unified detection and recognition for reading text in scene images

2.2.1 Model Likelihood
Substituting the parametric form (2.4) for the model likelihood in (2.14) we have
⎛
⎞
L (θ; D) ≡ ∑ ⎝ ∑ ( )
U C y (k)
C ,x(k) ; θ C − log Z ( x (k)) ⎠ (2.15)
k C∈C (k)
The set of functions {U C } C∈C (k) depends on the particular unknowns y (k) , and thus C
is indexed by the particular example k.
For certain forms of compatibility functions, it can be shown that the objective
function L (θ; D) is convex, which means that global optima can be found by gradient
ascent or other convex optimization techniques [13]. In particular, if the compatibility
functions are linear in the parameters, then the log likelihood (2.14) is convex.
Throughout this thesis, we use linear compatibility functions, which have the general
form
U C (y C ,x C ; θ C ) = θ C (y C ) · F C (x) , (2.16)
where F C : Ω → R d(C) is a vector of features of the observation, the dimensionality
of which depends on the particular set C. The parameter vector θ C ∈ R |YC|×d(C) is
conveniently thought of as a function θ C : Y C → R d(C) that takes an assignment y C
and returns an associated set of weights for the features F C .
Taking the gradient of the objective (2.15) with respect to the parameters yields
∇ θ L (θ; D) = ∑ ∑ ( ( )
∇ θC U C y (k)
C ,x(k) ; θ C −
k C∈C
[ (k) ( ) ])
E C ∇θC U C yC ,x (k) ; θ C | x (k) ; θ C (2.17)
= ∑ ∑ ( [ ])
FC (x) − E C FC (x) | x (k) ; θ C . (2.18)
k C∈C (k)
where E C indicates an expectation with respect to the marginal probability distribution
p (y C | x, θ, I). Equation (2.17) is the gradient for general compatibility functions,
while (2.18) is for linear compatibilities (2.16).
To calculate the log likelihood and its gradient, and thus find the optimal model θ,
we will need to be able to calculate log Z (x), the so-called log partition function, and
the marginal probabilities of each y C . In general, these both involve combinatorial
sums, so approximations must be made. Most of these are described in Section 2.3.2,
but we describe two here that are more closely related to the log-likelihood and the
objective function.
2.2.1.1 Parameter Decoupling
One simple approximation that may be made is to decouple the parameters in
θ during training. For instance, in the example described at the end of §2.1.2 and
shown in Figure 2.1, there are two types of compatibility functions, one for recognizing
characters based on their appearance and another for weighting bigrams. If the
parameter vector is decomposed into the parameters for the recognition and bigram
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