Single Image Super-Resolution using Gaussian Process Regression

Single Image Super-Resolution using Gaussian Process Regression
Single Image Super-Resolution using Gaussian Process Regression

Single Image Super-Resolution using Gaussian Process Regression

He He and Wan-Chi Siu

Department of Electronic and Information Engineering

The Hong Kong Polytechnic University

{07821020d,enwcsiu@https://www.360docs.net/doc/9117661960.html,.hk}

Abstract

In this paper we address the problem of producing a high-resolution image from a single low-resolution image without any external training set.We propose a framework for both magni?cation and deblurring using only the orig-inal low-resolution image and its blurred version.In our method,each pixel is predicted by its neighbors through the Gaussian process regression.We show that when using a proper covariance function,the Gaussian process regres-sion can perform soft clustering of pixels based on their local structures.We further demonstrate that our algo-rithm can extract adequate information contained in a sin-gle low-resolution image to generate a high-resolution im-age with sharp edges,which is comparable to or even supe-rior in quality to the performance of other edge-directed and example-based super-resolution algorithms.Experi-mental results also show that our approach maintains high-quality performance at large magni?cations.

1.Introduction

The goal of super-resolution(SR)is to estimate a high-resolution(HR)image from one or a set of low-resolution (LR)images.This inverse problem is inherently ill-posed since many HR images can produce the same LR image. SR methods can be broadly categorized into three classes as in[19]:interpolation-based methods,learning-based methods and reconstruction-based methods.Interpolation-based methods(e.g.,[9,13,21]are fast but the results are lack of?ne details.Reconstruction based methods (e.g.,[2,11,15,18,10])apply various smoothness pri-ors(e.g.,[1,5,19])and impose the constraint that when properly downsampled,the HR image should reproduce the original LR image.Alternatively,in learning-based meth-ods(e.g.,[14,7,4,22]),detailed textures are hallucinated by searching through a training set of LR/HR image or patch pairs.However,note that the training images need to be carefully selected,otherwise unexpected details can be found[6,12

].Figure1.(Adapted from[17])Graphical model of GP regression for image SR.Squares show the observed pixels and circles repre-sent the unknown Gaussian?eld.Inputs are neighbors(predictors) of the target pixel which is at the center of each3×3patch.The thick horizontal line represents a set of fully connected nodes.

In this paper we emphasize on super-resolution using only the LR input image without any external dataset or exemplar image.Previously,Fattal[6]proposed an up-sampling method based on a unique relationship between the gradient?elds of LR and HR images.Subsequently, Sun et https://www.360docs.net/doc/9117661960.html,ed the Gradient Pro?le Prior[19]learned from a large number of natural images to obtain sharp edges at large magni?cations.Glasner et al.[8]exploited the recur-rence of patches and constructed LR/HR training pairs by searching through an image pyramid.

Our contribution is two-fold.Firstly,we propose a framework to learn the reverse mapping from a LR image to the corresponding HR image pixelwise,relying on lo-cal structures de?ned by each pixel’s neighborhood.Pixels in the HR image are?rst estimated by their nearest neigh-bors(regressors)in an initial upsampled image via Gaussian process regression(GPR).The result is then deblurred by learning from LR/HR patches obtained from its downsam-pled version and the LR input.This self-directed learning is ready to be extended to image deblurring and denoising as well.Secondly,we demonstrate that GPR can serve as a soft clustering method for pixels based on the similarity de?ned by its covariance function.Thus it could also be

used in other SR algorithms to replace the nearest-neighbor based search.In our case,the covariance function enforces that a pixel is mainly estimated from training pixels em-bedded in similar environments,which results in signi?cant improvement for edge recovery.

In the rest of this paper,we?rst state our motivation in Section2.Then we give a brief overview of GPR in Section 3.In Section4,we describe our algorithm and give justi?-cations.Section5presents our results and the comparison to those produced by other methods.Finally,we conclude the paper with some discussion and future developments of the proposed algorithm.

