Fondazione Bruno Kessler - Technologies of Vision

contains material from
Template Matching Techniques in Computer Vision: Theory and Practice
Roberto Brunelli © 2009 John Wiley & Sons, Ltd

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4.1 Validity of distributional hypotheses

The discussion of template matching as hypothese testing presented in Chapter TM:3 focused on the case of a deterministic signal corrupted by normal noise. The two hypotheses correspond to the case of a deterministic signal corrupted by normal noise and pure normal noise. What happens when we compare slightly displaced versions of the signal? In this case we are not comparing signal (possibly plus noise) with noise, but signal with signal, albeit displaced. What kind of distribution can we expect?

We will experimentally investigate this using two different templates: an eye and a hex nut (see Figure 4.1).

1 sampleimages     <- file.path(system.file(package="TeMa"), 
2 ...                               "sampleimages") 
3 # 
4 img1 <- ia.scale(as.animage(getChannels(read.pnm( 
5 ...                   file.path(sampleimages, "sampleFace_01.pgm")))), 
6 ...                  255) 
7 m1     <- animask(28,89,54,33) 
8 # 
9 img2 <- ia.scale( 
10 ...                        ia.readAnimage( 
11 ...                           file.path(sampleimages, 
12 ...                                     "cameraSimulation", 
13 ...                                     "ortho_iso12233_large.tif")), 
14 ...                              maxValue = 255)[[1]] 
15 m2   <- animask(485,288,231,225) 
16 # 
17 tm.plot("figures/detail1", ia.show(ia.get(img1,m1))) 
18 tm.plot("figures/detail2", ia.show(ia.get(img2,m2)))

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Figure 4.1: The two different templates used to investigate the distribution of pixel differences arising from misalignment and misalignment plus noise.

1 # We first focus on the effect of pattern misalignment: 
2 # we consider the distributions of pixel values arising 
3 # from displacing the patterns by a (Manhattan) distance 
4 # z=1 and z=2 
5 # 
6 # 
7 dtl    <- ia.get(img1, m1) 
8 img    <- img1 
9 m      <- m1 
10 # 
11 z      <- 1 
12 values <- c() 
13 # 
14 for (i in z*seq(-1,1,by=1)) { 
15 ...   for (j in z*seq(-1,1,by=1)) { 
16 ...     dimg   <- ia.sub(ia.get(ia.slide(img, z*i, z*j), m), 
17 ...                         dtl) 
18 ...     values <- c(as.real(dimg@data), values) 
19 ...   } 
20 ... } 
21 # 
22 tm.dev("figures/dsts_1") 
23  hist(values,breaks=100,freq=FALSE,xlim=c(-50,50), 
24 ...   main="Normal vs. Cauchy (z=1)") 
25  lines(seq(-50,50),dnorm(seq(-50,50), sd=sqrt(var(values))), lty=1) 
26  lines(seq(-50,50),dcauchy(seq(-50,50),scale=mad(values,constant=1)),lty=2) 
27  legend(20,0.15, c("normal", "Cauchy"), lty=1:2) 
28 dev.off() 
29 # 
30 z      <- 2 
31 values <- c() 
32 # 
33 for (i in z*seq(-1,1,by=1)) { 
34 ...   for (j in z*seq(-1,1,by=1)) { 
35 ...     dimg   <- ia.sub(ia.get(ia.slide(img, z*i, z*j), m), 
36 ...                         dtl) 
37 ...     values <- c(as.real(dimg@data), values) 
38 ...   } 
39 ... } 
40 # 
41 tm.dev("figures/dsts_2") 
42  hist(values,breaks=100,freq=FALSE,xlim=c(-50,50), 
43 ...   main="Normal vs. Cauchy (z=2)") 
44  lines(seq(-50,50),dnorm(seq(-50,50), sd=sqrt(var(values))), lty=1) 
45  lines(seq(-50,50),dcauchy(seq(-50,50),scale=mad(values,constant=1)),lty=2) 
46  legend(20,0.04, c("normal", "Cauchy"), lty=1:2) 
47 dev.off()

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Figure 4.2: Considering slightly translated versions of the template as valid instances to be detected corresponds to the introduction of a template variability whose effect is not dissimilar to the addition of noise. As our lens in the template detection process is the computation of a sum of pixel differences we may get a first appreciation of the resulting variability by looking at the distribution of the differences of aligned pixels. While the detailed distribution depends on the specific template and on the allowed misalignment spacing, many practical cases result in a distribution that is better modeled with a Cauchy distribution than with a normal one. The two plots present the distributions arising from two different misalignments: z=1 and z=2.

1 # We now consider a slightly different case: we consider 
2 # a single displacement value z=1 (plus z=0) and change the 
3 # amount of normal noise: noise=0 and noise=2 
4 # 
5 # 
6 noise <- 0 
7 img   <- img2 
8 m     <- m2 
9 dtl   <- tm.addNoise(ia.get(img,m),"normal",scale=noise,clipRange=c(0,255)) 
10 # 
11 z      <- 1 
12 values <- c() 
13 # 
14 for (i in z*seq(-1,1,by=1)) { 
15 ...   for (j in z*seq(-1,1,by=1)) { 
16 ...     dimg   <- ia.sub(tm.addNoise(ia.get(ia.slide(img, z*i, z*j), m), 
17 ...                             "normal",scale=noise,clipRange=c(0,255)), 
18 ...                  dtl) 
19 ...     values <- c(as.real(dimg@data), values) 
20 ...   } 
21 ... } 
22 # 
23 tm.dev("figures/dsts_3") 
24  hist(values,breaks=100,freq=FALSE,xlim=c(-50,50), 
25 ...   main="Normal vs. Cauchy (noise=0)") 
26  lines(seq(-50,50),dnorm(seq(-50,50), sd=sqrt(var(values))), lty=1) 
27  lines(seq(-50,50),dcauchy(seq(-50,50),scale=mad(values,constant=1)),lty=2) 
28  legend(20,0.15, c("normal", "Cauchy"), lty=1:2) 
29 dev.off() 
30 # 
31 noise  <- 2 
32 values <- c() 
33 # 
34 for (i in z*seq(-1,1,by=1)) { 
35 ...   for (j in z*seq(-1,1,by=1)) { 
36 ...     dimg   <- ia.sub(tm.addNoise(ia.get(ia.slide(img, z*i, z*j), m), 
37 ...                             "normal", scale=noise, clipRange=c(0,255)), 
38 ...                  dtl) 
39 ...     values <- c(as.real(dimg@data), values) 
40 ...   } 
41 ... } 
42 # 
43 tm.dev("figures/dsts_4") 
44  hist(values,breaks=100,freq=FALSE,xlim=c(-50,50), 
45 ...   main="Normal vs. Cauchy (noise=2)") 
46  lines(seq(-50,50),dnorm(seq(-50,50), sd=sqrt(var(values))), lty=1) 
47  lines(seq(-50,50),dcauchy(seq(-50,50),scale=mad(values,constant=1)),lty=2) 
48  legend(20,0.15, c("normal", "Cauchy"), lty=1:2) 
49 dev.off()

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Figure 4.3: The two plots report the distribution of pixel differences in the case of misalignment and no noise (left) and in the case of misalignment plus a small amount of normal noise (right). The Cauchy distribution due to pattern misalignement dominates the normal noise distribution. Of course, increasing the amount of noise would increase the Gaussianity of the final distribution.

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