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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 2, FEBRUARY 2007 349

Kernel Regression for Image Processing

and Reconstruction

Hiroyuki Takeda, Student Member, IEEE, Sina Farsiu, Member, IEEE, and Peyman Milanfar, Senior Member, IEEE

Abstract—In this paper, we make contact with the ﬁeld of non-

parametric statistics and present a development and generalization

of tools and results for use in image processing and reconstruc-

tion. In particular, we adapt and expand kernel regression ideas

for use in image denoising, upscaling, interpolation, fusion, and

more. Furthermore, we establish key relationships with some pop-

ular existing methods and show how several of these algorithms,

including the recently popularized bilateral ﬁlter, are special cases

of the proposed framework. The resulting algorithms and analyses

are amply illustrated with practical examples.

Index Terms—Bilateral ﬁlter, denoising, fusion, interpolation,

irregularly sampled data, kernel function, kernel regression, local

polynomial, nonlinear ﬁlter, nonparametric, scaling, spatially

adaptive, super-resolution.

I. INTRODUCTION

HE ease of use and cost effectiveness have contributed to

the growing popularity of digital imaging systems. How-

ever, inferior spatial resolution with respect to the traditional

ﬁlm cameras is still a drawback. The apparent aliasing effects

often seen in digital images are due to the limited number of

CCD pixels used in commercial digital cameras. Using denser

CCD arrays (with smaller pixels) not only increases the pro-

duction cost, but also results in noisier images. As a cost-efﬁ-

cient alternate, image processing methods have been exploited

through the years to improve the quality of digital images. In

this paper, we focus on regression methods that attempt to re-

cover the noiseless high-frequency information corrupted by the

limitations of imaging systems, as well as the degradations pro-

cesses such as compression.

We study regression, as a tool not only for interpolation of

regularly sampled frames (up-sampling), but also for restoration

and enhancement of noisy and possibly irregularly sampled im-

ages. Fig. 1(a) illustrates an example of the former case, where

we opt to upsample an image by a factor of two in each direc-

tion. Fig. 1(b) illustrates an example of the latter case, where

an irregularly sampled noisy image is to be interpolated onto a

Manuscript received December 15, 2005; revised August 1, 2006. This work

was supported in part by the U.S. Air Force under Grant F49620-03-1-0387

and in part by the National Science Foundation Science and Technology Center

for Adaptive Optics, managed by the University of California at Santa Cruz

under Cooperative Agreement AST-9876783. The associate editor coordinating

the review of this manuscript and approving it for publication was Dr. Tamas

Sziranyi.

The authors are with the Electrical Engineering Department, University of

California Santa Cruz, Santa Cruz CA 95064 USA (e-mail: htakeda@soe.ucsc.

edu; farsiu@soe.ucsc.edu; milanfar@soe.ucsc.edu).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Software implementation available at http://www.soe.ucsc.edu/~htakeda.

Digital Object Identiﬁer 10.1109/TIP.2006.888330

high resolution grid. Besides inpainting applications [1], inter-

polation of irregularly sampled image data is essential for ap-

plications such as multiframe super-resolution, where several

low-resolution images are fused (interlaced) onto a high-reso-

lution grid [2]. Fig. 2 represents a block diagram representation

of such super-resolution algorithm. We note that “denoising” is

a special case of the regression problem where samples at all de-

sired pixel locations are given [illustrated in Fig. 1(c)], but these

samples are corrupted, and are to be restored.

Contributions of this paper are the following. 1) We describe

and propose

kernel regression as an effective tool for both de-

noising and interpolation in image processing, and establish its

relation with some popular existing techniques. 2) We propose a

novel adaptive generalization of kernel regression with excellent

results in both denoising and interpolation (for single or multi-

frame) applications.

This paper is structured as follows. In Section II, we will

brieﬂy describe the kernel regression idea for univariate data,

and review several related concepts. Furthermore, the classic

framework of kernel regression for bivariate data and intuitions

on related concepts will also be presented. In Section III, we

extend and generalize this framework to derive a novel data-

adapted kernel regression method. Simulation on both real and

synthetic data are presented in Section IV, and Section V con-

cludes this paper. We begin with a brief introduction to the no-

tion of kernel regression.

II. C

LASSIC KERNEL REGRESSION AND

ITS PROPERTIES

In this section, we formulate the classical kernel regression

method, provide some intuitions on computational efﬁciency,

as well as weaknesses of this method, and motivate the devel-

opment of more powerful regression tools in the next section.

