BSc CSIT (TU) Science Image Processing (BSc CSIT, CSC413) Question Paper 2079 Nepal

Digital Image Processing (DIP) System

A digital image processing system acquires, processes, stores and displays images using a sequence of well-defined steps.

Fundamental Steps

1. Image Acquisition – Capturing the image using a sensor (camera/scanner) and digitizing it (sampling + quantization).
2. Image Enhancement – Improving subjective quality (e.g. contrast stretching, histogram equalization, sharpening).
3. Image Restoration – Removing/undoing degradation using objective mathematical models (e.g. inverse, Wiener filtering).
4. Colour Image Processing – Processing in colour models (RGB, HSI, CMYK).
5. Wavelets and Multiresolution Processing – Representing images at multiple resolutions; basis of compression.
6. Compression – Reducing storage/transmission size (JPEG, run-length, Huffman).
7. Morphological Processing – Extracting shape components using dilation, erosion, opening, closing.
8. Segmentation – Partitioning an image into objects/regions (edge, threshold, region growing).
9. Representation and Description – Representing boundaries/regions and extracting features (descriptors).
10. Object Recognition – Assigning labels to objects based on descriptors.

A knowledge base guides and constrains every stage.

Components of a DIP System

• Image sensors – sensor + digitizer that convert light into a digital signal.
• Specialized image-processing hardware – ALU/front-end for fast arithmetic-logic operations.
• Computer – general-purpose machine controlling the system.
• Image-processing software – modules/library to perform specific tasks.
• Mass storage – short-term, online and archival storage for large image data.
• Image displays – monitors to view results.
• Hardcopy devices – printers, film cameras to record output.
• Network/cloud – for transmission of images.

Elements of Visual Perception

• Structure of the human eye – Light enters through the cornea, passes the pupil/lens and is focused on the retina. The retina contains two photoreceptors:
  • Cones (~6–7 million, concentrated at the fovea) – responsible for photopic (bright-light) and colour vision; high acuity.
  • Rods (~75–150 million) – give scotopic (dim-light) vision; sensitive to low light but not colour.
• Image formation in the eye – The lens changes shape (accommodation) to focus objects; image size on the retina is obtained by similar triangles.
• Brightness adaptation – The eye adapts over a huge intensity range but only discriminates a small range at once; perceived brightness is non-linear (Weber's law).
• Simultaneous contrast – Perceived brightness of a region depends on its background.
• Mach bands – The eye over/undershoots intensity at edges, enhancing perceived edges.

These perceptual limits (intensity discrimination, spatial resolution, optical illusions) directly guide how images should be sampled, quantized and enhanced.

Histogram Equalization

Histogram equalization is a contrast-enhancement technique that redistributes pixel intensities so the output histogram is approximately uniform (flat), spreading intensities across the full available range. It maps the input level $r_k$ to a new level using the cumulative distribution of the histogram:

where $L$ = number of gray levels, $MN$ = total pixels, $n_j$ = pixel count at level $j$.

Worked Example (3-bit image, $L = 8$, levels 0–7)

A representative distribution for a $64$-pixel ($MN = 64$) 3-bit image:

Steps

1. Compute the histogram (count $n_k$ for each level).
2. Compute probabilities $p_r(r_k)=n_k/MN$.
3. Compute the cumulative distribution (running sum of $p_r$).
4. Multiply by $(L-1)=7$ and round to the nearest integer to get the mapped level $s_k$.
5. Replace every pixel of level $r_k$ with $s_k$.

Result Mapping

The equalized histogram uses the full range more evenly, giving better contrast. (For any specified distribution, apply the same five steps.)

Frequency-Domain Filtering

In frequency-domain filtering, the image is transformed with the 2-D Discrete Fourier Transform (DFT), multiplied by a filter transfer function $H(u,v)$, and transformed back:

Procedure: (1) pad the image; (2) center the transform by multiplying by $(-1)^{x+y}$; (3) compute $F(u,v)$; (4) multiply by $H(u,v)$; (5) inverse DFT; (6) take the real part and crop.