2.Motivation

Our work is driven by the observation of structural re-dundancy in natural images.Previously,this self-similarity property has been exploited in image denoising by Non-Local Means[3],which was later generalized to a regular-ization method for inverse problems in[16].Recently,it has been statistically shown in[8]that small patches(e.g.,5×5) of edges,corners and textures tend to recur both within and across image scales.In our case,the similarity between two pixels is de?ned as the difference between their local ge-ometrical structures.For instance,pixels along the same edge tend to have similar neighborhoods whose intensities change fast in the direction perpendicular to that of the edge. Likewise,pixels in a smooth region have relatively invariant intensities within the neighborhood.Therefore,neighbors of a pixel indicate the local feature it is embedded in and can be used for prediction in a regression-based model.

GPR provides an elegant non-parametric Bayesian ap-proach for inference in the function space directly.By en-coding the high-level information that pixels having sim-ilar neighborhoods are strongly correlated,we essentially predict the target pixel from training examples whose local structures are similar to it(see Section4.2).Due to the ge-ometric duality[13]between LR and HR images,the input LR image serves as a suf?cient training dataset for extract-ing the local structural information.

3.Gaussian Process Regression

Gaussian Process(GP)de?nes a distribution over func-tion f,where f is a mapping from the input space X to R, such that for any?nite subset of X,its marginal distribution P(f(x1),f(x2),...f(x n))is a multivariate normal distri-bution,where x an input vector.In this section we give a brief review of GPR for implementation purpose;further details can be found in[17].

Gaussian Process is parameterized by its mean function m(x)and covariance function k(x i,x j)such that

f|X~N(m(x),K(X,X))(1)or equivalently,

f(x)~GP(m(x),k(x i,x j)).(2) where rows of the design matrix X are input vectors,f is a vector of function values and K(X,X)denotes the n by n matrix of covariances such that K ij=k(x i,x j).

In GPR we impose a GP prior on the target function value.Let y denote one observation with Gaussian noise and we have the GPR model

y=f(x)+ε,ε~N(0,σ2n).(3) The joint distribution of the training outputs y and the test outputs f?with a zero mean function is

y

f?

~N

0,

K(X,X))+σ2n I K(X,X?)

K(X?,X)K(X?,X?)

,

(4) where X and X?are design matrices for training data and test data respectively.Conditioning f?on the observation y, we can derive the predictive distribution

f?|X,y,X?~N(ˉf?,V(f?)),where(5)

ˉf

?

=K(X?,X)[K(X,X)+σ2n I]?1y,(6) V(f?)=K(X?,X?)?

K(X?,X)[K(X,X)+σ2n I]?1K(X,X?).(7) The notation K(X?,X),K(X,X?),K(X?,X?)for ma-trices of covariances are similar to the previous notation K(X,X).

Equation6shows that the mean prediction is a linear combination of the noisy observation y.Generally,GPR encodes the assumption that close inputs are likely to give similar outputs,hence examples similar to the test point are assigned higher weights.By specifying a proper covari-ance function,we can de?ne the similarity based on domain knowledge.

4.GPR for Super-resolution

In our regression-based framework,patches from the HR image are predicted pixelwise by corresponding patches from the LR image(Section4.1).GPR provides a way for soft-clustering of the pixels based on the local structures they are embedded in(Section4.2).Given the intuitive in-terpretation of hyperparameters of the covariance function in our case,we can optimize their values through MAP es-timation(Section4.3).

4.1.Single Image SR

Figure1shows a chain graph representation of GPR for image SR in our setting,where each3×3patch from the input image forms a predictor-target training pair.Thus in

(a)Training

targets(b)

Neighbors(c)Blurred HR

patch

(d)Training

targets(e)

Neighbors(f)Deblurred HR patch

Figure2.Single image SR framework.(a)and(d)are input LR

patches,in which the crossed points are target pixels for training

produced by simple grid sampling.(b)is the same patch as(a),

where neighbors are taken as the intensity values of the crossed

points’surrounding pixels.(e)is obtained by blurring and down-

sampling(c),the output of stage one.Neighbors are taken in the

same way as in stage one from(e).(f)shows the deblurred result

of the second stage.

