Basics for Image Processing

Kernels

A kernel (filter mask) is a fixed-size array of numerical coefficients with an anchor point, typically at the center. The kernel is convolved with the image to produce filtered output.

align:

center

height:

300

width:

450

Gaussian Pyramids

A Gaussian pyramid is built by repeatedly blurring (Gaussian average) and downsampling an image. Each level contains a local average corresponding to a pixel neighborhood at the level below. Used extensively in texture synthesis and multi-scale analysis.

Laplacian Pyramids

A Laplacian pyramid stores the difference between successive Gaussian pyramid levels rather than the blurred images themselves. Only the smallest (coarsest) level is stored directly — all higher levels are difference images that enable reconstruction of the full-resolution image. Used in image compression and blending (e.g., Laplacian pyramid blending for seamless composites).

Sobel Operator

The Sobel operator computes image gradients by convolving with directional derivative kernels. It is commonly used in edge detection.

\[\min _ { s ^ { x } ,s ^ { y } ,\alpha } \sum _ { i = 1} ^ { k } E \left( \alpha _ { i } \mathbf { s } _ { \mathbf { i } } ^ { \mathbf { X } } + \left( 1- \alpha _ { i } \right) \mathbf { s } _ { \dot { i } } ^ { y } \right)\]