Unit 1: Fundamental of Image Processing [8 Hours]

Digital Image Processing (DIP) deals with representing images in digital form and processing them using computer-based techniques. This unit builds the foundation of DIP by explaining pixels, image representation, human visual perception, sampling and quantization, color, image file formats, and basic image operations. It also introduces perceptual properties such as brightness, contrast, hue, saturation, and the Mach band effect, which explain how both computers and humans interpret visual information.

Tested Topics

1. Basic relationship between pixels — 1x (2023)
2. Image operations — 1x (2025)
3. Mach band effect — 1x (2023)
4. Sampling and Quantization — 1x (2023)

1.1 Image Representation

Definition

An image can be represented mathematically as a two-dimensional function f(x, y), where x and y specify the spatial position and f(x, y) gives the intensity or color value at that position.

A digital image is a rectangular grid of tiny picture elements called pixels. Each pixel stores a numerical value describing how bright or what color that small part of the image should appear.

Detailed Explanation

In mathematical form:

f(x, y) = intensity value at coordinate (x, y)

An ordinary continuous image may contain continuously varying spatial positions and intensity values. A computer cannot directly store infinitely many values. Therefore, the image is converted into a finite grid of discrete positions and discrete intensity or color values.

Pixels and Coordinates

A pixel, short for picture element, is the smallest individually addressable element of a digital image. Every pixel has:

• Spatial coordinates: identify where the pixel is located.
• Intensity or color value: describes the visual value stored at that location.

For example, a small grayscale image may be represented as:

Here, a low value represents a darker pixel and a high value represents a brighter pixel if the image uses the common 8-bit grayscale range from 0 to 255.

Major Forms of Digital Images

• Binary image: each pixel has only two possible values, commonly 0 and 1, representing two states such as black and white.
• Grayscale image: each pixel represents a level of brightness. An 8-bit grayscale image normally provides 256 possible intensity levels.
• Color image: each pixel contains multiple color components. For example, a Red, Green, Blue (RGB) image stores separate red, green, and blue values.

Image as a Matrix

A digital image can also be treated as a matrix:

f(x, y) → matrix of pixel values

If an image has M rows and N columns, it contains:

Total pixels = M × N

For example, a 1920 × 1080 image contains:

1920 × 1080 = 2,073,600 pixels

1.2 Basic Relationship Between Pixels

Definition

Basic relationships between pixels describe how pixels are positioned, connected, and measured relative to other pixels in a digital image. Important relationships include neighborhood, adjacency, connectivity, regions, boundaries, and distance.

A pixel does not exist independently from the pixels around it. Processing techniques often need to know which pixels touch each other, which ones belong to the same object, and how far one pixel is from another.

Detailed Explanation

Consider a pixel p located at coordinates (x, y). The pixels surrounding p can be classified according to their positions.

4-Neighbors of a Pixel

The 4-neighbors of p are the pixels immediately above, below, left, and right of it.

N₄(p) = {(x − 1, y), (x + 1, y), (x, y − 1), (x, y + 1)}

They can be visualized as:

Diagonal Neighbors

The diagonal neighbors lie diagonally around p:

N_D(p) = {(x−1, y−1), (x+1, y−1), (x−1, y+1), (x+1, y+1)}

8-Neighbors

The 8-neighbors include both the four direct neighbors and four diagonal neighbors.

N₈(p) = N₄(p) ∪ N_D(p)

Adjacency

Two pixels are adjacent when they are considered neighbors and their intensity values satisfy a specified condition. Let V represent the set of pixel values being considered, such as V = {1} in a binary image.

• 4-adjacency: pixels p and q are adjacent if q belongs to N₄(p) and both pixel values belong to V.
• 8-adjacency: pixels p and q are adjacent if q belongs to N₈(p) and their values belong to V.
• Mixed adjacency: also called m-adjacency. It allows diagonal adjacency only when the two pixels do not share a common 4-neighbor whose value belongs to V. This avoids ambiguous diagonal connections.

Connectivity and Path

A path from pixel p to pixel q is a sequence of pixels:

p = p₀ → p₁ → p₂ → ... → p_n = q

where every consecutive pair is adjacent according to the selected adjacency rule.

