unified detection and recognition for reading text in scene images
unified detection and recognition for reading text in scene images
unified detection and recognition for reading text in scene images
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An alternative method <strong>for</strong> prediction is called maximum posterior marg<strong>in</strong>al (MPM)<br />
estimation:<br />
ŷ i = arg max<br />
y i ∈Y i<br />
p (y i | x, θ, I), ∀i ∈ V. (2.31)<br />
It corresponds to choos<strong>in</strong>g the label <strong>for</strong> each unknown that maximizes the probability<br />
with all other label<strong>in</strong>gs marg<strong>in</strong>alized. This can often be a more effective way of consider<strong>in</strong>g<br />
the probabilities of all the label<strong>in</strong>gs, rather than simply the maximum (jo<strong>in</strong>t)<br />
label<strong>in</strong>g, as <strong>in</strong> MAP. Marg<strong>in</strong>alization, however, suffers from the same computational<br />
complexity problems. Sum-product loopy belief propagation, described next, can be<br />
used to approximate these marg<strong>in</strong>als.<br />
In practice, the MAP estimate tends to be conservative, try<strong>in</strong>g to give the most<br />
correct labels, while MPM tends to give higher <strong>detection</strong> rates. These methods are<br />
used variously <strong>in</strong> our experiments.<br />
2.3.2 Belief Propagation<br />
Because they all <strong>in</strong>volve comb<strong>in</strong>atorial sums or search spaces, we require approximations<br />
to compute the log partition function <strong>for</strong> the likelihood objective (2.15),<br />
the marg<strong>in</strong>al probabilities <strong>for</strong> its gradient (2.18), or to make predictions with MAP<br />
or MPM, Fortunately, there is one family of algorithms that h<strong>and</strong>les all of these.<br />
Namely, the loopy sum-product or belief propagation algorithm. In short, it exploits<br />
the nature <strong>in</strong> which the jo<strong>in</strong>t probability factorizes <strong>in</strong>to a product of local functions.<br />
Recall the bipartite “factor graph” between unknowns y, represented by nodes <strong>in</strong><br />
the graph, <strong>and</strong> the compatibility functions (or factors, when they are exponentiated)<br />
over the unknowns, as shown <strong>in</strong> the example of Figure 2.1 on page 22. The algorithm<br />
operates by iteratively pass<strong>in</strong>g messages between the nodes represent<strong>in</strong>g the unknowns<br />
<strong>and</strong> the factors. When these messages converge to a stable fixed po<strong>in</strong>t, the results<br />
are equivalent to m<strong>in</strong>imiz<strong>in</strong>g the so-called Bethe free energy, a variational method <strong>for</strong><br />
approximate <strong>in</strong>ference from statistical physics [136]. When the factor graph is a tree,<br />
the sum-product algorithm [60] efficiently per<strong>for</strong>ms exact <strong>in</strong>ference. In most cases,<br />
our graphs are not trees, <strong>and</strong> thus the results are only approximate.<br />
Let N (i) ≡ {C ∈ C | i ∈ C} be the family of <strong>in</strong>dices <strong>for</strong> the factors that neighbor<br />
the ith unknown. This corresponds to the set of factors hav<strong>in</strong>g y i as an argument.<br />
The set of edges <strong>in</strong> a factor graph are thus<br />
E (C) ≡ {(i, C) | i ∈ V ∧ C ∈ N (i)}. (2.32)<br />
For all edges <strong>in</strong> the factor graph (i, C) ∈ E (C), the node-to-factor messages have the<br />
general <strong>for</strong>m<br />
∏<br />
m i→C (y i ) ∝ m C ′ →i (y i ) (2.33)<br />
C ′ ∈N(i)\C<br />
so that the message from a node to a factor is the product of all the messages to<br />
that node from all other neighbor<strong>in</strong>g factors. The result<strong>in</strong>g functional message is<br />
normalized (i.e., sums to 1 over y i ) <strong>for</strong> numerical stability. Note that <strong>for</strong> parsimony,<br />
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