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Single Image Super-resolution with DetailEnhancement based on Local Fractal Analysis of

GradientHongteng Xu, Guangtao Zhai, Member, IEEE, and Xiaokang Yang Senior Member, IEEE

Abstract— In this paper, we propose a single image super-resolution and enhancement algorithm using local fractal anal-ysis. If we treat the pixels of a natural image as a fractal set,the image gradient can then be regarded as a measure of thefractal set. According to the scale invariance (a special case ofbi-Lipschitz invariance) feature of fractal dimension, we will beable to estimate gradient of high-resolution image from that ofthe low-resolution one. Moreover, the high-resolution image canbe further enhanced by preserving the local fractal length ofgradient during the up-sampling process. We will show thata regularization term based on the scale invariance of fractaldimension and length can be effective in recovering details ofhigh resolution image. Analysis will be given on the relation anddifference among the proposed approach and some other state ofthe art interpolation methods. Experimental results show that theproposed method has superior super-resolution and enhancementresults as compared to other competitors.

Index Terms— Super-resolution; enhancement; gradient; frac-tal; bi-Lipschitz invariance

I. INTRODUCTION

Single image super-resolution is a classic image processingproblem having both academic and industrial values. Super-resolution can be viewed as a scale-enhancement technique,namely to estimate a high-resolution image from a low-resolution one. The basic type of super-resolution is probablythe interpolation-based algorithm. Currently, bi-linear and bi-cubic [13] interpolation algorithms are still the most widelyused methods in practice. Besides these classic solutions, manyother interpolation methods, such as the auto-regression basedalgorithm in [38] and ICBI algorithm in [8], are proposedrecently. These interpolation methods are based on parametricmodels and are found to be quite effective in suppressing inter-polation artifacts. Besides parametric models, non-parametricmodel are also applied. A super-resolution method usingkernel regression is proposed in [28].

Another type of super-resolution method is example-based,with the underlying assumption that inheritance relation-ship exists between patches from low-resolution and high-resolution images. Therefore, by synthesizing high-resolution

Hongteng Xu, Guangtao Zhai and Xiaokang Yang(correspondence au-thor) are with the institute of image communication and information pro-cessing, Shanghai Jiao Tong University, Shanghai, China, 200240. Email:hongtengxu/zhaiguangtao/xkyang@sjtu.edu.cn. This work was supported inpart by NSFC (60932006, 61025005, 61001145), RFDP (20090073110022)and the 111 Project (B07022).

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patches matching to the low-resolution ones, the high-resolution image can be reconstructed. The example-basedmethod is first proposed in [7] and later is developed in [27],[37], [36], [9], [19], [11], [6]. In [7], [36], [37], [11], thehigh-resolution example patches are learned from a large dataset of high quality images. However, the inherent universalityassumption of image patches can be problematic for differenttype of contents. Although this type of algorithms usuallyworks well for some specific class of images, understandably,their overall performance depends heavily on the choice ofthe exemplary data set. For example, obvious artifacts can beobserved in the results of [11] when image content does notmatch with the exemplary data set. To reduce the dependenceon contents of the exemplary set, self-example-based methodis proposed in [19], [9], [6], where data set are formedby resizing the original image into different scales. Due tothe searching and fractional interpolation steps incurred, thecomputational complexity of [19], [9], [6] is high.

Towards better performances, many reconstruction-basedmethods have been proposed over the years [14], [24], [10],[39], [15], [4], [25], [23], [26], [16]. Generally, the degrada-tion of image is modeled by point diffusion function (PDF)in these reconstruction-based methods. The initial estimatesfrom interpolation-based algorithms are combined with someprior knowledge to get the final high-resolution results. Back-projection [14] or convex projections (POCS) [24], [10] canbe used to estimate high-resolution images. According to thedifference among those prior assumptions, various regulariza-tion terms can be used, such as the local gradient enhancementterm in [4], [26], [25] and the global gradient sparse constraintin [23], [16]. One noticeable advantage of reconstruction-based methods is that they can be easily integrated with othercommon image processing tasks, such as de-noising [21], de-convolution [18], enhancement [4], [5] and so on.

As aforementioned, regularization on image gradient is acommon approach to image super-resolution [4], [25], [26],[17], [18], [23]. In [17], [18], [23], global statistical distribu-tion of the gradients is modeled as Laplacian. In [4], [25], thegradient of local patch is assumed to obey general Gaussiandistribution and the super-resolution result is generated byraising the sharpness parameter. All of existing gradient reg-ularization type of algorithms are derived from the statisticalpoint of view. An obvious drawback of those statistical typeof methods is that the configuration of model parameters isempirical, i.e. it depends on the training samples.

Being quite different from those previous methods, in this

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paper, we look into the super-resolution problem from ageometrical viewpoint and propose a joint super-resolutionand enhancement algorithm based on local fractal analysisof image gradient. Our approach is based on the observationthat most natural images can be represented as a fractal setand image gradient is a natural measure of the set. Thefundamental of our approach is the invariance of bi-Lipschitztransform of fractal dimension, which suggests that the up-sampling processing does not change the fractal dimension ofan image. This assumption provides an effective yet practicalinter-scale prior knowledge for natural image up-sampling. Itensures that the reconstructed high-resolution image has thesame local fractal dimension with the low-resolution one.