Two pixels are connected if a valid path exists between them. Connectivity allows an image-processing system to identify separate objects or components in an image.

Region and Boundary

A region is a connected set of pixels. Pixels belonging to one object can therefore form a region.

The boundary of a region consists of those pixels in the region that have at least one neighboring pixel outside the region.

For example, in a binary image containing a white circle on a black background:

connected white pixels → circle region → outer white pixels touching black pixels → boundary

Distance Measures

Distance measures describe how far apart two pixels are. Let:

p = (x, y) and q = (s, t)

Euclidean distance:

D_E(p, q) = √[(x − s)² + (y − t)²]

City-block distance:

D₄(p, q) = |x − s| + |y − t|

Chessboard distance:

D₈(p, q) = max(|x − s|, |y − t|)

For p = (2, 3) and q = (5, 7):

• Euclidean distance = √[(2−5)² + (3−7)²] = √25 = 5
• City-block distance = 3 + 4 = 7
• Chessboard distance = max(3, 4) = 4

1.3 Elements of a Digital Image Processing System

Definition

A Digital Image Processing system is a collection of hardware and software components that acquire an image, convert it into digital form, process it, store it, display it, and communicate or print the result.

The complete system is more than an image-processing program. A camera or scanner first captures the scene, hardware converts it into data, software processes that data, and other devices store or present the output.

Detailed Explanation

A typical DIP system contains the following elements.

1. Image Sensing and Acquisition

An image sensor receives energy reflected or emitted from a scene and converts it into an electrical signal. A digitizer then converts the signal into numerical pixel values.

Scene → optical system → sensor → electrical signal → digitization → digital image

Examples include digital cameras, scanners, medical imaging sensors, and satellite sensors.

2. Specialized Image Processing Hardware

Specialized hardware performs operations that require high speed. It may handle image acquisition, filtering, compression, graphics processing, or other low-level numerical operations.

3. Computer

A general-purpose computer controls the overall system and executes processing tasks. Its processor and memory determine how efficiently large images and complex algorithms can be handled.

4. Image Processing Software

Image processing software contains algorithms used for operations such as enhancement, restoration, compression, analysis, segmentation, and visualization.

5. Mass Storage

Digital images can require significant storage capacity. Storage is commonly needed at three levels:

• Short-term storage: temporary memory used while an image is being processed.
• Online storage: disks or solid-state storage holding actively used images.
• Archival storage: long-term storage for large image collections.

6. Image Display

Displays convert digital pixel values into visible output. Monitors are used to examine the original image, intermediate processing results, and final output.

7. Hardcopy Devices

Hardcopy devices create permanent physical output. Examples include printers and photographic output systems.

8. Networking

Networking allows digital images to be transmitted between systems. This is essential when acquisition, processing, storage, and interpretation occur at different locations.

Typical Data Flow

Image acquisition → digitization → computer processing → storage → display or hardcopy → transmission when required

1.4 Elements of Visual Perception and Simple Image Formation Model

Definition

Visual perception is the process by which the human visual system receives light and interprets it as shapes, brightness, colors, edges, and objects. A simple image formation model describes the observed image intensity as the interaction between illumination falling on an object and the amount of that illumination reflected by the object.

An image does not depend only on the object being viewed. It also depends on the light illuminating the object and on how much light the object's surface reflects toward the observer or imaging sensor.

Detailed Explanation

Human Visual System

The human eye receives visible light and focuses it onto the retina. Important structures include:

• Cornea: the transparent front surface that begins focusing incoming light.
• Iris: controls the size of the pupil.
• Pupil: opening through which light enters the eye.
• Lens: changes shape to focus light on the retina.
• Retina: light-sensitive surface where the optical image is formed.
• Optic nerve: carries visual signals from the retina toward the brain.

Rods and Cones

The retina contains two important types of photoreceptor cells:

Brightness Adaptation and Discrimination

The human eye can operate across a very wide range of illumination levels, but it cannot distinguish all of those levels simultaneously. It adapts to the general brightness of the current environment. This ability is known as brightness adaptation.

Brightness discrimination is the ability to notice a small difference between two nearby intensity levels. The visible difference depends partly on the surrounding brightness, which is why identical pixel values can appear different when placed on different backgrounds.