Low frequencies (near the origin) carry slowly varying intensity; high frequencies carry edges and noise. A low-pass filter (LPF) keeps low frequencies (smoothing/blurring).

Let $D(u,v)=\sqrt{(u-M/2)^2+(v-N/2)^2}$ be the distance from the centre and $D_0$ the cutoff.

Ideal Low-Pass Filter (ILPF)

Sharp cutoff — passes everything inside the circle, blocks everything outside. Causes severe ringing (ripple artifacts) because its spatial counterpart is a sinc function.

Butterworth Low-Pass Filter (BLPF) of order $n$

Smooth transition controlled by order $n$. Low $n$ → gentle, no ringing; high $n$ → approaches ILPF with some ringing. A good compromise between smoothing and artifacts.

Gaussian Low-Pass Filter (GLPF)

Very smooth transition; its inverse transform is also Gaussian (always positive) so it produces no ringing at all, but gives slightly more blurring than Butterworth at the same $D_0$.

Comparison

Conclusion: Ideal gives the sharpest cutoff but worst ringing; Gaussian is artifact-free but blurs most; Butterworth offers a tunable balance and is most often used in practice.

Brightness refers to the overall lightness or darkness of an image — it is controlled by adding a constant to every pixel value, $g(x,y)=f(x,y)+b$. Increasing brightness shifts the whole histogram to the right (lighter); decreasing shifts it left (darker).

Contrast refers to the difference between the lightest and darkest regions, i.e. the spread of intensity values. It is changed by scaling, $g(x,y)=a\cdot f(x,y)$ (with $a>1$ increasing contrast). High contrast means a wide histogram spread; low contrast means values clustered together (flat, washed-out look).

Bit-Plane Slicing

An 8-bit grayscale pixel is represented by 8 bits. Bit-plane slicing decomposes the image into 8 binary images (bit planes), one per bit position, where each plane contains the value of that single bit for every pixel.

• Bit-plane 0 = least significant bit (LSB); bit-plane 7 = most significant bit (MSB).
• For pixel value $r$, the bit of plane $i$ is $\lfloor r/2^i\rfloor \bmod 2$.

Observations

• The higher-order planes (4–7) contain the visually significant data — most of the recognizable image structure.
• The lower-order planes (0–3) mainly carry fine detail and noise.

Uses

• Image compression – discarding low-order planes reduces storage with little visible loss.
• Analyzing relative importance of each bit to image content.
• Watermarking/steganography – hiding data in the LSB plane.

Example: value $131 = 10000011_2$ contributes a 1 to planes 7, 1 and 0, and 0 to the others.

High-Pass Filter (HPF)

A high-pass filter is a filter that passes high spatial frequencies (rapid intensity changes such as edges, fine detail and noise) while attenuating low frequencies (smooth, slowly varying regions). It is the complement of a low-pass filter:

In the spatial domain it uses a kernel with a positive centre and negative surround whose coefficients sum to zero, e.g. the Laplacian mask:

  0  -1   0
 -1   4  -1
  0  -1   0

Uses

• Edge detection / sharpening – emphasizes boundaries where intensity changes sharply.
• Detail enhancement – brings out fine texture and small features.
• High-boost / unsharp masking – sharpened image = original + scaled high-pass output.
• Feature extraction – pre-processing step for segmentation and recognition.

Because it removes the slowly varying background, the output of a pure HPF has a mostly dark, mean-zero appearance with bright edges.

Fourier Transform vs Discrete Cosine Transform (DCT)

The Discrete Fourier Transform (DFT) decomposes a signal/image into a sum of complex sinusoids (both sine and cosine), producing complex-valued coefficients (magnitude + phase). The Discrete Cosine Transform (DCT) expresses the data using only real cosine basis functions, producing real coefficients.

Key point: Because the DCT assumes an even (mirrored) extension, it avoids the boundary discontinuities of the DFT and gives better energy compaction, which is why JPEG uses the $8\times8$ DCT rather than the Fourier transform for compression.

Region Growing

Region growing is a region-based segmentation technique that groups pixels into larger regions starting from one or more seed points, based on a similarity (homogeneity) criterion.

