greater depth of field than single-sensor image acquisition, and it

can be performed using various configurations of sensors, such as parallel, concentric, or

overlapping strips. Sensor-strip image acquisition can be affected by various factors,

such as the size, resolution, sensitivity, and dynamic range of the sensors, and the

illumination, exposure, focus, and alignment of the scene or the object.

3. Sensor arrays: Image acquisition using sensor arrays involves the use of a

two-dimensional array of sensors to capture or generate an image of a scene or an

object. Sensor-array image acquisition can be used to acquire images with a wider field

of view, a higher resolution, or a greater depth of field than single-sensor or sensor-strip

image acquisition, and it can be performed using various configurations of sensors,

12)What is Hadamard transform? Explain in detail and Write its properties.

The Hadamard transform is a linear transformation that is used to decompose a

signal or an image into a set of orthogonal or uncorrelated components, and it is

named after the French mathematician Jacques Hadamard. The Hadamard

transform is a type of discrete Fourier transform (DFT), and it is related to other

DFTs, such as the discrete cosine transform (DCT) and the discrete sine transform

(DST). The Hadamard transform is widely used in various fields, such as image

processing, communications, data compression, and signal processing, and it has

several properties that make it attractive for these applications.

Here are some properties of the Hadamard transform:

 1. Orthogonality: The Hadamard transform is an orthogonal transformation,

which means that the transform matrix is an orthogonal matrix, and the

transformed coefficients are orthogonal or uncorrelated. Orthogonality is a

desirable property of transforms, because it preserves the energy or variance

of the signal or the image, and it reduces the redundancy or correlation of the

coefficients.

2. Symmetry: The Hadamard transform is a symmetric transformation, which

means that the transform matrix is a symmetric matrix, and the transformed

coefficients are symmetric or self-inverse. Symmetry is a desirable property of

transforms, because it simplifies the computation and analysis of the

transform, and it reduces the complexity and storage requirements of the

coefficients.

3. Completeness: The Hadamard transform is a complete transformation, which

means that it can represent any signal or image exactly, up to a scale factor,

and it can reconstruct the original signal or image exactly, up to the same

scale factor. Completeness is a desirable property of transforms, because it

ensures that the transformed signal or image contains all the information

13)Explain about KL Transform and Write its properties

The Karhunen-Loeve (KL) transform, also known as the principal component analysis

(PCA) or the eigenvector transform, is a linear transformation that is used to

decompose a signal or an image into a set of orthogonal or uncorrelated

components, and it is named after the Finnish mathematician Erkki Karhunen and

the French mathematician Jean-Pierre Loeve. The KL transform is a type of singular

value decomposition (SVD), and it is related to other SVDs, such as the Latent

Semantic Analysis (LSA) and the Latent Dirichlet Allocation (LDA). The KL transform

is widely used in various fields, such as image processing, machine learning, data
analysis, and signal processing, and it has several properties that make it attractive

for these applications.

Here are some properties of the KL transform:

1. Orthogonality: The KL transform is an orthogonal transformation, which

means that the transform matrix is an orthogonal matrix, and the transformed

coefficients are orthogonal or uncorrelated. Orthogonality is a desirable

property of transforms, because it preserves the energy or variance of the

signal or the image, and it reduces the redundancy or correlation of the

coefficients.

2. Normality: The KL transform is a normal transformation, which means that the

transform matrix is a normal matrix, and the transformed coefficients are

normal or unitary. Normality is a desirable property of transforms, because it

simplifies the computation and analysis of the transform, and it preserves the

energy or variance of the signal or the image.

14)Define the following two properties of 2D-DFT: i) Convolution ii) Correlation

The two-dimensional discrete Fourier transform (2D-DFT) is a linear transformation

that is used to decompose a two-dimensional signal or image into a set of

complex-valued frequency coefficients, and it is a generalization of the

one-dimensional discrete Fourier transform (1D-DFT). The 2D-DFT is widely used in

various fields, such as image processing, signal processing, communications, and

data analysis, and it has several properties that make it attractive for these

applications.