Simple Image Formation Model

A common image formation model is:

f(x, y) = i(x, y) × r(x, y)

where:

• f(x, y): observed image intensity at position (x, y).
• i(x, y): illumination falling on the scene.
• r(x, y): reflectance of the object at that position.

Illumination is non-negative:

0 < i(x, y) < ∞

Reflectance represents the fraction of incident illumination reflected by the surface:

0 ≤ r(x, y) ≤ 1

A value of r close to 0 represents a surface that reflects little light, while a value close to 1 represents a highly reflective surface.

Worked Example

Suppose the illumination at a point is 200 units and the surface reflectance is 0.4.

f(x, y) = i(x, y) × r(x, y)

f(x, y) = 200 × 0.4 = 80

The resulting image intensity at that position is therefore 80 units under the model.

1.5 Sampling and Quantization

Definition

Sampling converts the spatial coordinates of a continuous image into discrete positions, while quantization converts continuous intensity or color values into a finite number of discrete levels.

Sampling answers, “How many locations of the image will be recorded?” Quantization answers, “How many possible values can each recorded location have?” Both are necessary to represent an image digitally.

Detailed Explanation

This topic builds directly on 1.1 Image Representation. A continuous image must be made discrete in two ways before a computer can store it.

Continuous image → sampling of coordinates → quantization of values → digital image

Sampling

Sampling divides the continuous x-y image plane into a finite grid.

If an image is sampled into M rows and N columns, the resulting image contains:

Number of pixels = M × N

Increasing the number of samples normally increases spatial resolution, meaning that finer spatial details can be represented.

Effect of Sampling

• High sampling rate: more pixels represent the scene, allowing finer detail.
• Low sampling rate: fewer pixels represent the scene, so small details may disappear or become distorted.

If the sampling rate is too low for the spatial detail present in an image, an artifact called aliasing can occur. Fine patterns may then appear as false patterns or jagged structures.

Quantization

Quantization assigns each sampled point to one of a finite number of permitted intensity or color levels.

If each pixel uses k bits, the number of available levels is:

L = 2^k

Therefore:

• 1 bit → 2 levels
• 2 bits → 4 levels
• 4 bits → 16 levels
• 8 bits → 256 levels
• 9 bits → 512 levels

Increasing the number of quantization levels improves intensity resolution because smaller differences in brightness or color can be represented.

Effect of Quantization

• More levels: smoother intensity transitions and finer tonal representation.
• Fewer levels: visible bands or abrupt changes may appear in areas that should change gradually.

Image Storage Requirement

For an M × N image having L possible values per pixel:

k = log₂L

where k is the number of bits required for one pixel.

The total uncompressed image data size is:

B = M × N × k bits

where:

• M: number of rows.
• N: number of columns.
• k: bits per pixel.
• B: total number of bits.

Worked Example: 2023 Question

An image is represented by 512 possible colors and has dimensions 1028 × 1028. Assuming each pixel directly stores one of the 512 possible values:

Step 1: Find bits per pixel.

512 = 2⁹

Therefore:

k = 9 bits per pixel

Step 2: Find total number of pixels.

1028 × 1028 = 1,056,784 pixels

Step 3: Find total storage in bits.

B = 1,056,784 × 9 = 9,511,056 bits

Step 4: Convert to bytes.

9,511,056 ÷ 8 = 1,188,882 bytes

This is approximately 1.13 mebibytes of raw pixel data when 1 mebibyte = 1,048,576 bytes. File headers, metadata, padding, and compression are not included.

1.6 Color Fundamentals and Models

Definition

Color is a visual sensation produced when light of different spectral compositions stimulates the human visual system. A color model is a mathematical system for representing colors as numerical components.

A computer cannot store “red” or “blue” as human sensations. It represents a color using numbers, such as the amounts of red, green, and blue required to reproduce it.

Detailed Explanation

Fundamentals of Color

Visible color is associated with the portion of electromagnetic radiation that the human eye can perceive. The color perceived from an object depends on the spectral properties of the illumination, the object's interaction with that light, and the human visual system.

Three useful perceptual properties are:

• Hue: identifies the dominant perceived color family, such as red, green, or blue.
• Saturation: describes the purity or vividness of a color.
• Brightness or intensity: describes the perceived amount of light.

