Pattern Anal Applic DOI 10.1007/s10044-012-0316-4
SHORT PAPER
Exemplar-based color image inpainting: a fractional gradient function approach Anamandra Sai Hareesh • V. Chandrasekaran
Received: 16 December 2011 / Accepted: 21 December 2012 Ó Springer-Verlag London 2013
Abstract Image inpainting is the art of recovering a plausible image from images which are generally incomplete due to various factors, including degradation due to ageing, damage due to wear and tear and missing image details due to occlusion. In such situations, there is a need to predict the missing image information without introducing undesirable artifacts. Original contribution in this direction is due to a seminal paper by Criminisi et al. This has led to a number of novel contributions in terms of patch filling prioritization and associated metrics to measure color and structure. In this paper, we propose a fast and simple technique based on a novel gradient function and its generalization via fractional derivatives to evaluate the filling order prioritization. Results demonstrate superior and robust performance over all the recent advances in the domain of exemplar-based methods quoted in the literature.
There are several scenarios where inpainting comes to rescue. Let us consider one such situation. The increase in computing power and available disk storage space over the last few decades have created new possibilities for image and movie inpainting. Old photographs that are threatened by bleaching, can be preserved digitally. Old celluloid movies, that become more and more damaged every time they are exhibited, can be digitized and preserved. Typical damages are scratches or stains in photographs, peeled-off coatings, or dust particles burned into celluloid movies. All these flaws create regions where the original image information is lost. Manual restoration of images or single movie frames is possible but this process will be time consuming. Therefore it is desirable to automate this process. Several so-called inpainting algorithms have been developed to achieve this goal.
Keywords Fractional approach Exemplar-based inpainting Structure propagation
2 Inpainting definition
1 Introduction The most important challenge in the inpainting domain is in regard to large object removal and region-filling thereafter. Problems get even more complex when the inpainting domain is in color. To an observer, the inpainted image must look authentic without bearing any trace of being tampered with.
A. Sai Hareesh (&) V. Chandrasekaran Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Puttaparthi, India e-mail:
[email protected]
Let X denote a complete image domain, often a rectangular area on our computer screen, or more generally a finite Lipschitz domain in R2. Certain factors, such as object occlusion in visual fields or packet loss in wireless communication, result in a subset D of X in which the image data is missing or inaccessible. Let u denote the original complete image on X, and u0 denote the observed or measured portion of u, which can be noisy or blurry or damaged on D, i.e., u(x) = u0(x) for all x 2 X=D (the data we use to fill D is precisely outside D). The goal of inpainting is to recover u on the inpainting domain D as faithfully as possible from the available data u0 on the X=D: Inpainting domain and other details are shown in Fig. 1.
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[1], he presents several arrangements of simple geometric figures (mostly polygons and/or circles) partially overlapping and occluding each other. By asking several persons who participated in the test to complete the occluded figure, he inferred that most persons tend to prefer completions following some simple principles. The following three guidelines are amenable for our problem setting: 1.
2. Fig. 1 Inpainting problem
Because most objects are not transparent, humans experience the occlusion effect almost all the time. Still, we see the world as perfectly ordered and integrated rather than as a cluttered landscape of independent discrete pieces. This is nature’s answer to the inpainting problem.
3 Guidelines for inpainting If a human observer views an image he/she immediately recognizes and interprets its contents. Even if parts of the image are missing or occluded, the human visual system very reliably identifies separable objects or regular patterns and completes the image. Thus for a professional inpainter, due to his/her experience and training it is possible to imagine how the restored image should look like (e.g., see Fig. 2). For a computer it is difficult to cope with image contents at such a high level of cognition. Thus one is forced to look for simpler and low level image features which should be considered when performing an inpainting task. In the following we discuss some possible guidelines regarding the pertinent question of how an image should be inpainted to look ‘‘correct and complete’’ for the human visual system. We recall some ideas from the work of the psychologist, Gaetano Kanizsa. In chapter 5 of his book
3.
Continuation of direction: straight or smoothly curved lines are preferred over lines that frequently or abruptly change their direction Completeness, closedness: lines forming closed shapes are preferred over lines leaving open gaps Tendency to convexity: convex shape are preferred over concave ones.
Although these principles have been derived from plain homogeneous figures (or even only outlines) we may try to apply these guidelines over to images, since the strongest visual impression in an image comes from its edges. Images are assumed to consist of objects which are (in some sense) considered as homogeneous regions separated by edges. This point of view gives rise to the following situations, depending on the size and the position of the inpainting domain, i.e., the area of an image where new contents have to be created: 1.
2.
3.
The inpainting domain is completely included into one patch: in this case we need to fill the inpainting domain with the appearance of the surrounding. The inpainting domain covers parts of two different patches, i.e., an object boundary crosses through it: in this case the main problem consists in finding a reasonable course of the boundary. When constructing the boundary Kanizsa’s principles should be respected. The inpainting domain covers more than two objects: if two or more of the objects again overlap with each other, then T-junctions lie within the inpainting domain. Otherwise this problem reduces in principle to the two-object inpainting problem.