Because fractal is an efficient model of texture, fractalanalysis has been used as an effective texture descriptor intexture segmentation and classification [30], [32], [33], [35],[34]. By introducing fractal analysis into the super-resolutionalgorithm, the texture part of our super-resolution result is freefrom both over-smoothness and other artifacts such as ringingeffect and etc. Another advantage of the proposed methodis that it is directly extendable to image enhancement byincorporating a scale-invariance constraint on the local fractallength. Therefore, our proposed fractal based approach can bethought of as not only a scale-enhancement method but alsoa detail-enhancement technique.

The rest of the paper is organized as follows: SectionII gives a brief review of fractal and introduces the imagemodel based on local fractal analysis on image gradient. Theproposed super-resolution and enhancement algorithms aredetailed in Section III. Section IV gives experimental resultsand Section V concludes the paper.

II. FRACTAL-BASED IMAGE MODEL

A. A Brief Review of FractalA typical fractal can be constructed by the following strat-

egy. Firstly, we decompose a geometry G into N similarcopies of itself, each scaled down by a factor s. The similardecomposition process can be applied on each of the N copies.Then by repeating the similar decomposition infinitely, we canget a fractal F, as exemplified in Figure 1. Generally, F is a”Mathematical monster” which is unable to be measured inthe same measure space of G. To explicitly analyze fractal,Mandelbrot [20] firstly proposed the concept of fractal dimen-sion based on Hausdorff measure. The principle of fractaldimension is about a power law based self-similarity thatobjects measurements obey at various scales. To the fractalF mentioned above, there always exists a power law betweenthe quantities N and s, i.e. N(s) ∝ sD. Here D is definedas fractal dimension, which is larger than the topologicaldimension of F. Note that the definition of fractal dimensionhere is not mathematically strict but it suffices our lateranalysis. Interested readers are referred to [20], [2] for morecompleted theory of fractal.

Although perfect fractal does not exist in nature, fractalanalysis still has a wide range of applications in many fields.After extending the self-similarity from a strict geometricalconcept to a statistical one, most natural phenomenons can bemodeled as fractal over a range of scales.

Fig. 1. An illustration of the construction process of fractal. By replacingthe central part of line with a wedge-shaped fold line iteratively, we get thefamous fractal, Von Koch curve. Under conventional measure space, it hasinfinite length with zero area. The fractal dimension is logN

− log s= 1.262.

B. Fractal Analysis in Practical Applications

The fractal analysis of image is about describing the featureof image using the fractal dimension, which is intrinsically de-termined by the measure of fractal. Because direct calculationof fractal dimension using the Hausdorff measure is difficult,the concept of fractal dimension has been generalized andadapted for natural image descriptions. Many different variantshave been proposed in [22], [30], [32], [33], [35], [34].

For a natural image f(x), x ∈ R2 is the coordinate of theimage and the union of all the x is a set denoted as X. In[32], [33], [35], [34], X can be seen as a union of fractal setsdirectly. Each fractal set is composed of points having the sameintensity value, and the measure of each set is the numberof points. The fractal dimension of each set can be takentogether to form a multi-fractal-spectrum (MFS) vector. TheMFS vector is an illumination and projective invariant featuredescriptor of texture, which is suitable to give a global analysison texture image. Moreover, the MFS vector can be collectedunder different measures [34] for higher description power.On the other hand, natural image can also be modeled as anintensity surface (X, f(x)), in which X is regarded as a fractalset while f(x) is a measure of the fractal set. Being differentfrom the MFS vector, this model gives a local fractal analysisof image and the fractal dimension is calculated locally basedon pixel or patch of image.

The fractal based image model has been widely used intexture analysis, e.g. using the histogram of all the local fractaldimensions as a feature, image segmentation and textureclassification were conducted respectively in [22] and [30].However, some problems are still open for future study. Oneproblem is that those existing variants of fractal dimension caneasily cause confusion of the concept. For example, the localfractal dimension in [30] is the local density function in [33],which is used to segment image into several point sets. In fact,the density of fractal has a strict definition in [2], which hasa close relationship to fractal dimension. This point will beclarified later in our work.

Another problem with existing fractal based methods is thatthe model has been almost only used in texture analysis. Ex-cept the work in [22], [1], all methods mentioned above focuson textures. Although in [22], [1], fractal analysis is used inimage segmentation and compression, the performances of thecorresponding methods are unsatisfactory. As a consequence,some improvements are needed to use the fractal model inother image processing applications.

First, the local fractal analysis should be taken more seri-

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ously. Because texture usually shows global self-similarity, thefeature of texture can be described sufficiently well by globalfractal analysis. However, besides textures, nature images alsocontain large smooth and edge regions, which suggests themeasure of nature image have local adaptivity. Therefore, localfractal analysis method may be more suitable to nature imagethan the global analysis for natural image representation,analysis and compression [1].

Second, the selection of fractal measure should be deter-mined according to the character of application. Althoughthe intensity surface of image has been a popular choice intexture classification, there is no evidence showing that it isthe optimal measure in other applications. In fact, some othermeasures based on the gradient or the Laplacian result ofimage have been introduced in texture description in [33], [34]and achieved better classification results. To super-resolutionproblem, compared with image intensity, image gradient is amore significant feature that directly reflects the degradationof textures and edges [4], [25], [17], [18], [23]. The inter-scalechange of gradient is critical to the reconstruction of imagedetails. So, we think gradient is a more suitable measure offractal set in super-resolution problem.