Additive and Subtractive Color

Additive color creates colors by combining emitted light. The primary components are red, green, and blue. Combining all three at high intensity produces white.

Subtractive color describes systems in which pigments or inks absorb portions of incident light. Printing commonly uses cyan, magenta, yellow, and black components.

Red, Green, Blue Color Model

The Red, Green, Blue (RGB) color model represents each color using red, green, and blue components. It is widely used in cameras, monitors, phones, and computer graphics.

In an 8-bit-per-channel representation, each component commonly ranges from 0 to 255.

• (255, 0, 0) → red
• (0, 255, 0) → green
• (0, 0, 255) → blue
• (255, 255, 255) → white
• (0, 0, 0) → black

Cyan, Magenta, Yellow and Black Color Model

The Cyan, Magenta, Yellow, Black (CMYK) color model is mainly used in printing. Cyan, magenta, and yellow inks absorb different components of visible light. A separate black component is used because practical printing benefits from a dedicated dark ink for strong blacks, fine detail, and economical reproduction.

For normalized ideal RGB and Cyan, Magenta, Yellow (CMY) values:

C = 1 − R

M = 1 − G

Y = 1 − B

Hue, Saturation, Intensity Color Model

The Hue, Saturation, Intensity (HSI) color model describes color in terms that are closer to human perception.

• Hue: basic color type.
• Saturation: purity of the color.
• Intensity: overall light level.

Separating intensity from chromatic information can be useful in image analysis because brightness may be processed independently from color characteristics.

Hue, Saturation, Value Color Model

The Hue, Saturation, Value (HSV) color model also separates the color type and purity from a brightness-like component called value. It is commonly used in color selection interfaces and various computer-vision operations.

Luminance-Chrominance Models

Some color systems separate brightness-related information from color-difference information. For example, the YCbCr representation uses a luminance-related component Y and two chrominance components. Such separation is widely useful in digital image and video representation.

1.7 File Formats

Definition

An image file format defines how digital image data and related information are organized and stored in a file.

Two files may display the same picture while storing it differently. One format may prioritize small size, another exact reproduction, another transparency, and another animation or professional editing.

Detailed Explanation

An image format may specify:

• How pixel values are arranged.
• How colors are represented.
• Whether compression is used.
• Whether transparency is supported.
• What metadata can be stored.
• Whether multiple frames or images can exist in one file.

Lossless and Lossy Compression

Lossless compression reduces file size while allowing the original encoded image data to be reconstructed exactly.

Lossy compression reduces file size by permanently discarding some information considered less important for the intended reproduction.

Common Image File Formats

Choosing a File Format

The appropriate format depends on the image and the required output:

• Photographic image with small delivery size → JPEG or a modern compressed format such as WebP.
• Graphic requiring transparency or exact preservation → PNG.
• Simple frame-based web animation → GIF.
• High-quality publishing or archival workflow → TIFF when its features fit the workflow.

1.8 Image Operations

Definition

Image operations are mathematical or logical transformations applied to one or more images to modify pixel values, spatial positions, appearance, or image information.

An operation may change every pixel independently, use nearby pixels to calculate a new value, combine two images, or move pixels to new locations.

Detailed Explanation

An image operation normally accepts one or more input images and produces an output image:

Input image or images → processing operation → output image

Important classes of image operations include point operations, arithmetic operations, logical operations, neighborhood operations, and geometric operations.

1. Point Operations

In a point operation, each output pixel depends only on the corresponding input pixel.

g(x, y) = T[f(x, y)]

where T is a transformation applied independently to each pixel.

Examples include:

• Brightness adjustment.
• Contrast modification.
• Thresholding.
• Image negative transformation.

For brightness addition:

g(x, y) = f(x, y) + c

where c is a constant.

2. Arithmetic Operations

Arithmetic operations mathematically combine pixel values.

• Addition: combines information from images or increases pixel values.
• Subtraction: reveals differences between images.
• Multiplication: can scale intensity values or apply masks.
• Division: may be used for ratio-based processing and normalization.

For two images f and h:

g(x, y) = f(x, y) − h(x, y)

Image subtraction is useful in change detection. If two images of the same scene are aligned, unchanged areas tend to produce small differences while changed areas produce larger values.