An efficient algorithm should successfully handle all the three cases simultaneously. Note that, we always consider the inpainting domain to be simply connected or a disjoint union of simply connected regions.
4 Literature review
Fig. 2 Painting restored by J. Suvee (Louvre)
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A number of inpainting techniques have been proposed in the literature. The majority of these techniques have dedicated their efforts in the direction of texture synthesis, structural extension and color matching. They can be broadly classified into four classes:
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1. 2. 3. 4.
Statistical-based approaches Variational or PDE-based methods Fragment or exemplar-based methods Hybrid approaches.
4.1 Brief overview of existing methods In statistical-based approaches, images are modelled as samples drawn from certain random fields. The primary goal of random field modelling is to estimate the probability distribution function. Classic models are inspired mainly by Gibbs fields in statistical mechanics, in which local energy constraints are modelled in the same way as in Ising crystals [2]. Random fields can also be learned from an image database through techniques such as filtering and non-parametric estimation by the maximum entropy principle [3]. PDE-based methods on the other hand predict the missing image information by extrapolation of neighbouring pixels using certain criteria like the recovery of edges, or variational principles. Typically, the region to be inpainted is assumed to be coherent with the neighbouring regions through topology preservation. Inpainting is achieved by the process of diffusion of image intensity values in the local neighborhood modelled by PDEs. These local methods work very well for small regions but result in blurring for large holes. While the PDE-based approaches succeed in inpainting thin edges they lose out on correctly synthesizing the texture. The exemplar-based approaches are gaining popularity on account of their computational efficiency and visually pleasing results. Texture synthesis algorithms generate large image regions from sample textures. They operate essentially on one pixel at a time and determine its value by looking for similar areas in the available image data. The fragment or exemplar-based algorithms in some sense be considered as generalized texture synthesis. Instead of copying single pixels, whole blocks are transferred into the inpainting domain, thereby the resulting inpainting connects smoothly and is similar to the available image. Fragment-based inpainting algorithms are strongly dependent on available image contents that can be copied into the inpainting domain, even more than pixel-based texture synthesis algorithms. ‘‘Fundamentally new’’ contents cannot be created which can pose problems if samples of an ‘‘obviously correct’’ inpainting fragment do not exist in the image. Exemplar-based methods are a combination of texture synthesis and inpainting. The challenge is to fill in the hole that is left behind in a visually plausible way. In the past, this problem has been addressed by two classes of algorithms: (1) ‘‘texture synthesis’’ algorithms for generating large image regions from sample textures, and (2)
‘‘inpainting’’ techniques for filling in small image gaps. The former works well for ‘‘textures’’-repeating two-dimensional patterns with some stochasticity; the latter focus on linear ‘‘structures’’ which can be thought of as one-dimensional patterns, such as lines and object contours. Exemplar-based or fragment-based methods have proven their efficiency in restoration of texture as well as local geometry, but they also fail to reconstruct non-local geometric features such as long edges. Paper by Criminisi et al. [4] presents a novel and efficient algorithm that combines the advantages of these texture synthesis and inpainting approaches. In their paper, the authors have used a data term priority which is determined in the direction of isophotes. Later Wu and Ruan [5] has proposed a new data term prioritization scheme driven by cross-isophotes which determine the contrast between isophotes. The greedy search algorithm implemented in the above two exemplar-based approaches is computationally intensive. This is addressed by Chen et al. [6] by using a search strategy applied to a restricted range only. A DCT-based exemplar matching together with an efficient KD-tree guided search is proposed by Kwok and Wang [7] to locate an approximate nearest neighbour. Driori et al. in their article [8] have incorporated a pyramid image approximation and adaptive image fragments to obtain satisfactory results. Jianbing Shen et al. [9] have proposed a gradient-based image completion by solving a Poisson equation. In their paper average gradients computed over patches touching the inpainting boundary are used to decide the filling order. Then, based on an exponential similarity metric which combines color and gradient, a suitable inpainting patch is selected from the source region. Then the gradient information from the best patch selected for inpainting is used to fill the gradient image. From this completed gradient image, the inpainted image is recovered by solving a second order Poisson differential equation. Before getting into the contribution of this paper in the exemplar-based methods, let us get into the realms of hybrid approaches in the next subsection, which combines the techniques of exemplar- and PDE-based approaches for inpainting. 4.2 Hybrid approaches Until now we have mainly treated two types of algorithms: those that try to extrapolate image data in a sensible way into the inpainting domain (PDE-based algorithms) and algorithms that look for appropriate regions in the available image data that are copied into inpainting domain (texture, fragment synthesis). Both have their strengths and weaknesses. In the last few years, the image inpainting world has seen a paradigm shift from these stand-alone approaches to a unified approach that aims at combining their strengths.
