BSc CSIT (TU) Science Image Processing (BSc CSIT, CSC413) Question Paper 2077 Nepal

Image Enhancement in the Spatial Domain

Spatial-domain enhancement operates directly on the pixels of an image. A general transformation is written as:

where $f$ is the input image, $g$ the output image, and $T$ an operator defined over a neighbourhood of $(x,y)$. When the neighbourhood is a single pixel ($1\times1$), $T$ becomes an intensity (gray-level) transformation $s = T(r)$, where $r$ and $s$ are the input and output intensities.

Point Processing Techniques

1. Image Negative

Reverses the intensity levels, producing a photographic negative. Useful for enhancing white/grey detail embedded in dark regions (e.g. medical X-rays):

where $L$ is the number of gray levels (e.g. $L=256$ for 8-bit images).

2. Log Transformation

It maps a narrow range of low intensities into a wider range and compresses high intensities. It is used to expand dark pixel values and to display data with a large dynamic range (e.g. Fourier spectra). $c$ is a scaling constant.

3. Power-Law (Gamma) Transformation

• $\gamma < 1$ brightens the image (expands dark levels), similar to the log curve.
• $\gamma > 1$ darkens the image (expands bright levels).

Gamma transformations are used for contrast manipulation and gamma correction of display devices.

Histogram Processing

The histogram of an image plots the frequency $h(r_k) = n_k$ of each intensity level $r_k$ (or the normalized probability $p(r_k) = n_k/MN$). It describes the global tonal distribution; dark images cluster at low levels, bright images at high levels, and high-contrast images spread across the full range.

Histogram Equalization redistributes intensities so the output histogram is approximately uniform, improving contrast. The transformation is the cumulative distribution function (CDF):

Histogram Specification (Matching) maps the histogram to a desired target shape rather than a flat one, giving finer control over the result.

Summary

Spatial enhancement improves visual quality directly on pixels. Point operations (negative, log, power-law) remap intensities by a fixed function, while histogram processing uses the image's own statistics to enhance contrast.

Butterworth Filter (Frequency-Domain)

Filtering in the frequency domain multiplies the Fourier transform $F(u,v)$ of an image by a filter transfer function $H(u,v)$:

The Butterworth filter is a smooth filter whose transfer function provides a gradual transition between passed and attenuated frequencies, controlled by an order $n$. Unlike the ideal filter (sharp cut-off, causing ringing) and the Gaussian filter (no ringing), the Butterworth filter is a compromise: low ringing and a tunable sharpness via $n$.

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

Butterworth Low-Pass Filter (BLPF) — Smoothing

• Passes low frequencies and attenuates high frequencies.
• Low frequencies represent smooth, slowly varying intensity; high frequencies represent edges and noise. Removing high frequencies smooths/blurs the image and reduces noise.
• A larger order $n$ makes the filter approach the ideal LPF (more ringing); $n=1$ behaves like a Gaussian (no ringing).

Butterworth High-Pass Filter (BHPF) — Sharpening

• Passes high frequencies and attenuates low frequencies.
• Since edges and fine detail are high-frequency content, the BHPF sharpens the image by emphasizing edges and boundaries while suppressing the slowly varying background.

Comparison

Summary

The Butterworth filter offers a smoothly controllable transition (order $n$) between ideal and Gaussian behaviour: the low-pass version smooths, and the high-pass version sharpens an image.

Image Segmentation

Segmentation partitions an image into meaningful regions (objects and background) so that pixels in each region share some property. Formally the image $R$ is divided into regions $R_1,\dots,R_n$ such that:

• $\bigcup_{i=1}^{n} R_i = R$ (complete coverage)
• Each $R_i$ is connected, $R_i \cap R_j = \varnothing$ for $i \ne j$
• A predicate $P(R_i)$ is TRUE for each region and FALSE for the union of any two adjacent regions.

Segmentation approaches rely on two basic properties of intensity: discontinuity (edge-based) and similarity (region-based, thresholding).

1. Edge-Based Segmentation (Discontinuity)

Detects boundaries where intensity changes abruptly.