3. Logical Operations

Logical operations manipulate pixels according to bitwise or binary logic. Common operations include:

• AND: retains information where selected conditions are true in both inputs.
• OR: combines regions satisfying either input condition.
• Exclusive OR (XOR): highlights differing logical states.
• NOT: complements binary values.

These operations are particularly useful with binary images and masks.

4. Neighborhood Operations

In a neighborhood operation, an output pixel is determined from the input pixel together with surrounding pixels.

A small matrix called a kernel, mask, or filter is moved across the image.

Typical uses include:

• Smoothing noise.
• Sharpening details.
• Detecting edges.

The neighborhood relationships described in 1.2 Basic Relationship Between Pixels provide the spatial basis for these operations.

5. Geometric Operations

Geometric operations modify the spatial positions of pixels.

Common operations are:

• Translation: moves an image horizontally or vertically.
• Rotation: turns an image through an angle.
• Scaling: increases or decreases image dimensions.
• Reflection: flips the image across an axis.
• Shearing: slants the image geometry.

Geometric transformations frequently require interpolation because transformed coordinates may fall between existing pixel positions. Interpolation estimates the new pixel value from nearby known values.

6. Operations Based on Number of Input Images

Worked Walkthrough

Suppose a grayscale pixel has value 100 and a brightness operation adds 40:

g = f + c = 100 + 40 = 140

The transformed pixel becomes brighter because its intensity has increased from 100 to 140.

If another pixel already has value 240, adding 40 mathematically gives 280. In an 8-bit image restricted to 0–255, an implementation using clipping would store the maximum permitted value:

min(255, 240 + 40) = 255

1.9 Brightness

Definition

Brightness is the visual attribute associated with how light or dark an image or image region appears.

When pixel intensity values are raised, the image generally appears brighter. When they are lowered, it generally appears darker, subject to the image representation and display conditions.

Detailed Explanation

Brightness is related to intensity but should be understood as a perceptual property. The numerical intensity stored in an image contributes to perceived brightness, while human perception and surrounding illumination also affect the final visual impression.

Brightness Transformation

A basic digital brightness adjustment is:

g(x, y) = f(x, y) + c

where:

• f(x, y): original pixel value.
• g(x, y): transformed pixel value.
• c: brightness adjustment constant.

If c is positive, pixel values increase. If c is negative, they decrease.

For an 8-bit image:

f = 80, c = 30 → g = 110

When a result exceeds the permitted range, it is commonly clipped to the allowed maximum or minimum.

Brightness and Illumination

Brightness also connects to 1.4 Visual Perception and Simple Image Formation. Increasing illumination can make the recorded or perceived image brighter even when the object's reflectance remains unchanged.

1.10 Contrast

Definition

Contrast is the degree of difference between the brightness or intensity levels present in an image.

An image has stronger contrast when dark regions and bright regions are clearly separated. When most values occupy a narrow intensity range, the image often appears flat or washed out.

Detailed Explanation

Contrast determines how easily differences between image regions can be distinguished.

For a basic intensity range:

Intensity range = maximum intensity − minimum intensity

A wider useful spread of pixel values can provide stronger tonal separation, although perceived contrast also depends on spatial context.

Contrast Scaling

A simple linear operation can be written as:

g(x, y) = a × f(x, y)

where a is a scaling constant.

• If a > 1, differences from low to high values can be increased, subject to clipping and the chosen transformation.
• If 0 < a < 1, the numerical intensity range is compressed.

Brightness vs. Contrast

1.11 Hue

Definition

Hue is the color attribute that distinguishes basic color families such as red, green, yellow, and blue.

Hue answers the everyday question, “What color is it?” It describes the type of color rather than how bright or how vivid that color appears.

Detailed Explanation

Hue builds on 1.6 Color Fundamentals and Models. In representations such as HSI and HSV, hue is stored separately from saturation and the brightness-related component.

For example:

• A dark red and a bright red can share a similar hue.
• A pale red and a vivid red can also share a similar hue while having different saturation.

This separation is useful because processing can select or modify a color family without treating brightness as the same property.

1.12 Saturation

Definition

Saturation describes the purity, vividness, or colorfulness of a hue compared with a less colorful or more neutral appearance.

