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Figure 43 graphs c t for the two signals from figure

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Unformatted text preview: for a set of parameters that cause the inputs to model the outputs. Function learning attempts to nd the best parameter vector v; alignment attempts to nd the best transformation T . As we did for function learning, we can draw an analogy with sample entropy. The log likelihood of T is proportional to the conditional entropy of the image given the model, 1 log`T  = , N havT X  j uX ; T; ; q; F  : a 4.14 This is not the EMMA estimate of entropy, but the conditional entropy of v under the assumption that v is conditionally Gaussian. For the problems described in Section 3.1 it was possible to show that entropy optimization led to maximum mutual information solutions. For this problem however, we cannot claim that maximizing log likelihood is equivalent to maximizing the mutual information between v and u. The mutual information I vT X ; uX  = hvT X  , hvT X  j uX ; T; ; q; F  ; 4.15 includes both a conditioned and unconditioned entropy. For some types of transformations hvT X  may change as T is varied. In these cases minimizing conditional entropy is not equivalent to maximizing mutual information. One must also maximize the unconditioned entropy. In our simple example, where only translation is varied and the signals are periodic, unconditioned entropy does not change as T is varied. Returning to the rst synthetic example, we can plot C T  from 4.12 versus translation. Figure 4.3 graphs C T  for the two signals from Figure 4.1 we have assumed periodic boundary conditions on the signals. We can see that the cost has a very strong minimum at the true translation of 0 pixels. 81 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT 1400 u(x) v(x) Difference Squared 1200 Intensity 1 0.5 0 -0.5 -1 1000 800 600 400 200 0 0 100 200 300 Position 400 -150 -75 0 Position 75 150 Figure 4.3: On the left is a plot of image and model that are identical except for noise. On the right is a plot of C T  versus translation. There is a signi cant minimum at the correct aligning translation of 0 pixels. u(x) v(x) 1400 Difference Squared Intensity 1500 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 1300 1200 1100 1000 900 800 0 100 200 300 Position 400 -150 -75 0 Position 75 150 Figure 4.4: On the left is a plot of image and model that are related non-linearly. On the right is a plot of C T  versus translation. There is no minima at the aligning translation of 0 pixels. In fact minima exist at incorrect translations. Correlation works very well at matching together u and v when the imaging model and exogenous parameters are known. In many cases however we may be faced with a situation where F and q are unknown. In some cases alignment problems can still be solved by assuming that the imaging function is the identity function. This assumption is not e ective when aligning the non-monotonically related signals shown in Figure 4.2. Figure 4.4 graphs C T  versus translation for these two signals. Notice that each of the actual minima are at incorrect translations. In general C T  cannot be used to align signals related by an unknown non-linear transformations. C T  can however be generalized to work with signals that have been transformed linearly. Rather than minimize the squared di erence between signals, we can instead minimize the squared di erence between signals that have been normalized. A normalized signal 82 4.1. ALIGNMENT AI-TR 1548 4 u(x) v(x) Intensity 2 0 -2 -4 -6 0 100 200 300 Position 400 Figure 4.5: Graph of ux and vx = 3ux , 2 versus x. is one with a mean of zero and a standard deviation of one and can be computed as E b ux = ux , XuX  : u  4.16 The normalized version of a signal is invariant to multiplicative and additive changes to the original. The sum of the squared di erences between the normalized signals, NC T , can be computed directly as one minus the Normalized correlation between the signals u and v. Normalized cost is de ned as: , u  E 4.17 NC T  = 1 , Ea uX vT X  EavTXX a vT X  :  au X a As a shorthand we have abbreviated sums over the coordinates x as expectations and variances. Normalized cost can be used on signals like the ones shown in Figure 4.5 F u = 3u , 2. A plot of NC T  versus translation is identical to Figure 4.3. In some cases, normalized cost can be applied to signals transformed by non-linear monotonic functions. Note however that the two signals shown in Figure 4.2 are related by a non-monotonic function and cannot be accommodated in this way. In these examples translation does not e ect the mean or the standard deviation of the signals. As a result, normalized cost will not produce a convincing minimum where cost alone does not. 83 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT We may still wish to align models and images that are related by non-monotonic functions. In this case alignment can be performed by jointly searching over the space of possible imaging functions, exogenous parameters and transformations. Probability can be used to m...
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