Equation3,the observed y is the intensity of the pixel at the

center of a3×3patch and x is an eight-dimensional vector

of its surrounding neighbors.In order to adapt to different

regions of the image,it is partitioned into?xed-sized and

overlapped patches(e.g.,30×30),and the algorithm is run

on each of them separately.The patch-based results are then

combined to give the whole HR image.

We predict the HR image using a two-stage coarse-to-

?ne approach,which is consistent with the image formation

process.As shown in[2,18],the imaging process can be

modeled as a degradation process of the continuous spatial

domain depending on the camera’s Point Spread Function

(PSF).After discretization,this process can be further ex-

pressed as

L=(f?H)↓d,or equivalently,(8)

?H=f?H and L=?H↓d(9)

where L and H denote the LR and HR image respectively,

?H denotes the blurred HR image,f is the blur kernel and↓d

denotes the downsampling operator with a scaling factor d.

As a reasonable inversion of the degradation process,in the

?rst stage we intend to upsample the LR image without loss

of details and obtain?H.In the second stage we use L and

its blurred version?L(downsampled from?H)to simulate

the blurring process,thus re?ne the estimates to recover H.

The above ideas can be implemented by Algorithm1.

Figure2gives a real example of the resolving process.

Figure2(a),2(b)and2(c)show the?rst stage,where both

the training targets sampled in Figure2(a)and their neigh-

Algorithm:SRGPR(L)

?H←Upsample(L)

?L←?H↓d(Blur and downsample)

H←Deblur(?H,L,?L)

return H

Function:Upsample(L)

Bicubic interpolation:H b←L↑d

Partition L into n overlapped patches P1,...,P n

for p L=P1,...,P n do

Sample pixels in p L to obtain the target vector y

Put the eight neighbors of each element of y in

X NL as a row vector for training

Train a GPR model M using{y,X NL}

Put the eight neighbors of each pixel of H b in

X NH

b

as a row vector for prediction

p?

H

←M(X NH

b

)

end

return?H constructed from p?

H

Function:Deblur(?H,L,?L)

Partition?L into n overlapped patches P1,...,P n

corresponding to those in L

for p?

L

=P1,...,P n do

Obtain the same target vector y in p L

Put the eight neighbors in p?

L

of each element of

y in X

N?L

as a row vector for training

Train a GPR model M using{y,X

N?L

}

Put the eight neighbors of each pixel of?H in

X

N?H

as a row vector for prediction

p H←M(X

N?H

)

end

return H constructed from p H

Algorithm1:Super-resolution using GPR.

bors in Figure2(b)come from the LR input patch.Neigh-

bors for predicting pixels in the HR image are obtained by

upsampling Figure2(a)using bicubic interpolation.Fig-

ure2(d),2(e)and2(f)show the second stage,where the

training targets2(d)are the same as those in the?rst stage,

while the neighbors come from the blurred patch(Fig-

ure2(e)).Figure2(f),the deblurred result shows sharper

edges than Figure2(c)from stage one.

4.2.Covariance Functions

The covariance function plays the central role in GPR

as it encodes our assumptions about the underlying process

by de?ning the similarity between functions.We model the

image as a locally stationary Gaussian Process and choose

the squared exponential covariance function:

k(x i,x j)=σ2f exp

?

1

2

(x i?x j) (x i?x j)

2

,(10)

(a)Test

point(b)Training

patch(c)Covariance matrix

Figure3.Covariance Function.(b)is a patch from the LR input image and(a)is its corresponding patch from the upsampled image (bicubic interpolation).The crossed point in(a)is a test point x?to be estimated from the traning set X in(b).(c)shows its covariance matrix,where each pixel’s intensity value is proportional to the covariance between x?and the training target at the same location in(b),evaluated by the covariance function,Equation10

.

whereσ2

f represents the signal variance and de?nes the

characteristic length scale.Therefore,in our case similar-ity is based on the Euclidean distance between two vectors of the eight neighboring pixels’intensities,since the differ-ence of intensity values around the target pixel indicates its location(e.g.edges or smooth regions)and the direction of the edge if it is an edge-pixel.

