dilate

Dilate is a verb that means to make something wider or larger, especially a hollow organ, a vessel, or a geometric figure. In mathematics and related fields, dilation denotes a transformation or operation that scales objects about a point or region.

In geometry, dilation is a similarity transformation that expands or contracts a figure with respect to a fixed center by a scale factor k > 0. A point P maps to P' along the line through the center C such that CP' = k·CP. If k > 1 the image is larger; if 0 < k < 1 it is smaller; k = 1 leaves the figure unchanged. Dilation preserves shape and straight lines, and maps lines through the center to themselves. In coordinates, P' = C + k(P − C).

In digital image processing, morphological dilation grows foreground regions by applying a structuring element. The result is the union of translated copies of the element over the image; equivalently, A ⊕ B. Dilation fills small holes, connects nearby components, and expands boundaries, and is typically used alongside erosion in sequences to form opening and closing operations.

In physiology and medicine, dilation refers to the widening of a hollow structure, such as blood vessels (vasodilation), airways, or pupils (pupillary dilation). It can occur naturally or be induced by reflexes or pharmacological agents. Abnormal dilation can be a feature of certain conditions and is often investigated in clinical settings.