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this procedure. We can evaluate pv j u; T; N; q; F when F and q are unknown by integrating
out the unknown variables. The probability of the image would then be,
ZZ Y
pv j u; T; =
p = vT xa , F uxa; q pF pq dF dq :
4.18
x a 2a This equation integrates over all possible imaging functions and all possible sets of exogenous
variables. We are not aware of any approach that has come close to evaluating such an
integral. It may not be feasible. Another possible approach is to nd the imaging function
and exogenous variables that make the image most likely,
Y
pv j u; T; N max p = vT xa , F uxa; q pF pq :
4.19
F;q
xa 2a Here we have assumed that the integral in Equation 4.18 is approximated by the component
of the integrand that is maximal. The approximation is a good one when a particular F and
q are much more likely than any other.
Using 4.19 we can de ne an alignment procedure as a nested search: 1 given an
estimate for the transformation, nd F and q that make the image most likely; 2 given
estimates for F and q, nd a new transformation that makes the image most likely. Terminate
when the transformation has stabilized. In other words, a transformation associates points
from the model with points in the image; for every ux there is a corresponding vT x. A
function F and parameter vector q are sought that best model the relationship between ux
and vT x. This can be accomplished by training" a function to t the collection of pairs
fvT xa; uxag. Algorithms for nding F and q are very similar to the those for density
approximation and learning described in Chapter 3. Notice also that that alignment with an
unknown imaging model is very similar to entropy maximization. Entropy maximization is
a nested search for a density estimate and parameters. Alignment is a nested search for an
imaging model and a transformation. We will return to this analogy shortly.
Many of the pitfalls of density approximation as described in Chapter 2 apply to function
approximation as well. Before we can hope to learn the function F we must rst make a set
of assumptions about the form of F . Without these assumptions discontinuous estimates for
84 4.1. ALIGNMENT AITR 1548 F , which t the data perfectly well but are very unlikely, can prevent convergence. One way to prevent, or discourage, this behavior is to formulate a strong prior probability over the
space of functions, pF .
In many cases the search for an imaging function and exogenous parameters can be
combined. For any particular F and q, another function Fq ux = F ux; q can be
de ned. Combining functions like this is a common technique in both shape from shading"
and photometric stereo" research. Both techniques compute the shape of an object from
the shading that is present in an image or images. Rather than independently model the
exogenous variable the lighting direction and imaging function the re ectance function
a combined function is represented and manipulated. The combined function is called a
re ectance map Horn, 1986. It maps the normals of an object directly into intensities.
The three dimensional alignment procedure we will describe manipulates a similar combined
function.
How might Equation 4.19 be approximated e ciently? It seems reasonable to assume
that for most real imaging functions similar inputs should yield similar outputs. In other
words, that the unknown imaging function is continuous and piecewise smooth. An e cient
scheme for alignment could skip the step of approximating the imaging function and attempt
to directly evaluate the consistency of a transformation. A transformation is considered
consistent if points that have similar values in the model project to similar values in the
image. By similar we do not mean similar in physical location, as in jxa , xbj, but similar
in value, juxa , uxbj and jvT xa , vT xbj. One adhoc technique for estimating
consistency is to pick a similarity constant and evaluate the following sum:
X
Consistency1T = , vT xb , vT xa2 ;
4.20
where the sum is over xa 2 a and xb 2 b such that juxb,uxaj and xa 6= xb. Consistency
is awed in a number of ways. For instance, there are no obvious clues of picking . We can
replace the all or nothing" nature of the test with a more gradual discrimination:
X
Consistency2T = ,
g uxb , uxavT xb , vT xa2 ;
4.21
xa 6=xb where g is a Gaussian with standard deviation, . In order to minimize this measure, points
that are close together must be more consistent, and those further apart less so. Another
85 Paul A. Viola CHAPTER 4. MATCHING AND ALIGNMENT problem with any consistency measure is that it is too aggressive; consistency is maximized
by constancy. The most consistent transformation projects the points of the model onto a
constant region of the image. For example, if scale is one of the transformation parameters,
one entirely consistent transformation projects all of...
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 Spring '10
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 The Land, Probability distribution, Probability theory, probability density function, Mutual Information, Paul A. Viola

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