(a) Show that Steps 1 and 2 of the Marr-Hildreth algorithm can be implemented using four,...

(a) Show that Steps 1 and 2 of the Marr-Hildreth algorithm can be implemented using four, 1-D convolutions. (Hints: Refer to Problem and express the Laplacian operator as the sum of two partial derivatives, given by Eqs. 1 and 2, and implement each derivative using a 1-D mask, as in Problem.

1.

2.

10.10 The results obtained by a single pass through an image of some 2-D masks can be achieved also by two passes using 1-D masks. For example, the same result of using a 3 × 3 smoothing mask with coefficients 1/9 can be obtained by a pass of the mask [1 1 1] through an image. The result of this pass is then followed by a pass of the mask

The final result is then scaled by 1/9. Show that the response of Sobel masks (Fig) can be implemented similarly by one pass of the differencing mask [−1 0 1] (or its vertical counterpart) followed by the smoothing mask [1 2 1] (or its vertical counterpart).

FIGURE A 3 ×3 region of an image (the z’s are intensity values) and various masks used to compute the gradient at the point labeled z5.

10.18 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. 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.)

(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.

(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 see Problem.