Here are the definitions of two properties of the 2D-DFT:

 1. Convolution: Convolution is a mathematical operation that is used to combine

two signals or images, and it is defined as the integral of the product of the

two signals or images, as one is flipped and shifted over the other.

Convolution is a linear operation, and it is commutative, associative, and

distributive. Convolution is often used to model the effect of a linear system

or a filter on a signal or an image, and it is implemented using the convolution

theorem, which states that the 2D-DFT of the convolution of two signals or

images is equal to the product of the 2D-DFT of each signal or image.

2. Correlation: Correlation is a statistical measure that is used to quantify the

relationship or similarity between two signals or images, and it is defined as

the normalized cross-correlation of the two signals or images. Correlation is a

nonlinear operation, and it is not commutative, associative, or distributive.

Correlation is often used to detect or match patterns or features in a signal or

an image, and it is implemented using the correlation theorem, which states

that the 2D-DFT of the cross-correlation of two signals or images is equal to

the product of the 2D-DFT of each signal or image, with one of the signals or

images conjugated.

15)Illustrate the following mathematical operations on digital images i) Array versus

Matrix operations ii) Linear versus Nonlinear Operations

Digital images are represented as arrays or matrices of pixels, which are the smallest

units of an image, and they are used to store, process, and communicate the

intensity, color, or other characteristics of the image. Digital images can be

manipulated using various mathematical operations, which are functions that are

used to transform, combine, or analyze the pixels of the image. Mathematical

operations on digital images can be classified into array operations and matrix
operations, and linear operations and nonlinear operations, based on the type and

complexity of the operation.

Here are some examples of mathematical operations on digital images:

1. Array versus matrix operations: Array operations are mathematical operations

that are applied element-wise or independently to the pixels of the image, and

they do not require the use of a matrix or a tensor. Array operations are simple

and efficient, and they are often used to perform basic or element-wise

operations, such as addition, subtraction, multiplication, division, or negation.

Matrix operations are mathematical operations that are applied using a matrix

or a tensor, and they require the use of linear algebra or tensor algebra. Matrix

operations are more complex and powerful, and they are often used to

perform advanced or global operations, such as filtering, transformation,

projection, or decomposition.

2. Linear versus nonlinear operations: Linear operations are mathematical

operations that are linear with respect to the pixels of the image, and they

preserve the superposition or additivity of the pixels. Linear operations are

simple and predictable, and they are often used to perform basic or linear

operations, such as scaling, shifting, rotating, or mirroring. Nonlinear

operations are mathematical operations that are nonlinear with respect to the

pixels of the image, and they do not preserve the superposition or additivity of

the pixels. Nonlinear operations are more complex and versatile, and they are

often used to perform advanced or nonlinear operations, such as

enhancement, restoration, segmentation, or compression.

16)Describe the need of image transform? List out various transform used in image

processing.
Image transform is a mathematical operation that is used to convert an image from

one domain or representation to another domain or representation, and it is a

fundamental technique in image processing. Image transform is used to analyze,

manipulate, or communicate the content, structure, or features of an image, and it is

often used to optimize, simplify, or enhance the image for a specific task or

application. Image transform is based on various mathematical concepts, such as

functions, series, matrices, or vectors, and it is implemented using various

algorithms, such as interpolation, extrapolation, approximation, or optimization.

There are several types of image transform that are used in image processing, and

the specific transform used depends on the requirements and constraints of the

application. Some examples of image transform used in image processing are:

1. Spatial transform: Spatial transform is a transform that is based on the spatial

or positional coordinates of the pixels in the image, and it is used to change

the orientation, size, or position of the image. Spatial transform includes

operations such as translation, rotation, scaling, shearing, or warping, and it is

often used to align, register, or match images, or to correct geometric

distortion or perspective.

2. Frequency transform: Frequency transform is a transform that is based on the

frequency or spectral content of the pixels in the image, and it is used to

decompose the image into its frequency components, or to reconstruct the

image from its frequency components. Frequency transform includes

operations such as Fourier transform, wavelet transform, or Hadamard

transform, and it is often used to analyze, filter, or compress the image, or to

remove noise, blur, or artifacts.

