Question

What is the exact mathematical role of the homography matrix when transforming a coordinate from one image plane to another?

Accepted Answer

A homography matrix is a three by three invertible matrix that maps the coordinates of points from one planar surface to another in a projective space. To transform a coordinate from an initial image to a destination image, you represent the pixel position as a three-dimensional vector using homogeneous coordinates, which means adding a third value of one to the standard x and y pixel positions. You multiply this vector by the homography matrix, resulting in a new three-dimensional vector. Because this transformation occurs in projective space, the resulting coordinates are scale-invariant, meaning that the relative position of the point is preserved regardless of the magnitude of the coordinates. To convert these resulting values back into standard two-dimensional pixel coordinates, you divide the first two elements of the vector by the third element, which is known as dehomogenization. This process accounts for the perspective distortion that occurs when a camera views a flat surface from different angles, allowing for accurate alignment or warping of image perspectives.

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What is the exact mathematical role of the homography matrix when transforming a coordinate from one image plane to another?

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