According to the analysis above, we will apply local fractalanalysis in the image gradient domain and then design asuper-resolution and enhancement algorithm based on fractalanalysis.

C. Local Fractal Analysis of Image

In our work, image f(x) is modeled as (X, µ), where µ isa measure supported on fractal set X. Based on the theory offractal analysis, if X is a fractal set with dimension D, thedensity of x ∈ X under the measure µ is

d(x) = limr→0

µ(Br(x))

(2r)D. (1)

Here Br(x) is a ball centering at x with |B| = 2r, andµ(Br(x)) is the measure supported on Br(x). To a digitalimage, the density of point is constant in the whole image, sowe have a power law µ(Br(x)) ∝ (2r)D, and the local fractaldimension is

D(x) = limr→0

logµ(Br(x))

log 2r. (2)

According to the analysis in section II.B, we use the gradientof image as the measure µ in our work. The gradient of imagef(x) is defined as follows.

grad(f) =√

(∇xf)2 + (∇yf)2, (3)

where ∇x, ∇y are differential operators along x- and y-direction respectively. So, the measure of Br(x) is calculatedas

µ(Br(x)) =

∫Br(x)

Gr ∗grad(f)

‖grad(f)‖+ εdx. (4)

Here ε is a small constant which ensures the denominator tobe larger than zero. Gr is a Gaussian kernel, which is similarto the definition in [33], as follows.

Gr =1√

2πrσe

x2

r2σ2 . (5)

σ is a predefined constant.In digital situation, we can not calculate D from (2) directly.

The estimation of D can be gotten by linear regressionbased on sample points (r, µ(Br(x))) in log− log coordinatesystem. With different r, µ(Br(x)) can be calculated byGaussian filtering. The detail of the method as the schemeof Algorithm 1 shows.

Algorithm 1: Estimation of Fractal Dimension0. Given f(x), calculate grad(f).1. For r=1:R

Calculate Gr according to r.Based on (4), get µ(Br(x)) by convoluting Gr with grad(f).

2. To samples {(r, µ(Br(x)))}Rr=1, estimate D and L byminD,L

∑Rr=1 | log µ(Br(x))−D(x) log 2r − L(x)|2.

According to Algorithm 1, D is estimated as the slope offollowing linear function

logµ(Br(x)) = D(x) log 2r + L(x). (6)

Here D(x) is fractal dimension of Br(x) under the measureµ. eL(x) is the D-dimensional fractal value of µ when 2r = 1,which is interpreted as the D-dimensional fractal length underµ in [30]. It measures the size of fractal in a ball with unitsize. Because L(x) 7→ eL(x) is a bijection, in our work wedenote L(x) as local fractal length for convenience.

The most significant property of fractal dimension is itsinvariance to bi-Lipschitz transform (A mapping g : X 7→ Rm

is bi-Lipschitz transform if and only if there exists 0 < c1 6c2 6 ∞ so that c1‖x − y‖ 6 ‖g(x) − g(y)‖ 6 c2‖x − y‖,(x,y ∈ X).). Generally, the geometrical transforms includingrotation, affine, projection, and so on belong to bi-Lipschitztransform. To down- and up-sampling processing with factorn, we can get more meaningful conclusions because the down-and up-sampling are special cases of bi-Lipschitz transform,where c1 = c2 = n.

Theorem: Denote a signal (an image) as f(x), x ∈ X, theoriginal fractal dimension and length in point x are D and L.If we define a sampling function g(x) = nx, where n > 1means up-sampling while n < 1 means down-sampling, thenthe fractal dimension and length in point g(x) are

Dg = D, Lg = L−D log n. (7)

Proof: Given a point x, we have logµ(Br(x)) = D log 2r+L. Based on the definition of g(x), we know that g is invertibleand ‖g(x) − g(y)‖ = n‖x − y‖. The bi-Lipschitz constraintensures that Br(g(x)) = Br/n(x),

So we have

logµ(Br/n(x)) = D log 2r/n+ L

⇒ logµ(Br(g(x))) = D log 2r + L−D log n

⇒ Dg = D, Lg = L−D log n.

�According to the theorem above, the fractal dimension of

image should be scale-invariant. Theoretically, if we up-samplea low-resolution image fl with factor n, the correspondinghigh-resolution image fh has the same fractal dimension withfl and vice-versa. However, the practical situation is not ideal

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and the fractal dimension of interpolation result is differentfrom the ground truth. Figure 2 gives an illustration of thisphenomenon. We enlarge the image 2, 4, 8 times respectivelyby bi-cubic interpolation and calculate the local fractal dimen-sions of different patches. We find that the fractal dimensiondescends with the increase of the up-sampling factor, whichcan be seen from the decrease of the slope of regression result.

Fig. 2. Sample the patches and calculate local fractal dimensions accordingto the slope of regression lines. The red, green and blue regions in the imagecorrespond to the edge, smooth and texture regions respectively. With theincrease of the sampling factor n, the fractal dimension D reduces.

In our test, the decrease of fractal dimension is an universalphenomenon and holds for edges, smooth and texture regionsof images. This observation indicates that although the pointset X is up-sampled, the intensity of f(X) is not interpolatedaccurately and the measure value µ of X is also not accurateenough. As a matter of fact, traditional interpolation methoddoes not guarantee the invariance of fractal dimension. An-other important conclusion of the theorem above is that fractallength changes with scale factor n. During the up-samplingprocess, fractal length reduces.

