With reference to Eq. (10.2-23): (10.2-23)(a) Show that the average value of the Laplacian...

With reference to Eq. (10.2-23):

(10.2-23)

(a) Show that the average value of the Laplacian of a Gaussian operator, ∇2G(x, y),is zero.

---

(b) Show that the average value of any image convolved with this operator also is zero. (Hint: Consider solving this problem in the frequency domain, using the convolution theorem and the fact that the average value of a function is proportional to its Fourier transform evaluated at the origin.)

---

(c) Would (b) be true in general if we (1) used the mask in Fig. 10.4(a) to compute the Laplacian of a Gaussian lowpass filter using a Laplacian mask of size 3×3, and (2) convolved this result with any image? Explain. (Hint: Refer to Problem 3.16.)

3.16

(a) Suppose that you filter an image, f(x, y), with a spatial filter mask, w(x, y), using convolution, as defined in Eq. (3.4-2), where the mask is smaller than the image in both spatial directions. Show the important property that, if the coefficients of the mask sum to zero, then the sum of all the elements in the resulting convolution array (filtered image) will be zero also (you may ignore computational inaccuracies). Also, you may assume that the border of the image has been padded with the appropriate number of zeros.

(3.4-2)

---

(b) Would the result to (a) be the same if the filtering is implemented using correlation, as defined in Eq. (3.4-1)?

(3.4-1)

FIGURE 10.4 (a) Point detection (Laplacian) mask. (b) X-ray image of turbine blade with a porosity. The porosity contains a single black pixel. (c) Result of convolving the mask with the image. (d) Result of using Eq. (10.2-8) showing a single point (the point was enlarged to make it easier to see). (Original image courtesy of X-TEK Systems, Ltd.)