• Gradient (first-derivative) operators detect edges as points of maximum gradient magnitude:

Implemented with Sobel, Prewitt, Roberts masks.

• Laplacian (second-derivative) detects edges at zero-crossings; the Laplacian of Gaussian (LoG / Marr-Hildreth) first smooths to reduce noise sensitivity.
• Canny edge detector is the optimal multi-stage method: Gaussian smoothing → gradient → non-maximum suppression → hysteresis (double) thresholding.
• Detected edge pixels are then linked (local processing, Hough transform) into continuous boundaries.

Pros: good for sharp boundaries. Cons: sensitive to noise; edges may be broken/incomplete.

2. Region-Based Segmentation (Similarity)

Groups pixels that satisfy a similarity predicate.

• Region Growing: start from seed pixels and append neighbouring pixels whose properties (intensity, colour, texture) satisfy $P$, until no more pixels can be added.
• Region Splitting and Merging: start with the whole image; recursively split any region where $P$ is FALSE (e.g. quadtree), then merge adjacent regions whose union satisfies $P$.

Pros: produces connected, closed regions; more robust to noise than pure edge detection. Cons: sensitive to seed selection and predicate; can be computationally expensive.

Comparison

Summary

Edge-based methods locate object boundaries via intensity discontinuities, while region-based methods group similar pixels into homogeneous regions; the two are often combined for reliable segmentation.

Digital image: A digital image is a two-dimensional function $f(x,y)$ where $x,y$ are spatial coordinates and the amplitude $f$ at any point is the intensity (gray level), with both the coordinates and amplitudes being finite, discrete quantities. It is obtained by sampling (discretizing coordinates) and quantization (discretizing intensity), and is represented as an $M \times N$ matrix of numbers.

Pixel: A pixel (picture element) is a single element of that matrix — the smallest addressable unit of a digital image. Each pixel has a location $(x,y)$ and an intensity value (e.g. 0–255 for an 8-bit grayscale image, or an (R,G,B) triple for colour).

Gamma correction is a nonlinear intensity transformation that follows the power-law:

where $r$ is the input intensity (normalized to $[0,1]$), $s$ the output, $c$ a constant, and $\gamma$ the gamma exponent.

It compensates for the nonlinear response of display devices (CRT/LCD monitors), whose output luminance is approximately a power function of the input voltage. Without correction images appear too dark or too bright.

• $\gamma < 1$ : brightens the image / expands dark tones.
• $\gamma > 1$ : darkens the image / expands bright tones.
• $\gamma = 1$ : linear (no change).

Applying the inverse exponent corrects the display so the perceived brightness matches the intended values; it is also used for general contrast enhancement.

Histogram Equalization

Histogram equalization is a contrast-enhancement technique that redistributes pixel intensities so that the output histogram is approximately uniform, spreading values across the full available range.

Procedure:

1. Compute the histogram: count $n_k$ pixels at each level $r_k$ for $k = 0,1,\dots,L-1$.
2. Compute the normalized probability (PDF): $p(r_k) = n_k / MN$, where $MN$ is the total number of pixels.
3. Compute the cumulative distribution function (CDF): $\;\text{cdf}(r_k) = \sum_{j=0}^{k} p(r_j)$.
4. Map each level using:

5. Round $s_k$ to the nearest integer and replace each pixel's value accordingly.

Result: intensities that were clustered in a narrow band are stretched across the range, increasing global contrast. It is automatic (no parameters) but applies globally and can over-enhance noise; adaptive (local) histogram equalization addresses this.

Image Sharpening

Image sharpening enhances fine detail, edges, and transitions in intensity, making an image appear crisper. It is the opposite of smoothing: smoothing averages (integration), whereas sharpening uses differentiation to emphasize regions of rapid intensity change.

Filters Used

Spatial domain (high-pass / derivative filters):

• Laplacian (second derivative):

The sharpened image is $g = f - \nabla^2 f$ (or $f + \nabla^2 f$ depending on mask sign). A common $3\times3$ mask is:

 0  -1   0
-1   5  -1
 0  -1   0

• Gradient (first derivative): Sobel / Prewitt operators emphasize edges via $|\nabla f|$.
• Unsharp masking & High-boost filtering: subtract a blurred (low-pass) version from the original to obtain a detail mask, then add it back: $g = f + k(f - \bar f)$.