The principle of our super-resolution method is that if scaleinvariance of fractal set is preserved, the super-resolution resultwill be more accurate than traditional ones. Moreover, in thenext section, we will show that after adding the invarianceconstraint on fractal length, details of the super-resolutionresult can be enhanced simultaneously.

III. SUPER-RESOLUTION AND DETAIL ENHANCEMENTBASED ON LOCAL FRACTAL ANALYSIS

A. Joint Super-resolution and Enhancement based on Invari-ance of Fractal Dimension and Length

For a point x in the low-resolution image fl, we canestimate its fractal dimension Dori(x) and fractal lengthLori(x) respectively with the help of Algorithm 1 in sectionII.C. According to the theorem in Section II.C, we can assumethat the fractal dimension and length of the point y = nx inthe high-resolution image fh should obey (7). So, we have

logµ(Br(y)) = Dori(x) log 2r + Lori(x)− log nDori(x). (8)

(8) gives a strong priori to fh, which is used to estimate thegradient of fh. After interpolation, we get an approximationof fh, denoted as f̂h. In the gradient domain, we can get thedegraded local fractal dimension D̂ and fractal length L̂ in y,as follows.

log µ̂(Br(y)) = D̂(y) log 2r + L̂(y). (9)

The principle of our super-resolution method is to ensurethe scale invariance of fractal dimension. From the geometricalview, we need to transform linear function (9) to (8) bychanging the slope and the intercept of (9). According to (8,9),we have

logµ(Br(y)) =Dori(x)

D̂(y))log µ̂(Br(y)) (10)

+Lori(x)− Dori(x)

D̂(y)L̂(y)−Dori(x) log n.

µ(Br(y)) = β(µ̂(Br(y)))α, (11)

α =Dori

D̂, β =

1

nDorieLori−

DoriD̂

L̂.

By multiplying D̂ with α, (11) keeps the invariance of localfractal dimension. Because we use gradient of image as ameasure in (4), when 2r = 1 we can estimate the gradientof gradr(fh(y)) in the Br(y) as

grad(fh(y))s =‖grad(f̂h(y))‖

‖grad(f̂h(y))α‖+ ε(grad(f̂h(y)))α. (12)

Here ε is a small constant to make sure the denominator tobe larger than zero, which is similar to that in (4). Accordingto the analysis in section II, α > 1, the gradient of imagebecomes sharper and the task of super-resolution is fulfilled.

On the other hand, from (7) we can find that the fractallength decreases during up-sampling, so β in (11) is oftensmaller than 1. An obvious explanation of this phenomenonis the decreasing of local density during up-sampling. Recall(1,7), we find that after up-sampling X with factor n, thedensity reduces with factor 1/nD, as following shows

d =dorinD⇒ eL =

eLori

nD⇒ L = Lori −D log n. (13)

However, in the super-resolution processing, the density ofpoint should be invariant. It is equivalent to adding invarianceconstraint on the fractal length. If we keep the fractal lengthinvariant with the change of scale, the β in (11) can berewritten as βe = eLori−

DoriD̂

L̂. And then (12) becomes

grad(fh(y))s = βe‖grad(f̂h(y))‖

‖grad(f̂h(y))α‖+ ε(grad(f̂h(y)))α. (14)

Generally βe > 1, which means that the dynamic range ofgradient is enlarged. So, we achieve joint super-resolution andenhancement based on scale invariance assumption of fractal.Figure 3 gives an illustration of the change of gradient.

After getting grad(fh)s, we can further get directional dif-ferential (∇xfh)s, (∇yfh)s and estimate the high-resolutionimage by solving the following optimization problem.

fh = arg minf‖G∗f− f̂h‖22+λ

∑�=x,y

‖∇�f−(∇�fh)s‖22. (15)

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Fig. 3. (a) The change of gradient after preserving the invariance offractal dimension and length. (b) The images in the first row are X4 super-resolution results using bi-cubic interpolation, preserving the invariance offractal dimension and preserving the invariance of both fractal dimension andlength. The images in the second row are the corresponding gradients. Forimproved the visual effect, the logarithm of gradient is shown.

∇�fh)s is gotten from the gradient estimated by (14). G is aGaussian kernel. The first term is a reconstruction constraintensuring that the final result is similar enough to the interpo-lation result. While the second term is the regularization onthe gradient of image based on fractal invariance.

Because (15) is a convex optimization problem, manyeffective solutions exist, such as the gradient descent algorithmin [25]. In [17], [18], the analytical solution of (15) is gottenquickly by a FFT-based method. Inspired by [17], [18], weapply this method and get the analytical solution of (15) asoutlined below.

fh = F−1(F (G)F (f̂h) + λ

∑�=x,y F (∇�)F ((∇�fh)s)

F (G)F (G) + λ∑

�=x,y F (∇�)F (∇�)).

(16)Here F and F−1 is Fourier Transform pair, and F̄ is theconjugate of F . Experimental results show that this methodcan accelerate of the proposed method substantially.

According to the analysis above, we summarize the pro-posed method as follows.

