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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 nonfunctional signals arises when both the model and the
image are functions of some third unmeasurable signal. Call this signal zx. There are
now two imaging functions, one that creates the model ux = Fuzx; qu, and another that
creates the image vT x = Fv zx; 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 nonlinear transforms. Using the original signal from Figure 4.1 we can de ne
Fuz = sin2z 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 ux and vx 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
nonoccluded 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 hvju can be small for two di erent reasons: hv 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 hv 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, hv 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 AITR 1548 380 280 150 75 0
Position 75 150 Figure 4.16: Graph of weighted neighbor likelihood versus translation where vx 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 ux and vx = ux2 + . 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 = sin2z and v = z2 as z varies from 1.5
to 1.5. 98 4.2. WEIGHTED NEIGHBOR LIKELIHOOD VS. EMMA AITR 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 nonfunctional
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 ux; vT x = d hux + d hvT x , d hux; vT x :
dT
dT
dT
dT
Since hux is not a function of T , it drops from our calculations. Using the EMMA entropy
estimate, d I ux; vT 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 gv vi , v, v ;
k
xk 2A gv vi Wuv wi; wj P guvgwi , wj w ;
xk 2A uv wi , k
ui uxi ; uj uxj ; uk uxk ;
vi vT xi ; vj vT xj ; vk vT 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 = DIAGuu1; vv1 ; 100 4.4. MATCHING AND MINIMUM DESCRIPTION LENGTH AITR 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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