Page 258 - Digital Analysis of Remotely Sensed Imagery

220 Cha pte r S i x

r−1,c r−1,c−1 r−1,c r−1,c+1

r,c−1 r,c r,c+1 r,c−1 r,c r,c+1

r+1,c r+1,c−1 r+1,c r+1,c+1

(a) (b)
FIGURE 6.11 Defi nition of neighborhood with a window size of 3 × 3 pixels
for pixel (r, c) (shaded). (a) Four-connectivity neighborhood; (b) eight-
connectivity neighborhood.

matrix, often called kernel coefficients, serve to weigh pixels in calcu-
lating the convoluted output. Different kernel values serve different
filtering purposes.
The convolution of the kernel with the two-dimensional (2D)
input image is essentially a matrix multiplication (Fig. 6.12). Since
the kernel is square, the working window must also be square. The
weights in the kernel dictate the influence of pixels in the corre-
sponding position. This operation is mathematically expressed as

d
d
(,
DN = 1 ∑ ∑ w DN i j) (6.8)
out ij in
W
i =1 j =1
d d
W = ∑∑ w ij (6.9)
j=1 i=1
245 233 247 274 344

269 240 251 260 332
W 11 W 12 W 13
W 21 W 22 W 23 305 268 230 234 259
W 31 W 32 W 33 305 258 310 259 276

331 218 454 386 557
(a) (b)

FIGURE 6.12 The spatial convolution concept in image spatial fi ltering. (a) A
3 × 3 kernel, or template, containing weights; (b) the array of pixel values in
the input image (only partial) as shown in Fig. 1.3. The operation is based on
moving window. After the boldfaced pixel is convoluted, the operation moves
on toward the right by one pixel at a time.