A highly saturated red looks strong and vivid. Reducing saturation makes the color appear increasingly dull or grayish without necessarily changing its basic hue.

Detailed Explanation

Saturation builds on 1.11 Hue. Hue identifies the basic color, while saturation indicates how strongly that color is expressed.

Consider two red objects:

• A vivid red traffic sign has relatively high saturation.
• A pale, grayish red surface has lower saturation.

Hue, Saturation, and Brightness

1.13 Mach Band Effect

Definition

The Mach band effect is a visual illusion in which the human eye perceives exaggerated bright and dark bands near the boundary between regions of different brightness, even though those extra bands are not physically present in the image.

When two areas with different intensity levels meet, the boundary can appear stronger than the actual intensity values would suggest. A narrow bright strip may seem to appear on the brighter side and a narrow dark strip on the darker side.

Detailed Explanation

The Mach band effect demonstrates that human vision does not perceive every intensity value independently. The appearance of one region is influenced by neighboring regions.

How the Effect Appears

Consider several adjacent vertical regions arranged from dark to bright, with each region itself having a constant intensity.

dark region → medium-dark region → medium region → bright region

The physical intensity within each individual region may remain constant. However, near the boundaries:

• The side next to a darker region may appear slightly brighter.
• The side next to a brighter region may appear slightly darker.

The observer therefore perceives narrow bands that do not correspond to actual extra intensity bands in the image.

Relation to Edge Perception

The visual system is highly responsive to local changes in intensity. This enhances the perception of boundaries and helps separate neighboring surfaces. The Mach band effect is one demonstration of this local contrast behavior.

It also connects to 1.10 Contrast: perceived contrast near a boundary can be greater than the numerical contrast suggested by examining each region independently.

Exam Questions

1. What are different image operations? Explain. (2025)
2. Explain sampling and quantization with reference to digital image processing, further also calculate the size of image which is represented by 512 colors with x and y coordinates given as 1028*1028. (2023)
3. Write short notes on: b) Mach Band effect. (2023)
4. Define basic relationship between pixels. (2023)

Whole Chapter Summary

Image Representation

A digital image is a two-dimensional array of pixels represented by f(x, y). Coordinates identify pixel positions, while stored values represent intensity or color.

Basic Relationship Between Pixels

Pixels are related through neighborhood, adjacency, connectivity, regions, boundaries, and distance. The main neighborhood systems are 4-neighborhood, diagonal neighborhood, and 8-neighborhood.

Elements of a Digital Image Processing System

A complete DIP system includes image acquisition, processing hardware, computer, software, storage, displays, hardcopy devices, and networking. Together they convert a scene into useful digital output.

Elements of Visual Perception and Simple Image Formation

Human perception depends on the eye, retina, rods, cones, and adaptation to light. A basic image can be modeled by f(x, y) = i(x, y)r(x, y), combining illumination and surface reflectance.

Sampling and Quantization

Sampling discretizes spatial positions and determines the pixel grid. Quantization discretizes intensity or color values and determines the number of permitted levels.

Color Fundamentals and Models

Color is represented numerically using models such as RGB, CMYK, HSI, HSV, and YCbCr. Different models organize color information for display, printing, perception-oriented processing, or digital representation.

File Formats

Image formats determine how pixel data and associated information are stored. JPEG, PNG, GIF, TIFF, BMP, and WebP differ in compression, transparency, animation, quality, and intended use.

Image Operations

Image operations transform pixel values or positions. Major categories include point, arithmetic, logical, neighborhood, and geometric operations.

Brightness

Brightness describes perceived lightness or darkness. A basic digital brightness adjustment increases or decreases pixel intensity values.

Contrast

Contrast describes the difference among brightness or intensity levels. Stronger useful differences make regions and details easier to distinguish.

Hue

Hue identifies the basic color family, such as red, green, or blue. It is represented separately from brightness and saturation in several perceptual color models.

Saturation

Saturation describes how vivid or pure a color appears. High saturation produces strong color, while low saturation moves the appearance toward neutral gray.

Mach Band Effect

Mach bands are perceived bright and dark bands near intensity boundaries that are not necessarily present in the actual image values. They demonstrate the contextual nature of human brightness and edge perception.

Key Formulas and Points Sheet

Last-Minute Revision Sheet