3. Color transform: Color transform is a transform that is based on the color or

chromaticity of the pixels in the image, and it is used to change the color
 space, model, or appearance of the image. Color transform includes

operations such as RGB-to-HSV, RGB-to-CMYK, or RGB-to-YUV, and it is often

used to enhance, balance, or match the color of the image, or to adapt the

image to different devices, standards, or environments.

4. Morphological transform: Morphological transform is a transform that is

based on the shape or structure of the pixels in the image, and it is used to

extract, enhance, or analyze the morphological features of the image.

Morphological transform includes operations such as erosion, dilation,

opening, closing, or skeletonization, and it is often used to segment, classify,

or label the image, or to remove noise, clutter, or artifacts.

17)Explain the following terms: (i) Adjacency (ii) Connectivity (iii) Regions (iv)

Boundaries

Adjacency, connectivity, regions, and boundaries are important concepts in image processing

and computer vision, and they are often used to describe, analyze, or manipulate the spatial or

topological properties of an image or a scene.

Here are the definitions of these terms:

1. Adjacency: Adjacency is a relationship between pixels in an image, and it refers to the

proximity or nearness of the pixels. Adjacency can be defined in terms of distance,

orientation, or intensity, and it can be used to describe the local or neighborhood

structure of the image. Adjacency is often used to define the conditions or criteria for

pixel grouping, segmentation, or classification, or to identify or extract features or

patterns in the image.

2. Connectivity: Connectivity is a relationship between pixels in an image, and it refers to the

continuity or connectedness of the pixels. Connectivity can be defined in terms of

distance, orientation, or intensity, and it can be used to describe the global or topological
 structure of the image. Connectivity is often used to define the conditions or criteria for

pixel grouping, segmentation, or classification, or to identify or extract features or

patterns in the image.

3. Regions: Regions are sets of pixels in an image that share certain characteristics or

properties, and they are used to partition or segment the image into distinct or

meaningful areas. Regions can be defined in terms of size, shape, color, texture, or

intensity, and they can be used to represent, classify, or label the objects or structures in

the image. Regions can be represented as masks, contours, or shapes, and they can be

manipulated or analyzed using various techniques, such as merging, splitting, or filling.

4. Boundaries: Boundaries are sets of pixels in an image that separate or distinguish

different regions or objects in the image, and they are used to define or outline the shape

or contour of the regions or objects. Boundaries can be defined in terms of curvature,

gradient, or orientation, and they can be used to represent, classify, or label the objects or

structures in the image. Boundaries can be represented as curves, lines, or edges, and

they can be manipulated

18)State the following two properties of 2D-DFT i) Translation ii) Rotation

The two-dimensional discrete Fourier transform (2D-DFT) is a linear transformation

that is used to decompose a two-dimensional signal or image into a set of

complex-valued frequency coefficients, and it is a generalization of the

one-dimensional discrete Fourier transform (1D-DFT). The 2D-DFT is widely used in

various fields, such as image processing, signal processing, communications, and

data analysis, and it has several properties that make it attractive for these

applications.

Here are the definitions of two properties of the 2D-DFT:

1. Translation: Translation is a geometric transformation that is used to shift or

move an image or a signal from one position to another, and it is defined as

the addition of a constant or a displacement to the pixels or the samples of

 the image or the signal. Translation is a linear operation, and it is implemented

using the translation theorem, which states that the 2D-DFT of a translated

image or signal is equal to the product of the 2D-DFT of the original image or

signal and a complex exponential function. Translation is often used to

correct or align images or signals, or to extract or match features or patterns

in the images or signals.

2. Rotation: Rotation is a geometric transformation that is used to rotate or

orient an image or a signal around a fixed point or axis, and it is defined as the

multiplication of a complex exponential function by the pixels or the samples

of the image or the signal. Rotation is a linear operation, and it is implemented

using the rotation theorem, which states that the 2D-DFT of a rotated image or

signal is equal to the product of the 2D-DFT of the original image or signal and

a complex exponential function. Rotation is often used to correct or align

images or signals, or to extract or match features or patterns in the images or

signals.