Algorithm

1. Select seed pixel(s) representing each region of interest.
2. Examine the neighbours (4- or 8-connected) of the current region.
3. Append a neighbour to the region if it satisfies the similarity predicate — e.g. its intensity differs from the seed/region mean by less than a threshold $T$: $|f(x,y)-\mu_{region}| \le T$.
4. Repeat until no more pixels can be added.
5. Start a new region from an unassigned seed and continue until all pixels are labelled.

Characteristics

• Similarity may use intensity, colour, texture, etc.
• Advantages: good for noisy images where edges are hard to detect; produces connected regions; gives correct results when criteria are well chosen.
• Disadvantages: sensitive to seed choice and threshold; can be computationally expensive; may cause over/under-segmentation.

It is the conceptual opposite of region splitting and merging, which starts from the whole image and subdivides it.

Gradient Operator in Edge Detection

An edge is a location of sharp intensity change. The gradient measures the rate and direction of that change and is therefore the basis of first-derivative edge detection. For image $f(x,y)$:

• Magnitude (edge strength): $|\nabla f| = \sqrt{G_x^2 + G_y^2} \approx |G_x| + |G_y|$
• Direction (perpendicular to the edge): $\theta = \tan^{-1}\!\big(G_y/G_x\big)$

Use

• The gradient magnitude is large at edges and near zero in smooth areas, so thresholding the magnitude detects edges.
• $G_x$ and $G_y$ are computed by convolving with masks such as Sobel, Prewitt or Roberts operators.
• The gradient direction tells the edge orientation, used in non-maximum suppression (e.g. Canny).

Thus the gradient operator both locates edges (via magnitude) and orients them (via direction).

Morphological Gradient

The morphological gradient of an image $f$ with a structuring element $b$ is the difference between its dilation and its erosion:

where $\oplus$ is dilation and $\ominus$ is erosion.

• Dilation expands bright regions; erosion shrinks them. Their difference is large only where intensities change rapidly, i.e. at object boundaries.
• The result highlights edges/boundaries of objects, similar to a gradient operator but defined by set/shape operations.

Related variants

• Internal gradient: $f - (f \ominus b)$
• External gradient: $(f \oplus b) - f$

Use: edge extraction and boundary detection in binary and grayscale images.

Role of Quantization in Image Compression

Quantization maps a large (or continuous) range of input values to a smaller set of discrete output levels, intentionally discarding less important information to reduce the data needed to represent the image. It is the main source of size reduction in lossy compression.

Role / Importance

• Reduces psychovisual redundancy – removes detail the human eye cannot perceive (e.g. fine high-frequency variations).
• In transform coders like JPEG, the DCT coefficients are divided by a quantization table and rounded; many high-frequency coefficients become zero, which then compress very efficiently with run-length and entropy coding.
• It controls the compression-ratio vs quality trade-off: coarser quantization (larger step) → higher compression but lower quality; finer quantization → better quality, larger file.

Types

• Scalar quantization – each value quantized independently.
• Vector quantization – groups of values quantized together.

Note: Quantization is irreversible (lossy) — the discarded information cannot be recovered, so it is absent from lossless schemes.

Pseudo-Colour Image Processing

Pseudo-colour (false-colour) processing assigns colours to the gray (intensity) levels of a monochrome image based on a specified rule. The image carries no real colour information; colour is added artificially to make features easier to interpret by the human eye, which distinguishes far more colours than gray shades.

Principal Techniques

1. Intensity (density) slicing – The gray scale is partitioned into intervals (slices) by planes; each interval is mapped to a distinct colour. Simple but produces few colours.
2. Gray-level to colour transformation – Each pixel's intensity is passed through three independent transformation functions producing red, green and blue outputs:

The three are combined to form the colour image, giving smoother, richer colour mappings than slicing.

Uses / Importance

• Enhances visual interpretation by exploiting the eye's superior colour discrimination.
• Widely used in medical imaging (X-ray, MRI), remote sensing/satellite images, thermal/infrared imaging and scientific visualization to emphasize regions of interest.

Difference from full colour: pseudo-colour is derived from a single intensity channel, whereas true colour comes from real RGB sensor data.