Figure3(c)shows the covariance matrix K for one edge pixel x?(the crossed point)in Figure3(a)predicted by training points X in the LR training patch,where K ij= k(x?,X ij)and i,j correspond to pixel indices in Fig-ure3(b).Observe that high responses(red regions)from the training patch are largely concentrated on edges,which justi?es our choice of the covariance function.In addition, it is noted that high-responsive regions do not only include points on the same edge as the test point,but also other sim-ilar edges within the patch.Thus,the process can capture both local and global similarity within a patch.In general, pixels embedded in a similar structure to that of the tar-get pixel in terms of the neighborhood tend to have higher weights during prediction.

4.3.Hyperparameter Adaptation

So far,we have taken the parameter values in the co-variance function to be given.Here we explore the effect of varying these hyperparameters and illustrate how to set them through optimization.

In Bayesian statistics,hyperparameters refer to parame-ters in the prior distribution,which in our case are the sig-nal variance,σ2

f

,the characteristic length scale, in Equa-tion10and the noise variance,σ2n in Equation3.While sig-nal variance and noise variance are relatively self-explicit, the characteristic length scale can be regarded as parame-ters controlling the number of upcrossings of a level u in a one-dimensional GP.Assuming a zero mean function,it is shown in[17]that the characteristic length scale is

in-

(a)Test

point(b)Training

patch(c) =.50,σn=

.01

(d) =.05,σn=

.001(e) =1.65,σn=.14

Figure4.Hyperparameter adaptation.(a)is the upsampled patch (bicubic interpolation)used for prediction.(b)is the original low-resolution patch used for training.(c)to(e)show the contour plots of the weight matrix given the hyperparameters listed below each image.The hyperparameters used in(c)is obtained by MAP esti-mation while the other two sets of hyperparameters are chosen by hand for comparison.The signal varianceσ2f is?xed as0.162for all three settings.

versely proportional to the mean number of level-zero up-crossings,given the squared exponential covariance func-tion.

Figure4gives an interpretation of the hyperparameters in terms of the weighted linear combination of training tar-gets,y,in Equation6.Figure4(a)and Figure4(b)are the upsampled patch(bicubic interpolation)and the LR input patch respectively.To predict the crossed point as shown in Figure4(a),we use the mean of the predictive distri-bution(Equation6).Let the n by1weight vector w be K(X?,X)[K(X,X)+σ2n I]?1.The linear predictor can then be written as w T y.We reshape the vector w into a square matrix W and plot its contour.Each of W’s en-try is the weight of a corresponding training point in Fig-ure4(b).The signal variance is set by optimization of the marginal likelihood(further explained in subsequent para-graphs),which is the same in all three settings.In Figure 4(c),the hyperparameters are obtained by the MAP esti-mation.Figure4(d)shows the effect of a shorter length scale with low noise and Figure4(e)shows the result of a longer length scale with high noise.We can see that in Figure4(d)the contour map has several peaks,indicating a small number of highly correlated training points.In con-trast,the longer length scale in Figure4(e)over-smoothes the signal and results in large high-responsive regions.Gen-erally,this phenomenon stands for two possible views of a given signal:the former model assumes a quickly-varying ?eld and?ts to high-frequency details that could be noise; the latter model assumes a slowly-varying?eld thus some observed genuine data could be ignored.A non-optimal so-

lution either results in high-frequency artifacts in the image or a blurry image.

In Bayesian model selection,we usually use hierarchical models and impose a hyper-prior p(θ|H)over the hyperpa-rametersθ.Then the posterior can be expressed as follows:

p(θ|X,y,H)=

p(y|X,θ,H)p(θ|H)

p(y|X,θ,H)p(θ|H)dθ

.(11)

Depending on the model,the evaluation of the normalizing constant in Equation11may be intractable,thus calls for variational methods to approximate or Markov chain Monte Carlo(MCMC)methods to sample from the posterior.Al-ternatively,in the context of image SR we can take advan-tage of the above intuitive interpretation of the hyperparam-eters and solve the problem through MAP estimation.

