In the following, assume that G and f are discrete arrays of size n × n and M × N,respecti...

In the following, assume that G and f are discrete arrays of size n × n and M × N,respectively.

(a) Show that the 2-D convolution of the Gaussian function G(x, y)in Eq. (10.2-21) with an image f(x, y)can be expressed as a 1-D convolution along the rows (columns) of f(x, y)followed by a 1-D convolution along the columns (rows) of the result. (See Section 3.4.2 regarding discrete convolution.)

(10.2-21)

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(b) Derive an expression for the computational advantage of using the 1-D convolution approach in (a) as opposed to implementing the 2-D convolution directly. Assume that G(x, y)is sampled to produce an array of size n × n and that f(x, y)is of size M × N. The computational advantage is the ratio of the number of multiplications required for 2-D convolution to the number required for 1-D convolution.