Numerical Smoothing and Differentiation - Two-dimensional Convolution Coefficients

Two-dimensional Convolution Coefficients

Two-dimensional smoothing and differentiation can also be applied to tables of data values, such as intensity values in a photographic image which is composed of a rectangular grid of pixels. The trick is transform part of the table into a row by a simple ordering of the indices of the pixels. Whereas the one-dimensional filter coefficients are found by fitting a polynomial in the subsidiary variable, z to a set of m data points. the two-dimensional coefficient are found by fitting a polynomial in subsidiary variables v and w to a set of m × m data points. The following example, for a bicubic polynomial and m = 5, illustrates the process, which parallels the process for the one dimensional case, above.

The square of 25 data values, d₀ - d₂₄

v	-2	-1	0	1	2
w
−2	d0	d1	d2	d3	d4
−1	d5	d6	d7	d8	d9
0	d10	d11	d12	d13	d14
1	d15	d16	d17	d18	d19
2	d20	d21	d22	d23	d24

becomes a vector when the rows are placed one after another.

d=(d₀, d₀ ... d₂₄)T

The Jacobian has 10 columns, one for each of the parameters a₀₀ - a₀₃ and 25 rows, one for each pair of v and w values. Each row has the form

The convolution coefficients are calculated as

The first row of C contains 25 convolution coefficients which can be multiplied with the 25 data values to provide a smoothed value for the central data point of the 25.

A Matlab routine for computing the coefficients is available. 3-dimensional filters can be obtained with a similar procedure.