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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-

imizing a cost function. Section 2.1 shows how the GCRF

estimator can be motivated as a probabilistic model, while

Section 2.3 shows how it can be motivated as the minimum

of a quadratic cost function.

2.1. Basic Gaussian Conditional Random Field

Model

Before presenting the actual Gaussian Conditional Ran-

dom Field (GCRF) model used to denoise images, this sec-

tion will ﬁrst describe a GCRF model for images that suffers

from the problem of oversmoothing. Section 2.2 will then

show how this model can be improved to handle sharp edges

and give much better results.

This GCRF model is deﬁned by a set of linear fea-

tures that are convolution kernels. For a set of features,

. . . f

, the probability density of an image, X , condi-

tioned on the observed image, O, is deﬁned to be

p(X )=

exp

−

i=1

x,y

“



X ∗ f

)(x, y) − r

(x, y; O



”

(1)

where (X ∗ f

)(x, y) denotes the value at location (x, y) in

the image produced by convolving X with f

. We assume

that this image only contains those pixels where the entire

ﬁlter ﬁts in X . This corresponds to “valid” convolution in

MATLAB. The function r

(x, y; O) contains the estimated

value of (X ∗ f

)(x, y). For each feature f

, the function r

uses the observed image O to estimate the value of the ﬁlter

response at each pixel. In the rest of this paper, r

(x, y; O)

will be shortened to r

for conciseness.

The exponent:

i=1

x,y



(X ∗ f

)(x, y) − r



(2)

can be written in matrix form by creating a set of matrices

. . . F

. Each matrix F

performs the same set of linear

operations as convolving an image with a ﬁlter f

. In other

words, if

X is a vector created by unwrapping the image X ,

then F

X is identical to X ∗ f

. These matrices can then be

stacked and Equation 2 can be rewritten as

X − r)

X − r), (3)

where

F =













and r =













Equation 3 can be rewritten in the standard form of the

exponent of a multivariate normal distribution by setting

the precision matrix, Λ

−1

, to be F

F and the mean, µ, to

F )

−1

Note that the difference between (

X − µ)

−1

(

X − µ)

and Equation 3 is the constant r

r − µ

µ. However, this

value is constant for all X, so it only affects the normaliza-

tion constant, Z, and the distributions deﬁned using Equa-

tion 1 or Λ and µ are identical.

2.2. Utilizing Weights

One of the weaknesses of the GCRF model in Equation

1 is that it does not properly capture the statistics of nat-

ural images. Natural images tend to have smooth regions

interrupted by sharp discontinuities. The Gaussian model

imposes smoothness, but penalizes sharp edges. In appli-

cations like image denoising, optic ﬂow, and range estima-

tion, this violates the actual image statistics and leads to

oversmoothing.

One way to avoid this oversmoothing is to use non-

convex, robust potential functions, such as those used in

[18] and [12]. Unfortunately, the convenience of the

quadratic model is lost when using these models.

Alternatively, the quadratic model can be modiﬁed by

assigning weights to the various terms in the sum. For ex-

ample, if the ﬁlters, f

. . . f

, include derivative ﬁlters, the

basic GCRF model penalizes the strong derivatives around

sharp edges. This causes the images estimated fr om basic

GCRF models to have blurry edges. Using the observed

image to lower the weight for those derivatives that cross

edges will preserve the sharpness of the estimated image.

It should be noted that this example is illustrative. Section

3 will show how use training data to learn the relationship

between the observed image and the weights.

Adding weights to the exponent of Equation 1 can be

expressed formally by modifying Equation 2:

i=1

x,y

(x, y; O, θ) ((X ∗ f

)(x, y) − r

)

, (4)

where w

(x, y; O, θ) is a positive weighting function that

uses the observed image, O, to assign a positive weight to

each quadratic term in Equation 4.

In the denoising example that is described in Section

5, the weight function associated with each ﬁlter is based

on the absolute r esponse of the observed image to a set of

multi-scale oriented edge and bar ﬁlters. These ﬁlters will

be shown later in Figure 2. These ﬁlters are designed to

be sensitive t o edges. The underlying idea is that these ﬁl-

ter responses will enable the system to guess where edges

occur in the image and reduce the smoothness constraints

appropriately.

Assigning a weight to each quadratic term improves the

model’s ability to handle strong edges. For a term based on

a derivative ﬁlter, the weight could be increased in ﬂat re-

gions of the image and decreased along edges. This would

enable the model to smooth out noise, while preserving

sharp edges. Section 3 will discuss how to learn the pa-

rameters, θ, of this weighting function.

The potential difﬁculty in using this model is the fact that

the weights must be computed f rom the observed image.

This will be problematic if the observed image is too noisy

or the necessary information is not easy to compute. Fortu-

nately, as Section 5 will show, t he system is able to perform

well with high-noise levels when denoising images. This

indicates that good weights can still be computed in high-

noise problems.

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