A. Kernel Regression in 1-D

Classical parametric image processing methods rely on a spe-

ciﬁc model of the signal of interest and seek to compute the pa-

rameters of this model in the presence of noise. Examples of

this approach are presented in diverse problems ranging from

denoising to upscaling and interpolation. A generative model

based upon the estimated parameters is then produced as the

best estimate of the underlying signal.

In contrast to the parametric methods, nonparametric

methods rely on the data itself to dictate the structure of the

model, in which case this implicit model is referred to as a

regression function [3]. With the relatively recent emergence

of machine learning methods, kernel methods have become

well-known and used frequently for pattern detection and

discrimination problems [4]. Surprisingly, it appears that the

corresponding ideas in nonparametric estimation—what we

350 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 2, FEBRUARY 2007

Fig. 1. (a) Interpolation of regularly sampled data. (b) Reconstruction from irregularly sampled data. (c) Denoising.

Fig. 2. Image fusion often yields us irregularly sampled data.

call here kernel regression—are not widely recognized or used

in the image and video processing literature. Indeed, in the

last decade, several concepts related to the general theory we

promote here have been rediscovered in different guises, and

presented under different names such as normalized convolu-

tion [5], [6], bilateral ﬁlter [7], [8], edge-directed interpolation

[9], and moving least squares [10]. Later in this paper, we shall

say more about some of these concepts and their relation to

the general regression theory. To simplify this introductory

presentation, we treat the 1-D case where the measured data

are given by

(1)

where

is the (hitherto unspeciﬁed) regression function and

s are the independent and identically distributed zero mean

noise values (with otherwise no particular statistical distribu-

tion assumed). As such, kernel regression provides a rich mech-

anism for computing point-wise estimates of the function with

minimal assumptions about global signal or noise models.

While the speciﬁc form of

may remain unspeciﬁed, if

we assume that it is locally smooth to some order

, then in

order to estimate the value of the function at any point

given

the data, we can rely on a generic local expansion of the function

about this point. Speciﬁcally, if

is near the sample at ,we

have the

-term Taylor series

(2)

(3)

The above suggests that if we now think of the Taylor series

as a local representation of the regression function, estimating

the parameter

can yield the desired (local) estimate of the

regression function based on the data.

Indeed, the parameters

will provide localized information on the th deriva-

tives of the regression function. Naturally, since this approach

is based on local approximations, a logical step to take is to

estimate the parameters

from the data while giving

the nearby samples higher weight than samples farther away. A

least-squares formulation capturing this idea is to solve the fol-

lowing optimization problem:

(4)

where

is the kernel function which penalizes distance away

from the local position where the approximation is centered, and

the smoothing parameter

(also called the “bandwidth”) con-

trols the strength of this penalty [3]. In particular, the function

is a symmetric function which attains its maximum at zero,

satisfying

(5)

where

is some constant value.

The choice of the particular

form of the function

is open, and may be selected as a

Gaussian, exponential, or other forms which comply with the

above constraints. It is known [11] that for the case of classic

regression the choice of the kernel has only a small effect on

the accuracy of estimation, and, therefore, preference is given

Indeed the local approximation can be built upon bases other than polyno-

mials [3].

Basically, the only conditions needed for the regression framework are that

the kernel function be nonnegative, symmetric, and unimodal. For instance, un-

like the kernel density estimation problems [11], even if the kernel weights in

(6) do not sum up to one, the term in the denominator will normalize the ﬁnal

result.

TAKEDA et al.: KERNEL REGRESSION FOR IMAGE PROCESSING AND RECONSTRUCTION 351

to the differentiable kernels with low computational complexity

such as the Gaussian kernel. The effect of kernel choice for

the data-adaptive algorithms presented in Section III is an

interesting avenue of research, which is outside the scope of

this paper and part of our ongoing work.

Several important points are worth making here. First, the

above structure allows for tailoring the estimation problem to

the

local characteristics of the data, whereas the standard para-

metric model is generally intended as a more global ﬁt. Second,

in the estimation of the local structure, higher weight is given to

the nearby data as compared to samples that are farther away

from the center of the analysis window. Meanwhile, this ap-

proach does not speciﬁcally require that the data to follow a

regular or equally spaced sampling structure. More speciﬁcally,

so long as the samples are near the point

, the framework is

valid. Again this is in contrast to the general parametric ap-

proach which generally either does not directly take the loca-

tion of the data samples into account, or relies on regular sam-

pling over a grid. Third, and no less important as we will de-

scribe later, the proposed approach is both useful for denoising,

and equally viable for interpolation of sampled data at points

where no actual samples exist. Given the above observations, the

kernel-based methods are well suited for a wide class of image

processing problems of practical interest.