Algorithm 2: Joint Super-resolution and Enhancement Based onInvariance of Fractal Dimension and Length

0. Given fl and sampling factor n.Approximate f̂h by interpolation-based method.

1. Calculate grad(fl) and grad(f̂h) respectively.2. To a patch in grad(fl), denoted as Pl,

find the corresponding patch P̂h in grad(f̂h).3. Estimate Dori and Lori of Pl according to Algorithm 1.4. Estimate D̂ and L̂ of P̂h according to Algorithm 1.

5. Get α = DoriD̂

and βe = eLori−

DoriD̂

L̂.

6. According to (14), estimate Ph by Ph = βe‖P̂h‖‖P̂αh‖+ε

P̂αh .

7. Reconstruct grad(fh) through Ph.8. Get ∇xf and ∇yf from grad(fh).9. Get final result by (16).

B. Image Enhancement based on Maximization of FractalLength

In Algorithm 2, we focus on applying the scale invarianceproperty of fractal dimension and the invariance constraint

of fractal length to achieve super-resolution with detail en-hancement. In this subsection, however, we mainly analyzethe relationship between fractal length and the correspondingmeasure — because fractal length reflects the measurementvalue in the unit ball, the larger fractal length is, the largervalue of measure we get. In our work, the gradient of imageis the proposed measure, so enlarging fractal length leads tothe increase of gradient value and the enhancement of detailsin the image.

Inspired by the analysis above, if we increase each localfractal length to reach their global maximum, the details ofimage can be further enhanced. In other words, the proposedfractal analysis for image can be extended naturally to a pureenhancement problem.

For an arbitrary point x in the original image fo on thepoint set X, we first calculate its local fractal dimension D(x)and length L(x). We can enhance the details of the image byincreasing the local fractal length as:

log µ̂(Br(x)) = D̂(x) log 2r + max{L(x)}x∈X (17)⇒ µ̂(Br(x)) = emax{L(x)}x∈X−L(x)µ(Br(x))

⇒ grad(f(x))e = emax{L(x)}x∈X−L(x)grad(fo(x)).

Here grad(f(x))e represents the enhanced gradient of f , andthen we get the enhanced directional differential (∇xf)e and(∇yf)e. Finally, the enhanced image can be reconstructed bysolving following problem

fe = arg minf‖f − fo‖22 + λ

∑�=x,y

‖∇�f − (∇�f)e‖22. (18)

(18) is similar to (15), where the first term is the reconstructionconstraint to avoid over-enhancement, while the second termis the regularization of gradient.

Similarly, (18) can be solved by the FFT-based method in[17], [18]. We can summarize the enhancement strategy asfollows.

Algorithm 3: Detail Enhancement Based onMaximizing Fractal Length

0. Given fo, Calculate grad(fo).1. To each patch in grad(fo), denoted as Pi,

estimate Li of Pi according to Algorithm 1.2. Find the maximum among {Li}, denoted as Lmax.3. To each patch, enhance Pi by Pi = eLmax−LiPi.4. Construct grad(f)e through Pi.5. Get ∇xfe and ∇yfe from grad(f)e.6. Get final result fe by solving (18).

Differing from the works in [5], [12], which increase thedynamic range of intensity directly, the proposed methodachieves detail enhancement though fractal analysis on gra-dient. By enhancing gradient, the enhancement result doesnot change the contrast or illuminance of original image —purely details in the image, such as textures, are enhanced. Thesuperior performance of the proposed method will be shownin section IV.C.

IV. EXPERIMENTS AND DISCUSSIONS

To demonstrate the performance of the proposed method,we compare our approach with some other competitors. In the

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experiments, the f̂h in (15) is the X4 interpolation result of flby ICBI [8] directly. The Gaussian kernel G in (15) is a 9×9mask whose standard variance is 1.6 for X4 super-resolution.ε is set to be 0.01. σ in (5) is 1. The patch size is 5× 5. Theweight of the regularization term, λ = 1 in our work. Figure4-8 provides some comparison results for various methods1.

A. The Comparison with Local Gradient-based Methods

Similar to the methods in [4], [25], the proposed methodis also based on the local manipulation of gradient. However,there are several major differences among them. For example,the work in [4] does not use a strict constraint on the dynamicrange of gradient, and this may have the result of textureenhancement but usually at an uncontrolled degree. In Figure7, the super-resolution result of [4] is obviously over-bright.For the proposed method, however, both the degrees of super-resolution and enhancement can be well controlled by theinvariance of local fractal feature of image and the super-resolution result is free from over-enhancement.

(a)

(b)

Fig. 4. To each sub-figure, the images from left to right are original image,the X8 result in [25], the joint super-resolution and enhancement result gottenby the proposed method. The results of [25] are gotten directly from thecorresponding reference because of the lack of source code.

The gradient profile method in [25] has a close relationshipwith the proposed method. Both of them achieve super-resolution based on local gradient enhancement. Comparedwith [25], the proposed method has three advantages: 1) Thework in [25] tries to solve the single image super-resolutionproblem from the statistical point of view. The gradient of

1The super-resolution results of other methods are available fromthe following websites: http://www.cs.huji.ac.il/˜raananf/projects/lss_upscale/sup_images/index.html, http://www.cs.huji.ac.il/˜yoavhacohen/upsampling/, http://www.wisdom.weizmann.ac.il/˜vision/SingleImageSR.html.

image is assumed to obey general Gaussian distribution. Thesharpness of gradient is trained by a parametric method. How-ever, in our work, we solve the problem from a geometricalpoint of view. The gradient of image is modeled as a measureof fractal set and is sharpened based on the invariance of fractaldimension. In the work of [25], the sharpness parameter isregarded as a scale-invariant variable with an optimal value ofabout 1.6. This value is merely a training result using two largedata sets. On the other hand, in our work, the selection of thesharpness is adaptive, which is based on the scale invarianceof fractal dimension. Therefore, an obvious advantage of ourmethod is that the selection of parameter is adaptive and hasbackup from the fractal theory.

