sign detection in natural images with conditional random fields
sign detection in natural images with conditional random fields
sign detection in natural images with conditional random fields
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a region) and g = ( g j) is a vector of edge features (i.e., differences<br />
j=1...J<br />
between statistics of neighbor<strong>in</strong>g regions), so that F = ( )<br />
fy<br />
k )<br />
and<br />
k=1...K,y∈Y<br />
G =<br />
(g j y,y<br />
. Thus λ ∈ R K|Y| and µ ∈ R ′ J|Y|2 . When E =<br />
j=1...J,y,y ′ ∈Y×Y<br />
∅, the model uses no label context and is commonly called a <strong>conditional</strong><br />
maximum entropy classifier (hence, MaxEnt), or logistic regression.<br />
Tra<strong>in</strong><strong>in</strong>g and Inference<br />
Parameters for probabilistic models like CRFs are generally set by maximiz<strong>in</strong>g<br />
the likelihood of a data sample. Unfortunately, <strong>in</strong>ference for any <strong>random</strong><br />
field <strong>with</strong> the lattice topology is <strong>in</strong>tractable due to Z (an exponential sum).<br />
Markov cha<strong>in</strong> Monte Carlo (MCMC) (see e.g., [16]) is often used to approximate<br />
Z <strong>in</strong> similar generative models. However, <strong>in</strong> our <strong>conditional</strong> model Z is<br />
dependent on the image data x and must be estimated for each observation<br />
<strong>in</strong> the sample. A simpler approximation is to maximize the pseudo-likelihood<br />
(PL) [1], which is the product of the probabilities of nodes given their neighbor<strong>in</strong>g<br />
labels. The normalizers are then summations over labels at a s<strong>in</strong>gle<br />
node, rather than the possible label<strong>in</strong>gs of all nodes.<br />
A relatively new alternative to MCMC and PL for approximat<strong>in</strong>g likelihood<br />
is called tree reparameterization (TRP) [15]. Inference <strong>in</strong> graphical<br />
models <strong>with</strong>out cycles (unlike the lattice) is very efficient, i.e., due to the<br />
junction tree algorithm (e.g., [10]). An important consequence of the junction<br />
tree algorithm is that marg<strong>in</strong>al distributions are revealed on pairs of<br />
neighbor<strong>in</strong>g nodes, <strong>in</strong>duc<strong>in</strong>g an alternative factorization of the jo<strong>in</strong>t distribution.<br />
TRP operates by us<strong>in</strong>g junction tree to compute the exact marg<strong>in</strong>als on<br />
a spann<strong>in</strong>g tree of the cyclic graph. The spann<strong>in</strong>g tree’s factorization is then<br />
placed back <strong>in</strong>to the orig<strong>in</strong>al graph, and the process repeats <strong>with</strong> different<br />
spann<strong>in</strong>g trees until the parameterization converges, leav<strong>in</strong>g the marg<strong>in</strong>als.<br />
We demonstrate improved <strong>detection</strong> performance us<strong>in</strong>g TRP to approximate<br />
the likelihood over pseudo-likelihood. The likelihood function is convex and<br />
may be optimized globally via gradient ascent. Pseudo-likelihood is sensitive<br />
to <strong>in</strong>itialization, however, so node parameters are optimized first. We use the<br />
quasi-newton L-BFGS algorithm for maximization.<br />
To prevent tra<strong>in</strong><strong>in</strong>g procedures from overfitt<strong>in</strong>g parameters <strong>in</strong> <strong>conditional</strong><br />
models, a prior is <strong>in</strong>troduced, and the posterior is maximized rather than<br />
likelihood. We employ a diagonal zero-centered Gaussian prior on parameters<br />
[2] (similar to weight decay <strong>in</strong> neural networks or ridge regression); variances<br />
are experimentally determ<strong>in</strong>ed through cross-validation.<br />
Given the image data, our model simply yields a jo<strong>in</strong>t posterior distribution<br />
on label<strong>in</strong>gs. When <strong>in</strong>terested <strong>in</strong> pick<strong>in</strong>g a hard and fast label for each<br />
region of the image (node <strong>in</strong> the graph), the question becomes what to do<br />
<strong>with</strong> that distribution. A simple, oft-used answer is to f<strong>in</strong>d its maximum.<br />
That is, use maximum a posteriori (MAP) estimation:<br />
ŷ = arg max<br />
y∈Y |V | p (y | x) .