Returning to the estimation problem based upon (4), one can

choose the order

to effect an increasingly more complex local

approximation of the signal. In the nonparametric statistics lit-

erature, locally constant, linear, and quadratic approximations

(corresponding to

0, 1, 2) have been considered most

widely [3], [12]–[14]. In particular, choosing

, a lo-

cally linear, adaptive, ﬁlter is obtained, which is known as the

Nadaraya–Watson estimator (NWE) [15]. Speciﬁcally, this es-

timator has the form

(6)

The NWE is the simplest manifestation of an adaptive ﬁlter

resulting from the kernel regression framework. As we shall

see later in Section III, the bilateral ﬁlter [7], [8] can be

interpreted as a generalization of NWE with a modiﬁed kernel

deﬁnition.

Of course, higher order approximations

are also

possible. The choice of order in parallel with the smoothness

affects the bias and variance of the estimate. Mathematical ex-

pression for bias and variance can be found in [16] and [17], and,

therefore, here, we only brieﬂy review their properties. In gen-

eral, lower order approximates, such as NWE, result in smoother

images (large bias and small variance) as there are fewer degrees

of freedom. On the other hand, overﬁtting happens in regres-

sions using higher orders of approximation, resulting in small

bias and large estimation variance. We also note that smaller

values for

result in small bias and consequently large variance

in estimates. Optimal order and smoothing parameter selection

procedures are studied in [10].

Fig. 3. Examples of local polynomial regression on an equally spaced data set.

The signals in the ﬁrst and second rows are contaminated with the Gaussian

noise of SNR

=9[

]

and

6.5 [dB], respectively. The dashed lines, solid

lines, and dots represent the actual function, estimated function, and the noisy

data, respectively. The columns from left to right show the constant, linear, and

quadratic interpolation results. Corresponding RMSEs for the ﬁrst row experi-

ments are 0.0364, 0.0364, 0.0307 and for the second row are as 0.1697, 0.1708,

0.1703.

The performance of kernel regressors of different order are

compared in the illustrative examples of Fig. 3. In the ﬁrst ex-

periment, illustrated in the ﬁrst row, a set of moderately noisy

regularly sampled data are used to estimate the underlying

function. As expected, the computationally more complex

high-order interpolation

results in a better estimate

than the lower-ordered interpolators (

or ). The

presented quantitative comparison of the root-mean-square

errors (RMSE) supports this claim. The second experiment,

illustrated in the second row, shows that for the heavily noisy

data sets (variance of the additive Gaussian noise 0.5), the

performance of lower order regressors is better. Note that the

performance of the

and -ordered estimators

for these equally spaced sampled experiments are identical. In

Section II-D, we study this property in more detail.

B. Related Regression Methods

In addition to kernel regression methods we are advocating,

there are several other effective regression methods such as

B-spline interpolation [18], orthogonal series [13], [19], cubic

spline interpolation [20], and spline smoother [13], [18], [21].

We brieﬂy review some of these methods in this section.

In the orthogonal series methods, instead of using Taylor se-

ries, the regression function

can be represented by a linear

combination of other basis functions, such as Legendre polyno-

mials, wavelet bases, Hermite polynomials [10], and so on. In

the 1-D case, such a model, in general, is represented as

(7)

Variance of the additive Gaussian noise is 0.1. Smoothing parameter is

chosen by the cross validation method (Section II-E).

352 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 2, FEBRUARY 2007

Fig. 4. Comparison of the position of knots in (a) kernel regression and (b) classical B-spline methods.

The coefﬁcients are the unknown parameters we want

to estimate. We refer the interested reader to [13, pp. 104–107]

which offers further examples and insights.

Following the notation used in the previous section, the

B-spline regression is expressed as the linear combination of

shifted spline functions

(8)

where the

-order B-spline function is deﬁned as a times

convolution of the zero-order B spline [18]

where

else

(9)

The scalar

in (8), often referred to as the knot,deﬁnes the

center of a spline. Least squares is usually exploited to estimate

the B-spline coefﬁcients

The B-spline interpolation method bears some similarities to

the kernel regression method. One major difference between

these methods is in the number and position of the knots as illus-

trated in Fig. 4. While in the classical B-Spline method the knots

are located in equally spaced positions, in the case of kernel re-

gression the knots are implicitly located on the sample positions.

A related method, the nonuniform rational B-spline is also pro-

posed [22] to address this shortcoming of the classical B-Spline

method by irregularly positioning the knots with respect to the

underlying signal.

