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1995_Viola_thesis_registrationMI

# 2 conditional entropy hvju can be small for two di

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Unformatted text preview: y noise, and a portion of it occluded. Both joint entropy and mutual information are symmetric measures. A plot of EMMA entropy for the swapped signals is identical to Figure 4.13. A more complex example of non-functional signals arises when both the model and the image are functions of some third unmeasurable signal. Call this signal zx. There are now two imaging functions, one that creates the model ux = Fuzx; qu, and another that creates the image vT x = Fv zx; qv . Medical registration, which we describe in some detail in the next chapter, is a clear example of a two sensor problem. In medical registration one seeks an alignment of signals from two types of sensors for example a CT scan and an MRI scan. Neither gives perfect information about the object, and neither is completely predictable from the other. The two sensor problem can be simulated by transforming our original signal by two di erent non-linear transforms. Using the original signal from Figure 4.1 we can de ne Fuz = sin2z and Fv z = z2. The resulting aligned distribution should fall approximately along a line that looks like Figure 4.19. The actual distribution is shown in Figure 4.20. Here too EMMA shows a strong minimum at the correct alignment of model and image see Figure 4.21. 95 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT 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 4.15: Samples from the joint space of ux and vx where the image has been occluded. Though these signals are aligned the the weighted neighbor function is a terrible t to the data. The data is segregated into two parts: the linearly related part that arise from the non-occluded signal, and the constant part that projects to the occluded part. Mutual Information versus Joint Entropy Recall that conditional entropy is not a measure of the dependence between two signals see Section 2.2. Conditional entropy hvju can be small for two di erent reasons: hv is itself small, or v is dependent on u. Mutual information is a better measure of dependence. For the simple examples described in this chapter hv j u can be used alone as a measure of alignment. In more complex examples, where the model can change scale or project on a limited part of the image, hv must be taken into account. In general we will solve alignment problems by maximizing mutual information. 96 Log Likelihood 4.2. WEIGHTED NEIGHBOR LIKELIHOOD VS. EMMA AI-TR 1548 380 280 -150 -75 0 Position 75 150 Figure 4.16: Graph of weighted neighbor likelihood versus translation where vx has been occluded. Joint Entropy 2.2 2 1.8 1.6 -150 -75 0 Position 75 150 Figure 4.17: Graph of EMMA joint entropy estimate versus translation for the occluded pair of signals. 97 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 4.18: Samples from the joint space of ux and vx = ux2 + . The roles of the model and the image have been reversed. 2.5 sin(x) vs. x^2 2 1.5 1 0.5 0 -0.5 -1.5 -1 -0.5 0 0.5 1 1.5 Figure 4.19: Graph of the function de ned by u = sin2z and v = z2 as z varies from -1.5 to 1.5. 98 4.2. WEIGHTED NEIGHBOR LIKELIHOOD VS. EMMA AI-TR 1548 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1.5 -1 -0.5 0 0.5 1 1.5 Joint Entropy Figure 4.20: Samples from the joint space of from the simulated two sensory data. 2.2 2 -150 -75 0 Position 75 150 Figure 4.21: Graph of EMMA joint entropy estimate versus translation for the non-functional pair of signals. 99 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT 4.3 Alignment Derivation Let us derive the equations and algorithms used for alignment by maximization of mutual information. We wish to maximize the mutual information between model and image signals. This requires a search of the aligning transformation space. We will use the stochastic gradient descent algorithm described in Section 3.3. The derivative of mutual information is d I ux; vT x = d hux + d hvT x , d hux; vT x : dT dT dT dT Since hux is not a function of T , it drops from our calculations. Using the EMMA entropy estimate, d I ux; vT x  1 X X W v ; v  v , v T ,1 d v , v  4.33 dT NB xi2B xj 2A v i j i j v dT i j 1 X X W w ; w  w , w T ,1 d w , w  : 4.34 ,N uv i j i j j uv dT i B xi 2B xj 2A The following de nitions have been used: j Wv vi; vj  P gv vi , v, v  ; k xk 2A gv vi Wuv wi; wj  P guvgwi , wj w  ; xk 2A uv wi , k ui uxi ; uj uxj  ; uk uxk  ; vi vT xi ; vj vT xj  ; vk vT xk ; wi ui; vi T ; wj uj ; vj T ; and wk uk ; vk T : We assume that the covariance matrices of the component densities used in the approximation scheme for the joint density are block diagonal, , ,, uv1 = DIAGuu1; vv1 ; 100 4.4. MATCHING AND MINIMUM DESCRIPTION LENGTH AI-TR 1548 and we obtain an estimate for the derivative of the mutual information as follows d 1 XX dI = , ,d vi; vj T Wv vi; vj v 1 , Wuv wi; wj vv1 dT vi , vj  : dT N B xi 2B xj 2A If we are to i...
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