2) The algorithm in [25] is sensitive to gradient direction,because the gradient profile is defined according to the di-rection of gradient. The feature indicates that the approach isa pixel-based method. For each pixel of an image, we haveto calculate the gradient profile and then sharpen it. Undermost situations, the gradient profile is generated by directionalinterpolation. On the other hand, our proposed method doesnot take the direction of gradient into consideration andcalculate fractal dimension on a patch-by-patch manner. Thefractal dimension can be calculated by direct filtering, whichhas much lower computational complexity as compared to themethod of gradient profile as in [25]. Moreover, patch-basedalgorithm is more suitable for parallel computing.

3) The reconstruction step in our work is different from thatof [25]. The work in [25] used a gradient descent algorithm.In our work, we devise a transform-based method with ananalytic solution. The proposed non-iterative method is clearlymore faster than that in [25].

In Figure 4, we provide some comparison results betweenthe methods in [25] and the proposed method. We can find thatthe proposed method is able to achieve more visually pleasanteffects. Moreover, in the next sub-section we will provide theRoot-Mean-Square error (RMSE), SSIM2[31] and the run timecomparison for various methods in Table I-III, respectively.

B. Super-resolution with Detail Enhancement Using Algo-rithm 2

Besides gradient-based methods [4], [25], we also compareour approach with some other methods, including bi-cubic in-terpolation, the methods in [8], [23], [9], [11], [6] and GenuineFractals (a state-of-the-art commercial product). Figure 5-8gives X4 super-resolution results of these methods. We canfind that under most situations, our method generates resultscomparable to the state-of-the-art. The simultaneous super-resolution and enhancement results are given by combiningthe invariance of fractal length with that of fractal dimension.

Compared with interpolation based method [8] and recon-struction based method [23], the proposed method achieveclearer details for enlarged images by combining the invarianceof fractal dimension and that of fractal length. On the otherhand, compared with the example-based method in [9], [11],

2In our work, we sample image into overlapped patches and calculate theSSIM of each patch. The final SSIM is the expectation of all the patches’SSIMs.

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(a) (b)

(c) (d)

Fig. 5. (a, c) The original images. (b, d) The X4 super-resolution results gotten by different methods labeled in the corners of images.

[6], our method does not involve complex search step over alarge data set, and therefore has relatively lower computationalcomplexity.

The main advantage of our super-resolution approach is itsability to get vivid texture regions. Although some of thesalient edges appear to be sharper in the results of [9], [6]than in our results, our approach protect texture regions frombeing over-smoothed, such as the knitting hat of Child and thetexture of the wheel in Figure 5, the sculpture in Figure 7 andthe face of old man in Figure 8(a). As mentioned above, thefractal model is quite efficient for describing image textures, soin the super-resolution problem, enforcing scale invariance offractal dimension can protect the image textures efficaciously.Figure 8 illustrates the advantage of our approach over someother methods. In Figure 8(a), we compare our method with

bi-cubic and the methods in [4], [9]. We can find that thewrinkle of face is enhanced better by our approach. In Figure8(b), after X4 enlarging, the texture of the fur is over-smoothedin the results of Genuine Fractals and [6]. On the other hand,noise-like pattern can be found in the result of [11], whichis caused by the suboptimal selection of exemplary sets. Thesuper-resolution result of the proposed method is comparableto that of Genuine Fractals. When combining super-resolutionwith texture enhancement, the texture region of our results ismore natural than those of Genuine Fractals and [6], [11].

Besides subjective visual effects, we also compare theproposed method with other methods by objective qualitymeasurements, including RMSE and SSIM [31]. To comparethe super-resolution result with original high-resolution imageobjectively, we first need to get a low-resolution image from

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(a) (b)

(c)

Fig. 6. (a) The original image. (b) The X4 super-resolution results gotten by different methods labeled in the corners of images. (c) The enlarged resultscorresponding to the regions labeled in (b).

original one. Currently, there are two popular methods for thisstep — one is zooming out image by interpolation method(bi-linear or bi-cubic); the other is based on point spreadfunction (PDF) model, which obtains low-resolution image byGaussian filtering and down-sampling sequentially. Generally,the interpolation based methods [8], [38] are mainly based oninterpolation based method while the image degradation modelused in [25], [26], [9], [6] are all based on PDF. Here we takethe differences between the two methods into considerationand apply the proposed method on the low-resolution imagesgotten by them respectively.

Table I reports the RMSE and the SSIM of each methodson the low-resolution images gotten by bi-cubic interpolation

method3. The proposed super-resolution method based on in-variance of fractal dimension gets the RMSE result comparableto those of state of the art. However, when we add scale-invariance constraint of local fractal length to the proposedmethod, the details in the images, such as the knitting hat ofChild in the Figure 5 and the fur of koala in the Figure 8, areenhanced while the RMSE of image increases.