Cubic spline interpolation technique is one of the most pop-

ular members of the spline interpolation family which is based

on ﬁtting a polynomial between any pair of consecutive data.

Assuming that the second derivative of the regression function

exists, cubic spline interpolator is deﬁned as

(10)

where under the following boundary conditions:

(11)

all the coefﬁcients (

s) can be uniquely deﬁned [20].

Note that an estimated curve by cubic spline interpolation

passes through all data points which is ideal for the noiseless

data case. However, in most practical applications, data is

contaminated with noise and, therefore, such perfect ﬁts are no

longer desirable. Consequently, a related method called spline

smoother has been proposed [18]. In the spline smoothing

method, the afore-mentioned hard conditions are replaced with

soft ones, by introducing them as Bayesian priors which pe-

nalize rather than constrain nonsmoothness in the interpolated

images. A popular implementation of the spline smoother [18]

is given by

(12)

where

and can be replaced by either (8) or any orthogonal se-

ries (e.g., [23]), and

is the regularization parameter. Note that

assuming a continuous sample density function, the solution to

this minimization problem is equivalent to NWE (6), with the

following kernel function and smoothing parameter:

(13)

where

is the density of samples [13], [24]. Therefore, spline

smoother is a special form of kernel regression.

In Section IV, we compare the performance of the spline

smoother with the proposed kernel regression method, and, later

in Section V, we give some intuitions for the outstanding per-

formance of the kernel regression methods.

TAKEDA et al.: KERNEL REGRESSION FOR IMAGE PROCESSING AND RECONSTRUCTION 353

The authors of [9] propose another edge-directed interpola-

tion method for upsampling regularly sampled images. The in-

terpolation is implemented by weighted averaging the four im-

mediate neighboring pixels in a regular upsampling scenario

where the ﬁlter coefﬁcients are estimated using the classic co-

variance estimation method [25].

The normalized convolution method presented in [5] is very

similar to the classic kernel regression method (considering a

different basis function), which we show is a simpliﬁed ver-

sion of the adaptive kernel regression introduced in Section III.

An edge-adaptive version of this work is also very recently pro-

posed in [6].

We note that other popular edge-adaptive denoising or inter-

polation techniques are available in the literature, such as the

PDE-based regression methods [26]–[28]. A denoising exper-

iment using the anisotropic diffusion (the second-order PDE)

method of [26] is presented in Section IV; however, a detailed

discussion and comparison of all these diverse methods is out-

side the scope of this paper.

C. Kernel Regression Formulation in 2-D

Similar to the 1-D case in (1), the data measurement model

in 2-D is given by

(14)

where the coordinates of the measured data

is now the 2

1 vector . Correspondingly, the local expansion of the regres-

sion function is given by

(15)

where

and are the gradient (2 1) and Hessian (2 2)

operators, respectively, and

is the vectorization operator,

which lexicographically orders a matrix into a vector. Deﬁning

as the half-vectorization operator of the “lower-trian-

gular” portion of a symmetric matrix, e.g.,

(16)

and considering the symmetry of the Hessian matrix, (15) sim-

pliﬁes to

(17)

Then, a comparison of (15) and (17) suggests that

the pixel value of interest and the vectors

and are

(18)

(19)

As in the case of univariate data, the

s are computed from the

following optimization problem:

(20)

with

(21)

where

is the 2-D realization of the kernel function, and is

the 2

2 smoothing matrix, which will be studied more care-

fully later in this section. It is also possible to express (20) in a

matrix form as a weighted least-squares optimization problem

[10], [29]

(22)

where

(23)

(24)

(25)

with “diag” deﬁning a diagonal matrix.

Regardless of the estimator order

, since our primary in-

terest is to compute an estimate of the image (pixel values), the

necessary computations are limited to the ones that estimate the

parameter

. Therefore, the least-squares estimation is simpli-

ﬁed to

(26)

where

is a column vector with the ﬁrst element equal to one,

and the rest equal to zero. Of course, there is a fundamental

difference between computing

for the case, and

using a high-order estimator

and then effectively dis-

carding all estimated

s except . Unlike the former case,

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xuhuanzhuimeng

2013-09-12

对于学习机器学习，支持向量机等应用到核回归的，有一些启发。
不能毕业的小码农

2015-05-03

写的不错但是感觉处理速度有点慢 = = 自己不能直接用
qq_22191919

2015-12-14

很不错，正在学习
mingqianmm

2012-04-17

对于学习机器学习，支持向量机等应用到核回归的，有一些启发。
lifeissmile

2011-11-15

写得不错，有助于研究kernel regression 算法。可以边看文章，边理解代码。

gumuxiangyang

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