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3.2 Linear filtering 101

zero wrap clamp mirror

blurred zero normalized zero blurred clamp blurred mirror
Figure 3.13 Border padding (top row) and the results of blurring the padded image (bottom row). The normalized
zero image is the result of dividing (normalizing) the blurred zero-padded RGBA image by its corresponding soft
alpha value.

where the (sparse) H matrix contains the convolution kernels. Figure 3.12 shows how a
one-dimensional convolution can be represented in matrix-vector form.

Padding (border effects)

The astute reader will notice that the matrix multiply shown in Figure 3.12 suffers from
boundary effects, i.e., the results of filtering the image in this form will lead to a darkening of
the corner pixels. This is because the original image is effectively being padded with 0 values
wherever the convolution kernel extends beyond the original image boundaries.
To compensate for this, a number of alternative padding or extension modes have been
developed (Figure 3.13):

• zero: set all pixels outside the source image to 0 (a good choice for alpha-matted cutout
images);

• constant (border color): set all pixels outside the source image to a specified border
value;

• clamp (replicate or clamp to edge): repeat edge pixels indefinitely;

• (cyclic) wrap (repeat or tile): loop “around” the image in a “toroidal” configuration;

• mirror: reflect pixels across the image edge;

• extend: extend the signal by subtracting the mirrored version of the signal from the
edge pixel value.