Frequency domain (high-pass filters): Ideal, Butterworth, and Gaussian high-pass filters pass high frequencies (edges) and attenuate low frequencies, sharpening the image.

In all cases, high-frequency / high-pass operators are used because edges correspond to high spatial frequencies.

Haar Transform

The Haar transform is one of the simplest orthogonal (unitary) image transforms, derived from the Haar functions — the oldest known wavelet basis. It decomposes a signal/image into a low-frequency average (approximation) component and high-frequency difference (detail) components, making it a basic discrete wavelet transform.

Transform form: For an $N \times N$ image $F$, the transform is

where $H$ is the $N\times N$ orthogonal Haar matrix ($N$ a power of 2) whose rows are sampled Haar functions, and $H^{-1} = H^{T}$.

Basic $2\times2$ Haar matrix:

The first row computes the sum/average (low-pass) and the second the difference (high-pass).

Properties / Uses:

• Real, orthogonal, and very fast to compute (only additions/subtractions, no multiplications by irrational numbers beyond scaling).
• Provides both spatial and frequency (multi-resolution) localization, unlike the Fourier transform.
• Used in image compression, edge detection, feature extraction, and as the simplest example of wavelet-based processing.

Thresholding in Segmentation

Thresholding is a region-based segmentation technique that separates objects from the background based on intensity. A threshold value $T$ partitions pixels into classes:

Pixels above the threshold become foreground (object), the rest background, producing a binary image.

Types:

• Global thresholding: a single $T$ for the whole image (works when object and background have distinct, bimodal intensity peaks).
• Otsu's method: automatically selects $T$ by maximizing between-class variance.
• Adaptive / local thresholding: $T$ varies across the image to handle non-uniform illumination.
• Multiple thresholding: several thresholds separate more than two classes.

It is simple and fast but sensitive to noise and uneven lighting.

Opening vs Closing (Morphology)

Both are compound morphological operations built from erosion ($\ominus$) and dilation ($\oplus$) using a structuring element $B$ on a binary image $A$.

Opening — erosion followed by dilation:

Closing — dilation followed by erosion:

Both are idempotent ($A \circ B \circ B = A \circ B$) and preserve overall object size, unlike plain erosion/dilation. Opening tends to remove foreground noise; closing tends to remove background holes.

Run-Length Encoding (RLE)

Run-Length Encoding is a simple lossless image-compression technique that exploits spatial (interpixel) redundancy — long runs of identical pixel values. Instead of storing each pixel, it stores each run as a pair: (value, run-length).

Example (a row of pixels):

Original:  W W W W W B B W W W W W W W   (14 pixels)
RLE:       (W,5) (B,2) (W,7)            (3 pairs)

For binary images, only the run lengths need be stored (the value alternates between 0 and 1). It is the basis of compression in formats such as BMP, TIFF, and fax (CCITT) standards, and is used as a stage in JPEG (encoding runs of zero AC coefficients).

Pros: simple, fast, lossless, very effective on images with large uniform areas (binary/graphics). Cons: poor on noisy or highly detailed natural images, where it can even increase size.

Walsh Transform

The Walsh transform is a non-sinusoidal, orthogonal image transform whose basis functions are Walsh functions — square waves taking only the values $+1$ and $-1$. It is closely related to the Hadamard transform (the Walsh-Hadamard transform), differing only in the ordering of the basis functions (Walsh ordering = sequency order, by number of sign changes).

1-D Walsh transform of a sequence $f(x)$ of length $N = 2^n$:

where $b_i(x)$ is the $i$-th bit of $x$. The kernel values are $\pm1$.

Properties / Uses:

• Real, symmetric, orthogonal; the inverse has the same form (separable and easily extended to 2-D as $W = H F H$).
• Computed using only additions and subtractions (no multiplications), so it is much faster than the Fourier transform.
• Basis functions are ordered by sequency (analogous to frequency).
• Applications: image compression, feature extraction, and signal/image coding where computational simplicity matters.