Intensity Transformation and Spatial Filtering

Part 1: Intensity Transformations (Point Processing)

These techniques operate on a single pixel ( $1 \times 1$ neighborhood) at a time, mapping an input intensity $r$ to an output intensity $s$ .

Technique / Filter	Category	Mathematical Concept / Mechanism	Primary Purpose / Application
Image Negative	Linear	$s = L - 1 - r$	Reverses intensity levels. Used to enhance white/gray details embedded in large dark regions (e.g., medical X-rays).
Log Transformation	Logarithmic	$s = c \times \log (1 + r)$	Expands a narrow range of dark input values while compressing higher-level values. Used to reveal hidden details (e.g., Fourier spectrum).
Power-Law (Gamma)	Exponential	$s = c \times r^{γ}$	Lightens ( $γ < 1$ ) or darkens ( $γ > 1$ ) an image. Used for Gamma Correction to fix nonlinear monitor displays.
Contrast Stretching (Min-Max)	Piecewise-Linear	$I_{n e w} = (I - M i n) \frac{N e w M a x - N e w M i n}{M a x - M i n} + N e w M i n$	Stretches a narrow intensity range to span a wider target range. Used to normalize images with poor illumination.
Thresholding	Piecewise-Linear	$s = 1$ if $r \geq m$ , else $s = 0$	Converts grayscale to binary (black and white) based on a threshold $m$ . Used for image segmentation.
Intensity-Level Slicing	Piecewise-Linear	Highlights a specific range of intensities $[A, B]$ . Can either set all other pixels to black (binary mapping) or preserve their original values.	Highlighting specific features of interest, such as masses in medical X-rays or bodies of water in satellite imagery.
Bit-Plane Slicing	Bitwise / Piecewise	Decomposes an 8-bit image into 8 binary planes (Bit 0 to Bit 7). The Most Significant Bit (MSB, Bit 7) contains the most visual data; the LSB (Bit 0) looks like noise.	Analyzing the importance of bits for image compression and steganography (hiding data in the LSB).
Histogram Equalization	Statistical / Global	$s_{k} = (L - 1) \sum_{j = 0}^{k} p_{r} (r_{j})$ (Maps pixels using the Cumulative Distribution Function of the histogram).	Fully automatic contrast enhancement. Spreads out the most frequent intensity values to create a "flat" histogram, maximizing dynamic range.

Part 2: Smoothing Spatial Filters (Neighborhood Processing)

These filters use a moving mask (e.g., $3 \times 3$ ) to calculate a new value for the center pixel based on its surrounding neighbors. Their main goal is noise reduction and blurring.

Technique / Filter	Category	Concept	Primary Purpose / Application	Kernel Matrix (Mask) Example
Box Filter (Averaging)	Linear Spatial	All mask coefficients are equal (e.g., all $1$ s, divided by $9$ ). Replaces center pixel with the neighborhood average.	Blurring an image and reducing random noise. Drawback: Causes significant blurring of edges.	$\frac{1}{9} [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$
Weighted Average	Linear Spatial	Center mask coefficients have higher values than outer ones.	Smoothes the image while preserving slightly more detail/edges compared to a standard box filter.	$\frac{1}{16} [\begin{matrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{matrix}]$
Median Filter	Order-Statistic (Nonlinear)	Replaces the center pixel with the median value of the neighborhood.	Excellent for removing impulse noise (salt-and-pepper noise) with considerably less blurring than linear filters.	N/A (Uses Sorting/Ranking)
Max Filter	Order-Statistic (Nonlinear)	Replaces the center pixel with the maximum value of the neighborhood.	Finds the brightest points in an image; effectively removes "pepper" (dark) noise.	N/A (Uses Sorting/Ranking)
Min Filter	Order-Statistic (Nonlinear)	Replaces the center pixel with the minimum value of the neighborhood.	Finds the darkest points in an image; effectively removes "salt" (bright) noise.	N/A (Uses Sorting/Ranking)

Part 3: Sharpening Spatial Filters (Neighborhood Processing)

These filters use spatial differentiation to highlight rapid transitions in intensity, effectively sharpening edges and fine details.

Technique / Filter	Category	Concept / Math	Primary Purpose / Application	Kernel Matrix (Mask) Example
Laplacian Filter	Linear Spatial (2nd Derivative)	Isotropic mask (e.g., center $- 4$ or $- 8$ , surrounded by $1$ s). $\nabla^{2} f = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}}$	Highlights fine details and rapid intensity transitions. The result is usually added back to the original image to restore background tones.	$[\begin{matrix} 0 & 1 & 0 \\ 1 & - 4 & 1 \\ 0 & 1 & 0 \end{matrix}]$ or $[\begin{matrix} 1 & 1 & 1 \\ 1 & - 8 & 1 \\ 1 & 1 & 1 \end{matrix}]$ (for composite, add 1 or -1 depending on center)
Unsharp Masking	Image Arithmetic	$g (x, y) = f (x, y) + [f (x, y) - \bar{f} (x, y)]$	Subtracts a blurred version of the image from the original to create an edge mask, which is then added back to sharpen the image.	N/A (Subtracts a blurred image from the original)
Highboost Filtering	Image Arithmetic	$g (x, y) = f (x, y) + k \times [f (x, y) - \bar{f} (x, y)]$	Similar to Unsharp Masking, but multiplies the edge mask by a weight $k > 1$ to drastically increase the contribution of the edges.	N/A (Uses Unsharp mask logic with a multiplier $k > 1$ )
Roberts Cross-Gradient	Nonlinear Spatial (1st Derivative)	Uses $2 \times 2$ masks to compute diagonal differences.	Extracts edges by calculating the gradient magnitude. Very sensitive to noise due to its small mask size.	$G_{x} = [\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}]$ $G_{y} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]$
Sobel Operator	Nonlinear Spatial (1st Derivative)	Uses $3 \times 3$ masks to compute horizontal ( $G_{x}$ ) and vertical ( $G_{y}$ ) changes.	Extracts edges. Provides a slight smoothing effect due to the $3 \times 3$ size, making it less sensitive to noise than Roberts.	$G_{x} = [\begin{matrix} - 1 & - 2 & - 1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{matrix}]$ $G_{y} = [\begin{matrix} - 1 & 0 & 1 \\ - 2 & 0 & 2 \\ - 1 & 0 & 1 \end{matrix}]$