In our model,the marginal likelihood is

p(y|X)=

p(y|f,X)p(f|X)d f(12)

Given the likelihood y|f~N(f,σ2n I)(Equation3)and the GP prior over the latent function f(Equation1),this integral can be analytically solved,giving rise to the log marginal likelihood

log p(y|X,θ)=?1

2

y T K?1y y?

1

2

log|K y|?

n

2

log2π,

(13)

where K y=K(X,X)+σ2n I,representing the covariance matrix of the noisy observation y.The optimal solution θ?can be obtained by gradient descent with the following partial derivative

?L ?θi =

1

2

y T K?1

?K

?θi

K?1y?

1

2

tr(K?1

?K

?θi

),(14)

where L refers to the above log marginal likelihood.In our experiments,satisfactory results can be obtained by initial-

izing at0.223,σn at0.05andσ2

f to be the variance of the

training targets.

We conclude this section by giving the following ad-vantages of our patch-based GP regression for image SR. Firstly,recurring patterns(3×3neighborhood)in the LR patch can be captured for ef?cient training.Secondly,dif-ferent image regions(e.g.,30×30patches)can have a cus-tomized set of hyperparameters to adapt to its global https://www.360docs.net/doc/9117661960.html,stly,the O(N3)time complexity is reduced to O(Mn3),where M is the number of patches which grows linearly with the image size N and n is the number of train-ing points in one patch which is a constant.

5.Experiments

Implementation:We test our method on a variety of images with a magni?cation factor from2to10.In

most

(a)

GPP(b)Our result

Figure6.Super-resolution(8×)comparison of Gradient Pro?le Prior(GPP)reconstruction[19]and our

methods.

(a)10×our

result(b)10×detail synthesis Figure7.Super-resolution result of direct magni?cation(3×)and 10×comparison with the result from[20].

of our experiments we began by using the original image as LR input and magnify it with a scale factor of2.For further magni?cation,we used the previous output image as the input LR image and solved its HR image.In the second stage of our method,we made use of the original LR image for deblurring,which also serves as a constraint that when downsampled,the HR image should reproduce the LR im-age.We set the patch size to20×20for the original LR image and increase it according to the magni?cation factor in later magni?cations.For each patch,approximately2/3 area is overlapped with neighboring patches.After running our algorithm on each patch,we combined them by linear smoothing in the overlapping areas.When processing color images,we worked in the YIQ color space and only applied our SR method to the illuminance(Y)channel since hu-mans are more sensitive to illuminance changes.The other two chrominance channels(I and Q)are simply upsampled using bicubic interpolation.The three channels were then combined to form our?nal HR image.

Results:In Figure5,we compare our method with the bicubic interpolation and the Gradient Pro?le Prior recon-struction[19].The bicubic interpolation introduces ringing and jaggy artifacts,for example along edges on the wing

(a)Bicubic

(MSSIM=0.84)(b)GPP

(MSSIM=0.84)(c)Our result

(MSSIM=0.86)(d)Ground truth

Figure5.Super-resolution(3×)comparison of bicubic interpolation,Gradient Pro?le Prior(GPP)reconstruction[19]and our methods. The MSSIM scores with respect to the ground truth are listed below each

result.

(a)

Bicubic(b)Imposed edge

statistics(c)Patch

redundancy(d)Our result

Figure8.Super-resolution(4×)comparison of bicubic interpolation,reconstruction with imposed edge statistics[6],single image super-resolution using redundant patches across scales[8]and our method.

(region in the blue-bordered rectangle).The GPP recon-

struction alleviates the artifacts yet the overall result is still

blurry,as shown in the red-bordered rectangle.Our re-

sult generates sharp edges with rare artifacts and is most

close to the ground truth,in terms of both subjective visual

evaluation and the Mean Structural Similarity(MSSIM)in-

dex[23],which assesses the image quality from the per-

spective of image formation.Figure6further shows a com-

parison between GPP[19]and our methods under8×mag-

ni?cation.In the enlarged rectangle region,we can see that

the result of GPP is less realistic while our result shows

clear carved curve of the eye.