On the other hand, to the low-resolution images obtained byPDF model, the proposed method achieve encouraging results,which is reported in Table II. In both the RMSE and theSSIM, the proposed method has better performances than other

3“Child” seems to be the only public available image having both groundtruth and super-resolution results of all the test methods. On the other hand,because the source codes of [25] and [6] are unavailable on-line, the worksin [25] and [6] were achieved by ourselves, in which the configurations ofparameters were based on the discussion in [25] and [6].

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(a) (b)

(c)

Fig. 7. (a) The original image. (b) The X4 super-resolution results gotten by different methods labeled in the corners of images. (c) The enlarged resultscorresponding to the regions labeled in (b).

competitors. Some experimental results are shown in Figure9 as well. The details in the super-resolution result of theproposed method, such as the hair and the sweater, are morevivid than others’ results. The improvement of the RMSE andthe SSIM results is based on the change of image degradationmodel. According to (15, 18), the degradation model impliedin the proposed method is more similar to PDF model, whichleads to the good performance of the method in such situation.

Take the fact that most of reconstruction based methods andlearning based methods [27], [25], [26], [9], [6] apply PDFmodel into consideration, the results in Table II and Figure4-9 verify the effectiveness and superiority of the proposedmethod in super-resolution.

Similar to the work in [25], we also analyze the influenceof noise on the proposed method for the completeness ofour work. Figure 10 gives the X4 super-resolution results

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(a)

(b)

(c)

Fig. 8. (a) The original image and X4 results gotten by bi-cubic, [4], [9] and our joint super-resolution and enhancement method respectively. (b) Theoriginal image and X4 results gotten by Genuine Fractals, [11], our super-resolution method and joint super-resolution and enhancement method, respectively.(c) The enlarged results corresponding to the regions labeled in (b).

TABLE IRMSE AND SSIM RESULTS OF DIFFERENT METHODS ON THE IMAGES GOTTEN BY INTERPOLATION BASED RESIZING METHOD

Image \ Method Freeman[7] Fattal[6] Fattal[4] Glasner[9] Shan[23] Yoav[11] ICBI[8] Proposed: Invariant D / Invariant D & LRMSEChild 23.728 24.065 23.713 23.609 18.265 27.278 21.654 22.141/25.300Koala — 13.821 9.423 — 9.065 17.242 12.993 12.943/15.007

Kodak Images — 17.598 — — 14.572 — 17.410 17.388/19.506SSIMChild 0.953 0.951 0.954 0.953 0.971 0.939 0.960 0.958/0.942Koala — 0.972 0.974 — 0.982 0.950 0.974 0.972/0.969

Kodak Images — 0.965 — — 0.975 — 0.963 0.960/0.957

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(a) Comparison of different methods.

(b) Enlarged results.

Fig. 9. The comparisons of variant methods for image obeying PDF degradation model.

(a) Gaussian noise

(b) Quantization noise

Fig. 10. The influence of noise on the proposed method.

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TABLE IIRMSE AND SSIM RESULTS OF DIFFERENT METHODS ON THE IMAGES OBEYING PDF DEGRADATION MODEL

Image \ Method Fattal[6] Shan[23] Sun[25] ICBI[8] Invariant D Invariant D&LRMSE SSIM RMSE SSIM RMSE SSIM RMSE SSIM RMSE SSIM RMSE SSIM

Child 20.98 0.962 11.19 0.989 16.78 0.975 9.47 0.992 8.68 0.993 8.12 0.994kodim1 27.49 0.755 22.47 0.834 23.73 0.781 21.76 0.831 21.25 0.856 20.60 0.873kodim2 12.74 0.967 10.82 0.977 10.73 0.968 10.44 0.979 10.23 0.980 9.96 0.981kodim3 12.54 0.962 10.08 0.976 10.33 0.964 9.66 0.978 9.31 0.979 9.06 0.980kodim4 15.76 0.941 11.96 0.967 12.88 0.950 11.37 0.971 10.94 0.974 10.54 0.976kodim5 31.11 0.764 24.23 0.850 27.04 0.786 23.02 0.866 22.11 0.880 20.91 0.898kodim6 22.29 0.935 18.18 0.956 19.39 0.940 17.48 0.960 16.96 0.962 16.40 0.965kodim7 22.43 0.835 14.88 0.925 15.54 0.873 13.74 0.936 12.94 0.944 11.90 0.955kodim8 41.65 0.750 30.94 0.856 34.08 0.781 29.37 0.871 28.30 0.883 27.25 0.898kodim9 20.31 0.849 14.10 0.928 13.67 0.885 13.37 0.935 12.83 0.941 12.37 0.950kodim10 18.68 0.866 13.57 0.927 14.71 0.891 12.90 0.933 12.47 0.938 12.09 0.943kodim11 20.66 0.854 16.17 0.909 17.36 0.871 15.49 0.917 15.04 0.922 14.58 0.930kodim12 15.71 0.941 12.64 0.962 14.31 0.935 12.14 0.966 11.59 0.969 11.26 0.971kodim13 32.69 0.816 26.55 0.876 28.13 0.840 25.74 0.885 25.15 0.891 24.33 0.901kodim14 23.15 0.883 17.72 0.932 18.58 0.900 16.69 0.941 16.11 0.945 15.42 0.951kodim15 20.96 0.957 14.09 0.980 17.09 0.964 13.51 0.983 12.77 0.984 12.34 0.985kodim16 14.81 0.940 11.72 0.961 11.00 0.949 11.26 0.965 11.02 0.966 10.79 0.968kodim17 17.75 0.920 12.97 0.957 16.63 0.931 12.16 0.963 11.57 0.967 11.05 0.970kodim18 22.86 0.814 18.29 0.884 19.26 0.840 17.44 0.896 16.92 0.904 16.19 0.917kodim19 23.57 0.842 18.65 0.898 20.31 0.857 17.83 0.906 17.28 0.913 16.71 0.921kodim20 21.48 0.970 14.95 0.985 17.32 0.976 13.40 0.988 12.96 0.988 12.45 0.989kodim21 26.67 0.815 17.83 0.914 19.60 0.874 16.90 0.924 16.34 0.929 15.65 0.937kodim22 17.18 0.927 14.05 0.952 14.45 0.934 13.47 0.956 13.13 0.958 12.70 0.962kodim23 14.60 0.963 11.09 0.978 12.08 0.965 10.46 0.981 10.11 0.982 9.52 0.984kodim24 24.52 0.875 19.98 0.912 21.07 0.885 19.22 0.919 18.71 0.924 18.13 0.930

