Linear filters and convolution

Multiply each weight by its sample

Begin with a small shape: black cells store 0 and white cells store 1. Take the 3 × 3 square around the selected cell, add its nine values and divide by 9. The middle panel exposes every contribution. Four white cells out of nine, for example, produce 4/9, a gray result.

A kernel is another small table, containing weights. The mean has nine weights of 1 and divisor 9. The Gaussian approximation uses 1–2–1 / 2–4–2 / 1–2–1 and divisor 16, favoring cells near the center. Try grayscale, then RGB. Color uses the same computation separately for three channels; the channel control reveals one of them.

Linearity and array boundaries

Sobel uses negative and positive weights: constant values cancel while transitions remain. This measures change rather than a new brightness. Negative results are retained and use a separate display scale. The edges chapter develops both gradient directions and Canny.

In additional settings, correlation applies weights as drawn; convolution first rotates the kernel by 180°. A symmetric mean is unchanged; the asymmetric upper kernel reveals the difference. The displayed calculation already uses the selected convention.

For a fixed linear operator, filtering a sum equals summing filtered inputs and scale passes through. Clipping, rounding and data-dependent weights can break this. Intermediate negative values are therefore preserved.

Zero, repeat and reflection are different assumptions about scene continuation. Our reflection does not duplicate the edge cell. Periodic extension connects opposite borders and appears in Fourier methods. Comparisons need matching boundary rules.

Same filter, fewer computations

Box filtering gives equal weights; Gaussian and binomial filters reduce distant contributions. A separable kernel factors into two vectors: rows, then columns. Direct and factored paths below are computed independently and compared.

Select a result cell: the left view opens the horizontal pass, and the calculation takes its three vertical neighbours with weights 1, 2, 1, then divides by 4. The intermediate is therefore not an extra illustration: its values produce the result. Multiple repetitions display the last such transition.

An integral image obtains a rectangle sum from four prefix values. FFT can help large kernels but needs correct padding and circular-convolution handling. Direct calculation is often cheaper for small kernels; changing computation must not change the operation's meaning.