Problem

(a) Show that the Laplacian of a continuous function f(t, z) of continuous variables t and...

(a) Show that the Laplacian of a continuous function f(t, z) of continuous variables t and z satisfies the following Fourier transform pair [see Eq. (3.6-3) for a definition of the Laplacian]:
(3.6-3)
[Hint: Study entry 12 in Table 4.3 and see Problem 4.25(d).]
(b) The preceding closed form expression is valid only for continuous variables. However, it can be the basis for implementing the Laplacian in the discrete frequency domain using the M × N filter
for u = 0, 1, 2,…, M − 1 and v = 0, 1, 2,…, N − 1. Explain how you would implement this filter.
(c) As you saw in Example 4.20, the Laplacian result in the frequency domain was similar to the result of using a spatial mask with a center coefficient equal to −8. Explain the reason why the frequency domain result was not similar instead to the result of using a spatial mask with a center coefficient of −4. See Section 3.6.2 regarding the Laplacian in the spatial domain.

EXAMPLE 4.20: Image sharpening in the frequency domain using the Laplacian.

Figure 4.58(a) is the same as Fig. 3.38(a), and Fig. 4.58(b) shows the result of using Eq. (4.9-8), in which the Laplacian was computed in the frequency domain using Eq. (4.9-7). Scaling was done as described in connection with that equation.We see by comparing Figs. 4.58(b) and 3.38(e) that the frequency domain and spatial results are identical visually. Observe that the results in these two figures correspond to the Laplacian mask in Fig. 3.37(b), which has −8 a in the center (Problem 4.26).

FIGURE 4.58 (a) Original, blurry image. (b) Image enhanced using the Laplacian in the frequency domain. Compare with Fig. 3.38(e).

FIGURE 3.38 (a) Blurred image of the North Pole of the moon. (b) Laplacian without scaling. (c) Laplacian with scaling. (d) Image sharpened using the mask in Fig. 3.37(a). (e) Result of using the mask in Fig. 3.37(b). (Original image courtesy of NASA.)

FIGURE 3.37 (a) Filter mask used to implement Eq. (3.6-6). (b) Mask used to implement an extension of this equation that includes the diagonal terms. (c) and (d) Two other implementations of the Laplacian found frequently in practice.

(4.9-8)

(4.9-7)

4.26

(a) Show that the Laplacian of a continuous function f(t, z) of continuous variables t and z satisfies the following Fourier transform pair [see Eq. (3.6-3) for a definition of the Laplacian]:
(3.6-3)
[Hint: Study entry 12 in Table 4.3 and see Problem 4.25(d).]
(b) The preceding closed form expression is valid only for continuous variables. However, it can be the basis for implementing the Laplacian in the discrete frequency domain using the M × N filter
for u = 0, 1, 2,…, M − 1 and v = 0, 1, 2,…, N − 1. Explain how you would implement this filter.
(c) As you saw in Example 4.20, the Laplacian result in the frequency domain was similar to the result of using a spatial mask with a center coefficient equal to −8. Explain the reason why the frequency domain result was not similar instead to the result of using a spatial mask with a center coefficient of −4. See Section 3.6.2 regarding the Laplacian in the spatial domain.

4.25 The following problems are related to the entries in Table 4.3.

(a) Prove the validity of the discrete convolution theorem (entry 6) for the 1-D case.
(b) Repeat (a) for 2-D.
(c) Prove the validity of entry 7.
(d) Prove the validity of entry 12.

(Note: Problems 4.18,4.19, and 4.31 are related to Table 4.3 also.)

TABLE 4.3

Summary of DFT pairs.The closedform expressions in 12 and 13 are valid only for continuous variables.They can be used with discrete variables by sampling the closed-form, continuous expressions.