A median filter replaces an image signal value by the median of nearby values. Unlike a linear average, it can reject isolated extreme observations without averaging both sides of an image edge. The small-radius response depends on neighbourhood geometry and measure; a uniform disk-area median and a circle-boundary median have different time scales.
At a smooth point with nonzero gradient, rotate coordinates so . Write . A candidate median has level boundary in the disk. The area imbalance is ; equal half-areas force . Here is the tangential second derivative, giving the formula. The disk uses uniform area measure, not its boundary measure.

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A median filter is a non-linear digital filtering technique commonly used in image processing to reduce noise while preserving edges. It operates by moving a window (or kernel) over the image and replacing the value of each pixel with the median value of the pixels in the surrounding neighborhood defined by the window.
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