of the proposed method for noised images. In Gaussiannoise situation (variance is 0.01), when we maintain fractaldimension and length to be invariant for super-resolution withdetail enhancement, the noise in the result is more seriousthan that in the bi-cubic result. However, when it comes toquantization noise in compressed image, the proposed methodachieves better visual effect than bi-cubic’s results — theresolution of image is increased while the quantization noise isno more visible than that in the bi-cubic result. The robustnessto quantization noise implies that it is possible to embedthe proposed method into image/video decoding system forincreasing the resolution of images or video frames.

Moreover, Table III gives the run time of each algorithms.In the experiment, we achieve X2, X4 and X8 super-resolutionfor a 128×128 image. The test platform is MATLAB on IntelCore2 CPU with 3GB memory. Except ICBI [8], the proposedmethod beats others in speed. Besides Table III, we also givesthe curves of run time for different methods (X2 situation),as Figure 11 shows. Compared with the works in [25], [23],[6], the proposed method has the slower increasing rate of runtime to the change of image size.

C. Pure Detail Enhancement Using Algorithm 3

As abovementioned, our fractal image model can be ex-tended to image enhancement naturally, especially to textureenhancement. Compared with other popular image enhance-ment methods, including the bilateral filter in [29], the guidedfilter in [12] and the weighted least square (WLS) filter in[5], [3], the proposed method has two advantages. First, thegains of detail signals in [5], [3], [12] are global parametersand set manually, which can not be adapted according to thedifferent contents of image. On the other hand, the parameter

Fig. 11. The comparisons of variant methods on run-time for images withdifferent sizes in X2 super-resolution situation.

in our method is adaptively chosen, which corresponds to localfractal lengths and the global maximum of them. As a result,the gain of detail is adaptive to smooth, edge and textureregions respectively, and our method prevents the images fromover-enhancement. Second, our fractal based enhancementmethod has superior texture enhancement performance. Themethods in [5], [3], [12] enhance the intensity of imagedirectly. Although the edge-preserving enhancement can beachieved by designing filter carefully, these methods alwayschange the contrast of image Differing from them, the gainof detail in the proposed method is applied on the gradient ofimage, so that the intensity and the contrast of image are notchanged while the details of image become more visible.

Figure 12 gives a comparison of different enhancementmethods. The test images come from the BSDS300 data set

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TABLE IIIRUN TIME (SECOND) RESULTS OF DIFFERENT METHODS

Image \ Method Fattal[6] Shan[23] Sun[25] ICBI[8] Proposed: Invariant D / Invariant D & L128× 128, X2 56.72 6.27 18.24 1.19 2.40 / 2.54128× 128, X4 256.33 24.16 81.50 4.08 9.11 / 9.12128× 128, X8 1140.90 93.20 531.15 10.91 35.32 / 36.07

of Berkeley4. From the experimental results, it can be foundthat our approach achieves more vivid texture reconstructionthan other two methods. The proposed method enhances thetexture regions, such as the grass, the fur and the feather ofanimals, while maintains the contrast of image.

V. CONCLUSION

In this paper, a single image super-resolution and enhance-ment method is proposed based on local fractal analysis ofimage. We first establish a fractal model for image whichemploys the gradient as a fractal measure for the point set ofimage. By using the assumption of scale invariance of localfractal dimension and length, the problem of image super-resolution and enhancement can be solved jointly. Experi-mental results show that the fractal-based method providesgood results for both super-resolution and enhancement. It isalso shown that by constraining the local fractal lengths equalto their global maximum, our approach can enhance texturedetails sufficiently well. In this work image gradient is used asa sole measure of fractal, which can be insufficient for somecomplicated image contents, so in our future work, we plan toincorporate more measures into our fractal image model andextend our the model to video processing.

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 12. (a, c, e) Each sub-figure gives original image and enhancement results gotten by [12], [3], and our method respectively. (b, d, f) Each sub-figuregives enlarged local images corresponding to the regions labeled in (a, c, e) respectively. The parameters of other two methods are configured according tothe configuration in [12], [3].