Chapter 3: Image Processing

The linear point operator $g = \alpha f + \beta$ is a scalar affine map applied independently to each pixel. In vector terms, if the image is a vector $\mathbf{f} \in \mathbb{R}^{HW}$, then $\mathbf{g} = \alpha \mathbf{f} + \beta \mathbf{1}$ is an element-wise affine transform that shifts the mean by $\beta$ and scales the variance by $\alpha^2$. Contrast stretching is the special case that maps $[f_{\min}, f_{\max}]$ onto $[0, 255]$ via $\alpha = 255 / (f_{\max} - f_{\min})$, which is the unique affine map satisfying the two boundary conditions.

Gamma correction applies the power-law nonlinearity $g = 255 (f/255)^\gamma$, which is a monotonic concave function for $\gamma < 1$ and convex for $\gamma > 1$. The sRGB standard uses a piecewise function combining a linear segment near zero with a $\gamma \approx 2.2$ power law. In information-theoretic terms, gamma encoding allocates more code values to dark tones where human contrast sensitivity $dL/dI$ is highest, effectively performing a form of perceptual quantization that minimizes visible banding for a fixed bit depth.

The histogram $H(i)$ estimates the probability mass function $p(i) = H(i)/N$ of pixel intensities. Histogram equalization seeks a monotonic mapping $T$ such that the output CDF is approximately uniform: $T(i) = \lfloor (L-1) \cdot \text{CDF}(i) \rfloor$ where $\text{CDF}(i) = \sum_{j=0}^{i} p(j)$. This is an application of the probability integral transform — for continuous random variables, if $X \sim F$, then $F(X) \sim \text{Uniform}(0,1)$. The discrete version only approximates uniformity because the CDF is a step function.