Learning Gaussian Conditional Random Fields for Low-Level Vision
Marshall F. Tappen
University of Central Florida
Orlando, FL
mtappen@eecs.ucf.edu
Ce Liu Edward H. Adelson William T. Freeman
MIT CSAIL
Cambridge, MA 02139
{celiu, adelson, billf}@csail.mit.edu
Abstract
Markov Random Field (MRF) models are a popular tool
for vision and image processing. Gaussian MRF models
are particularly convenient to work with because they can
be implemented using matrix and linear algebra routines.
However, recent research has focused on on discrete-valued
and non-convex MRF models because Gaussian models
tend to over-smooth images and blur edges. In this paper,
we show how to train a Gaussian Conditional Random Field
(GCRF) model that overcomes this weakness and can out-
perform the non-convex Field of Experts model on the task
of denoising images. A key advantage of the GCRF model is
that the parameters of the model can be optimized efficiently
on relatively large images. The competitive performance of
the GCRF model and the ease of optimizing its parameters
make the GCRF model an attractive option for vision and
image processing applications.
1. Introduction
Markov Random Field (MRF) models are a popular tool
for solving low-level vision problems. In the MRF model,
the relationships between neighboring nodes, which often
correspond to pixels, are modeled by local potential func-
tions. The parameters of these functions can be effectively
learned from training samples. The power of learning and
inference in an MRF makes it a popular choice for low-level
vision problems including image denoising [12], stereo re-
construction [3], and super-resolution [18].
However, learning and inference in MRFs are, in gen-
eral, nontrivial problems. Because of the nonlinear and non-
convex potential functions used in many MRFs, sophisti-
cated sampling-based algorithms are often used for param-
eter learning [21, 12]. Unfortunately, sampling algorithms
can be slow to converge. For certain types of discrete-
valued graphs, there are efficient algorithms such as graph
cuts [3] and loopy belief propagation [5], but learning still
remains a hard problem.
Gaussian Markov random fields, which are MRFs where
the variables are jointly Gaussian, have a long history in
computer vision research [14]. Gaussian models are par-
ticularly convenient to work with because t he inference
in Gaussian models can be easily accomplished using lin-
ear algebra. Algorithms for numerical linear algebra are
well understood, and efficient implementations are avail-
able. Nevertheless, Gaussian Markov random fields can re-
sult in over-smoothed images when the potential functions
for neighboring nodes are homogeneous or isotropic, i.e.
identical everywhere.
Typically, the key to success with Gaussian Markov ran-
dom fields is to have the neighboring potential function de-
pendent on the input signal, as in [10, 17]. These inho-
mogeneous or anisotropic Gaussian MRFs can overcome
the weakness of the homogeneous ones by reconstructing
piecewise smooth image with desirable properties. Be-
cause the potential functions now depend on the signal these
MRFs are no longer generative models, but are instead con-
ditional models. Therefore, these Gaussian MRFs can also
be called Gaussian conditional random field (GCRF). In
GCRFs, the parameters that describe how each potential
function depends on the input signal is typically designed
empirically and hand-tuned to generate the desired results.
Although this might be feasible when there are few param-
eters, this approach does not scale-up when automatically
designing and optimizing larger, more general models.
In this paper, we show how the parameters of GCRF can
be efficiently for low-level vision tasks. We derive the learn-
ing algorithm by minimizing the error in the MAP solution
of the model for the training samples. This tractable method
of training, described in Section 3, enables the GCRF
model to improve on the results produced by the more com-
plex, non-convex Field of Experts model. The fact that
GCRF models are relatively easy to train and can perform
as well as more complicated models makes them poten-
tially useful for a number of low-level vision and image-
processing tasks. To demonstrate the convenience and
power of the model, example implementations are available
at http://www.eecs.ucf.edu/˜mtappen.
Section 2 describes the GCRF model. The training algo-
rithm is described in Section 3. Related work and models
are discussed in Section 4. The GCRF model is compared
to the Field of Experts model in Section 5.
2. Motivating the GCRF Model
The Gaussian Conditional Random Field (GCRF) model
can be motivated i n two ways: probabilistically as a Condi-
tional Random Field [6], or as an estimator based on min-
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