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SCIENCE CHINA

Technological Sciences

*Corresponding author (email: xcfeng@mail.xidian.edu.cn)

Progress of Projects Supported by NSFC

April 2013 Vol.56 No.4: 930–939

doi: 10.1007/s11431-013-5152-2

Two-direction nonlocal model for image interpolation

ZHANG XuanDe

1,2

, FENG XiangChu

, WANG WeiWei

& LIU GuoJun

Department of Applied Mathematics, School of Science, Xidian University, Xi’an 710071, China;

School of Mathematics and Computer, Ningxia University, Yingchuan 750021, China

Received July 22, 2012; accepted January 14, 2013; published online February 27, 2013

How to sufficiently exploit the self-similarity of natural images for image restoration has attracted extensive interest in the

field of image processing in recent years. In fact, the self-similarity implies two-direction similarity structures inherent in im-

ages, when a group of similar patches are rearranged to form a matrix, there exists similarity between both columns and rows

of this matrix. In this paper, we propose a two-direction nonlocal model (TDNL) to symmetrically exploit the two-direction

similarity structures in images, the model directly takes the similar patches as local adaptive dictionary to represent each patch

in the image and constrain the representation coefficients by Tikhonov regularization. TDNL can achieve the best results so far

and obtain significant gains over the existing methods, in terms of both peak signal to noise ratio (PSNR) measure and the vis-

ual quality when it is applied to the problem of image interpolation.

solar energy, image interpolation, image priors, two-direction similarity structure, two-direction nonlocal model

Citation: Zhang X D, Feng X C, Wang W W, et al. Two-direction nonlocal model for image interpolation. Sci China Tech Sci, 2013, 56: 930939, doi:

10.1007/ s11431-013-5152-2

1 Introduction

In engineering applications, the image formation process is

always modeled as

,

Ku n (1)

where f is the degraded observation, u is the unknown true

image, K is a linear operator modeling the degradation pro-

cess (e.g., blurring, down-sampling or the combination of

them, etc.), and n is the independent identically distributed

additive noise. The goal of image restoration is to estimate

the ideal image u from the degraded observation f, which is

a typical inverse problem.

Among the various image restoration problems, image

interpolation is an important instance, where K is a down-

sampling operator and n is assumed to be 0. With rapid de-

velopment of the hardware technology, the resolution of

display device is becoming higher and higher. For example,

currently the resolution of HDTV is up to 19201080, and

even Ipad2 can have a resolution of 1024768. However,

due to the limit of network bandwidth, often we can only

transmit image/video streams in a low resolution (LR).Then,

in the receiving end, the resolution of the received LR im-

age/video has to be enhanced in order to fit the resolution of

display devices. Therefore, image interpolation remains a

very important problem in the field of image processing.

Linear interpolators [1, 2] are commonly used in im-

age/video software and hardware products due to their rela-

tively low complexity. However, these interpolators were

designed on the basis of local smoothness assumption,

which results in the fact that they only prefer the smooth

region in images and may introduce jaggy artifacts around

edge and textures. With ever-increasing computation power

in the image processing field, more adaptive image interpo-

lation methods have been proposed in recent years.

Zhang X D, et al. Sci China Tech Sci April (2013) Vol.56 No.4 931

Edge-guided method is the representative category of

adaptive methods. This method first detects the local regu-

larity direction and then interpolates the missing pixels

along this direction [3‒13]. It shows certain improvements

over the linear interpolators, but identifying the local regu-

larity direction, which may be aliased during the subsam-

pling process (Figure 1), is a very challenging work. To

alleviate this problem, the authors of refs. [14, 15] proposed

to interpolate the image in multiple directions and then fuse

the directional interpolation results. Specifically, the multi-

ple results were fused by using minimum mean square es-

timation (MMSE) in ref. [14]. In ref. [15] the local regular-

ity in each direction was evaluated by the L1 norm of

wavelet coefficients lying in the elongated directional re-

gions and the fusing weights were estimated by solving a

weighted-L1 problem. Additionally, in ref. [16] the high

resolution (HR) image was reconstructed according to the

covariance matrix that was directly estimated from the LR

observations. Modeling the local contents of the image and

also learning the model parameters from the same local

content is another line of adaptive methods, in which both

autoregressive models [17, 18] and kernel regression model

[19] are the representatives.

Wavelets were also used in image interpolation. The idea

is to exploit the statistical similarity between different scales

of a wavelet-decomposed image. The interpolation is done

by predicting the HR details from the LR observation. Since

only the binary wavelet transform has the fast algorithm

(Mallat algorithm), the wavelet-based interpolators can only

be used to deal with the problem where the zooming factor

is a power of 2.

Regularization is the general method for image interpola-

tion. This method steers the process of HR image recon-

struction by enforcing the image priors in an optimization

problem. It was indicated in ref. [20] that enforcing several

different image priors in the same reconstruction process

has a good opportunity to achieve better performance.

In this paper, we employ the regularization method to

address the problem of image interpolation. The perfor-

mance of the regularization method mainly depends on the

Figure 1 The local regularity direction may be aliased during the sub-

sampling process. (a) Original image; (b) down-sampled image.

image priors involved. Self-similarity prior has become the

most important image prior in recent years, and the exploi-

tation of which has greatly enhanced the image restoration

performance. However, the concept of the self-similarity

has not been mathematically defined and is always intui-

tively used, what indeed do we mean by the phrase “self-

similarity” in image processing? In this paper, we assume

that the self-similarity implies the two-direction similarity

structures inherent in images. Here the two-direction simi-

larity means that, as a group of similar patches are arranged

in a matrix, there exists similarity among both columns and

rows of this matrix. Based on this assumption, we propose a

two-direction nonlocal model (TDNL) to symmetrically

exploit the two-direction similarity structures in images, the

model directly takes similar patches as a local adaptive dic-

tionary to represent each patch in the image and constrain

the representation coefficients by Tikhonov regularization.

The TDNL model can achieve the best results so far and

indicate significant gains over the existing methods, when it

is applied to the problem of image interpolation, in terms of

both peak signal to noise ratio (PSNR) measure and the

visual quality.

The rest of the paper is organized as follows. In Section 2,

we first present a classification and comparison of the ex-

isting image priors, then we analyze the two-direction simi-

larity structures inherent in images and formulate out the

two-direction nonlocal model. Section 3 discusses the solu-

tion algorithm of the model. Section 4 presents experi-

mental results and a comparison study. Conclusions are

given in Section 5.

2 Two-direction nonlocal model

2.1 Notations

We follow the style in refs. [21, 22] to define notations that

will be used in the following content. Denote the image u

 

[, ,, , ]

uu u u , where i indexes the pixel loca-

tion and

represents the total number of pixels. Let

be the operation that extracts the

ss patch centered

u from the image and vectorizes it into a



1s vector.

Let

C be the operation that extracts the group of patches

having a certain degree of similarity with

u , special-

ly,















,,,

iii i

Cu RuRu Ru

and each



,1,2,,

uj t

satisfies





M(,)

RuRu , where

M(,)

is the similarity

metric,



is the predefined similarity tolerance and t is

the number of patches extracted by

C . We call

Cu the

nonlocal matrix corresponding to

u (Figure 2). Note that

both

and

C can be represented as matrices, mean-

while, both

R and

CC are diagonal matrices.

The similarity metric is crucial for extraction of the non-

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