19)Derive the basis function for walsh transform

The Walsh transform is a linear transformation that is used to decompose a signal or

an image into a set of orthogonal or uncorrelated components, and it is named after

the American mathematician Joseph Walsh. The Walsh transform is a type of

Hadamard transform, and it is related to other Hadamard transforms, such as the

Haar transform and the Krawtchouk transform. The Walsh transform is widely used

in various fields, such as image processing, signal processing, data analysis, and

communications, and it has several properties that make it attractive for these

applications.

Here is the derivation of the basis function for the Walsh transform:
The Walsh transform is defined as the matrix multiplication of a signal or an image

by a Walsh matrix, and the Walsh matrix is defined as the Hadamard matrix of the

same size as the signal or the image. The Hadamard matrix is a square matrix of

size NxN, and it is defined as follows:

H(N) = [h(i,j)] = [(-1)^(i*j)]

where i and j are the row and column indices, respectively, and they range from 0 to

N-1.

The Walsh matrix is obtained by permuting the rows or the columns of the

Hadamard matrix, and it can be expressed as follows:

W(N) = [w(i,j)] = [h(p(i),p(j))]

where p(i) and p(j) are the permuted row and column indices, respectively, and they

determine the order of the rows and the columns in the Walsh matrix.

The Walsh transform is implemented using the fast Walsh transform (FWT)

algorithm, which is a fast and efficient algorithm for computing the Walsh transform,

and it is based on the properties of the Walsh matrix, such as symmetry,

orthogonality, and decomposability.

I hope this helps! Let me know if you have any questions or if you would like further

clarification.

20)Define 4,8-,madjacancy.Explain the lengths of shortest 4,8,mpaths between pixels with

examples

Adjacency is a relationship between pixels in an image, and it refers to the proximity or nearness

of the pixels. Adjacency can be defined in terms of distance, orientation, or intensity, and it can

be used to describe the local or neighborhood structure of the image. Adjacency is often used to
define the conditions or criteria for pixel grouping, segmentation, or classification, or to identify

or extract features or patterns in the image.

4-adjacency and 8-adjacency are types of adjacency that are used to describe the connectivity or

neighborhood of pixels in an image, and they are based on the number of neighbors or adjacent

pixels that a pixel has. 4-adjacency and 8-adjacency are used in different contexts or

applications, and they are implemented using different algorithms or techniques.

4-adjacency: 4-adjacency is a type of adjacency that is used to describe the connectivity or

neighborhood of pixels in a grayscale or binary image, and it is based on the four neighbors or

adjacent pixels that a pixel has. 4-adjacency is often used to define the conditions or criteria for

pixel grouping, segmentation, or classification, or to identify or extract features or patterns in the

image. 4-adjacency is implemented using the 4-connected neighborhood, which is a set of four

pixels that are adjacent to a pixel and that share a common edge or vertex with the pixel.

8-adjacency: 8-adjacency is a type of adjacency that is used to describe the connectivity or

neighborhood of pixels in a grayscale or binary image, and it is based on the eight neighbors or

adjacent pixels that a pixel has. 8-adjacency is often used to define the conditions or criteria for

pixel grouping, segmentation, or classification, or to identify or extract features or patterns in the

image. 8-adjacency is implemented using the 8-connected neighborhood, which is a set of eight

pixels that are adjacent to a pixel and that share a common vertex with the pixel.

The shortest 4-path and 8-path between pixels are the paths that have the minimum number of

pixels or the minimum distance between two pixels in an image, and they are used to measure

the distance or the similarity between the pixels. The shortest 4-path and 8-path between pixels

are often used to define the conditions or criteria for pixel grouping, segmentation, or

classification, or to identify or extract features or patterns in the image.

Here is an example of the shortest 4-path and 8-path between pixels in a grayscale image:

4-path:

[(0,0), (1,0), (2,0), (3,0), (3,1)]

8-path:

[(0