To further inspect the effectiveness of our method,we

show a comparison of our large10×SR result to that of

the single image detail synthesis with edge prior method

from[20]in Figure7(a)and Figure7(b).With the exam-

ple texture,the detail synthesis approach can produce real-istic details which are usually lost under large magni?ca-tion.Our results,however,demonstrate comparable quality in terms of the sharpness of edges,for example in regions of the antennae and the head.

Figure8compares our method with bicubic interpo-lation,the imposed edge statistics approach[6]and the single image SR approach using patch redundancy across scales[8].The image of the old man’s face has vari-ous edges of different scales.The imposed edge statistics method produces some unrealistic edges as in the case of Figure6(a).For instance,the forehead wrinkles show lin-eation effect.In the image of the chip connector,the long edge of the side is approximately constructed by only one pixel in the input LR image.Both the imposed edge statis-tics approach and the single image SR approach have stair-case or ringing artifacts along the major edge.By capturing the pixels’local structures and searching similar predictors globally in a patch,our method produces sharp edges with the least artifacts.

Figure9shows more results using our method.Images on the left are LR inputs shown by nearest-neighbor interpo-lation and images on the right are HR images with a mag-ni?cation factor of8.The results show that our method can produce sharp edges reliably at high magni?cation fac-tors.In Table6we give objective measurements(RMS and MSSIM[23])to evaluate our method in comparison to bicu-bic interpolation and New Edge Directed Interpolation[13] (NEDI).We notice that the performance of NEDI deterio-rates fast when the magni?cation factor is larger than2.The measurement shows that our method outperforms bicubic interpolation and NEDI with lower RMS error and higher MSSIM index.

Noisy input:In Figure10,we demonstrate that our SR method is resistant to minor noise in the input LR image. For noisy input,we setσn according to the noise level in the LR image and optimizeσf and by the MAP estimation, such that the model can correctly interpret noisy signals.We can see that in Figure10(b),most of the noise are smoothed out.

6.Discussions and Conclusions

In this paper we present a novel method for single image super-resolution.Instead of imposing sophisticated edge priors as in[6,19],we use a simple but effective covariance function to capture the local structure of each pixel through the Gaussian process regression.In addition,by using a non-parametric Bayesian model,the prediction process is able to adapt to different edges in the local image patch instead of setting a general parameter learnt from a large number of images.The experimental results show that our method can produce sharp edges with minimal artifacts.We propose a two-stage framework,where the?rst stage

aims

(a)

Input(b)Our results

Figure9.More Super-resolution(8×)results.

to recover an HR image based on the structural information extracted from the LR image and the second stage intends to re?ne the obtained HR image by learning from a training set constructed by the LR image itself.In comparison to the general training set as used in[7,24],our results capture most of the details presented in the LR image and the visual quality is comparable to some example-based methods(see Figure7(a)).

Though presented in the context of super-resolution,our framework of self-directed learning can be extended to

(a)Noisy LR

input(b)Our result

Figure10.Super-resolution(3×)result of noisy low-resolution image.

Image

RMS MSSIM

BC NEDI Our BC NEDI Our

Lena10.2511.129.650.800.790.82 Daisy18.2117.9917.020.830.830.85 Tower10.6911.699.680.760.750.79 Table 1.RMS errors and MSSIM scores of different super-resolution(4×)methods.BC stands for bicubic interpolation. NEDI stands for new edge-directed interpolation[13].

other image reconstruction scenarios such as denoising and deblurring by modifying the covariance function and the training pairs.Finally,the performance may be further im-proved through using more complex covariance function to include extracted features and to express explicitly the local structure of a pixel.

Acknowledgement

This work is supported by the Center for Signal Process-ing(G-U863),the Hong Kong Polytechnic University and the Research Grant Council of the Hong Kong SAR Gov-ernment(PolyU5278/08E:B-Q14F).

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