1995_Viola_thesis_registrationMI

Moreover in both of these graphs the joint

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Unformatted text preview: xjuX ; a = g vT x , F ux; a see Sections 2.3.1 and 4.1.2 for commentary on the equivalence of log likelihood and sample entropy. We can relax the constraint that v be conditionally Gaussian by using EMMA to estimate the conditional entropy: hvT xjuX   hux , hvT x; uX  : 4.32 The rst term is the entropy of the model. It is not a function of the transformation. Why are these two seemingly unrelated concepts, weighted neighbor likelihood and conditional entropy, so closely related? We can gain some intuition about this equivalence by looking at the joint distribution of u and v. For any particular transformation we can sample points uxa; vT xa T and plot them. Figure 4.8 shows the joint samples of the signals from Figure 4.1 when aligned. The thin line in the plot is the weighted neighbor function approximation of this data; it is a good t to the data. There is a noticeable clumping, or clustering, in the data. These clumps arise from the regions of almost constant intensity in the signals. There are four large regions of constant intensity and four clusters. When these almost identical signals are aligned they are strongly correlated. Large values in one signal corresponds to large values of the other. Conversely, small values in one correspond to small values in the other. Correlation measures the tendency of the data to lie along the line x = y normalized correlation measures the tendency of the data to lie along some line of positive slope. Figure 4.9 shows the joint samples of the two signals shown in Figure 4.2. These signals are not linearly related or even correlated, but they are functionally related. Weighted neighbor likelihood measures the quality of the weighted neighbor function approximation. In both of these graphs the points of the sample lie near the weighted neighbor function approximation. Moreover, in both of these graphs the joint distribution of samples is tightly packed together. Points are not distributed throughout the space, but lie instead in a small part of the joint space. This is the hallmark of a low entropy distribution. We can generate similar graphs for signals that are not aligned. Figures 4.10 and 4.11 show the same signals except for the fact that the image has been shifted 30 units. For these shifted signals the structure of the joint distribution is destroyed. The weighted neighbor function 89 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Figure 4.8: Samples from the joint space of ux and vx = ux + . A small black square is plotted for every pixel in the signals. The X-axis is the value of ux. The Y-axis is the value of vx. The clumping of points in clusters is caused by the regions of almost constant intensity in the images. The thin line plotted through the data is the weighted neighbor function estimate. approximation is a terrible t to the data. As a result the weighted neighbor likelihood of these signals is low. Alternatively we could look directly at the distributions. When the signals are aligned the distributions are compact. When they are misaligned the distributions are spread out and haphazard. Or in other words, aligned signals have low joint entropy and misaligned signals have high joint entropy. This suggests an alternative to weighted neighbor likelihood: the EMMA approximation of joint entropy. Graphed below are the EMMA estimates of joint entropy, hw, versus translation for each signal alignment problems discussed. Figure 4.12 shows a graph of joint entropy for the two signals that are nearly identical. Figure 4.13 shows a graph of joint entropy for the model and the non-linearly transformed image. In both case the graphs show strong minima at the correct aligning translation. 90 4.1. ALIGNMENT AI-TR 1548 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Figure 4.9: Samples from the joint space of ux and vx = ,ux2 + . 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Figure 4.10: Samples from the joint space of ux and vx = ux + 30 + . Unlike the previous graph these two signal are no longer aligned. 91 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 Figure 4.11: Samples from the joint space of ux and vx = ,ux + 302 + . The two signals are not aligned. 2.2 u(x) v(x) Joint Entropy Intensity 1 0.5 0 -0.5 2 1.8 1.6 -1 1.4 0 100 200 300 Position 400 -150 -75 0 Position 75 150 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 u(x) v(x) 2.2 Joint Entropy Intensity Figure 4.12: On the left is a plot of image and model that are identical except for noise. On the right is a plot of the estimated joint entropy versus translation. 2 1.8 0 100 200 300 Position 400 -150 -75 0 Position 75 150 Figure 4.13: On the left is a plot of image and model that are related non-linearly. On the right is a plot of estimated joint entropy versus translation. 92 4.2. WEIGHTED NEIGHBOR LIKELIHOOD VS. EMMA AI-TR 1548 4.2...
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This note was uploaded on 02/10/2010 for the course TBE 2300 taught by Professor Cudeback during the Spring '10 term at Webber.

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