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In this subsection we present some hybrid algorithms, which combine both techniques to join their advantages. All these algorithms are similar in principle but differ in certain technical details. Since these algorithms have many properties in common we frequently point out the differences and their consequences. Rane et al. [10] propose an algorithm specifically designed for filling missing blocks in a block-wise compressed image (i.e., JPEG). This algorithm is mainly an extension to [11]. Since many image compression schemes subdivide an image into blocks of n 9 n pixels, which are compressed independently, it seems natural to treat the same blocks as primitive elements in the inpainting process. Using a wavelet decomposition of adjacent blocks a missing block is classified either as edgy (containing an edge) or non-edgy (containing only texture). Depending on this classification edgy blocks are inpainted using the PDE-based algorithm from [12], and non-edgy blocks are filled with textured synthesized using the algorithm [13], whereby the eight neighbour blocks are used as sample texture. Thus each block is treated using either PDE inpainting or texture synthesis. In Bertalmio et al. [14] the authors extend the foregoing algorithm by first decomposing the image into a sum of a piecewise constant and an oscillatory image. For that purpose they use the algorithm the same authors developed in [15] which is based on the G-norm introduced by Meyer [16]. This piecewise constant image is inpainted using the PDE-based algorithm from [12], the oscillatory image is filled using the synthesis algorithm from [13]. Both image components are added to give the final inpainted image. Yamauchi et al. [17] decompose the image into low and a high frequency part. The low frequency part is inpainted using the linear heat equation, while the synthesis algorithm from [18] is used to handle the high frequency part. Grossauer [19] uses nonlinear diffusion filter of Perona– Malik type for decomposing the image into geometry and texture parts. For inpainting geometry part, the method proposed by Ginzburg–Landau [20] is used. After that postprocessing is done using coherence enhancing diffusion [21]. Segmentation is performed on the inpainted geometry image. For texture synthesis they employ the algorithm from [22]. Bugeau et al. [23] combine three building blocks, namely, copy-and-paste texture synthesis, geometric PDE’s, and coherence among neighbouring pixels in a unifying variational model, and provide an algorithm for image inpainting trying to approximate the minimum of the their proposed energy functional. The energy they propose contains three terms which combine all three main ideas just mentioned. The first term is a variational formulation of texture synthesis method used by Efros and Leung [13]. The second and third terms combine ideas of diffusion and coherence applied in the patch space. The algorithm uses a
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multi-resolution scheme, does automatic computation of weights for patch selection and involves Poisson editing as part of its post-processing step. 4.3 Our contribution to exemplar-based methods In this paper we propose a simple gradient-based approach, which does not require complex solutions to a Poisson process [9]. Based on several experiments on a large class of functions, we have chosen a novel function which is a linear combination of the gradient and logarithm of gradient to decide the filling order priority. Firstly, our proposed method does not require a dot product computation between the normal to the inpainting boundary and the isophotes as suggested in [4], as we consider that the patch filling priority order could be incorrect on account of varying shapes of the boundary and their associated normals. Secondly, we have not considered the use of average gradient in the neighbourhood of a boundary pixel as proposed in [9]; it is obvious that two different distributions can be centered on the same mean and the averaging operation tends to lose its discriminating power to decide the choice of inpainting location. Therefore we have intentionally chosen the local gradient at the pixel p on the boundary to be the sole guiding parameter for filling order priority. The results of the proposed method have demonstrated superior performance over the previous complex approaches under minimal noise conditions. Having realized the fact that any gradient-based function will exhibit deteriorating performance in the presence of noise, we have generalized the proposed gradient function by replacing it with fractional order derivatives. By suitably selecting the fractional order, we are able to demonstrate its robustness on synthetic and real world images. When the value of the fractional order r equals 1 it turns out to be the standard gradient function. The paper is organized as follows. Section 5 introduces the definition of the fractional order derivative and the motivation behind choosing the same. Section 6 provides the overview of our proposed method and in Sect. 7 sideby-side comparison of results of various methods found in the literature with our method.
5 Fractional derivative 5.1 Theory The concept of differentiation and integration of noninteger order is no means new. This goes back to Leibitnz note in his letter to L’Hopital dated 30 September 1695 in which the meaning of derivative of order one half is discussed. Subsequently systematic studies and contributions
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have been made by a number of famous mathematicians including Liouville, Riemann, Holmgren, Euler, Lagrange, etc. Fractional calculus refers to the theory of integrals and derivatives of arbitrary order which generalize and unify the concepts of integer-order differentiation and integration. The Grunwald–Letnikov derivative [24] of an integer order n and the r fold integral of the continuous derivative f(t) are particular cases of the general expression n r X r r ð1Þm f ðt mhÞ ð1Þ a Dt f ðtÞ ¼ lim h h!0 m m¼0 nh¼ta
This equation triggers the idea of a generalization of the notions of differentiation and integration by allowing r to be an arbitrary real or even complex number. Therefore, in this paper, a modified version of the Grunwald–Letnikov formulation which lends itself amenable for implementation in discrete domain on account of its binomial like coefficients has been used. Let u(x, y) be the image function. A right sided (noncausal) discrete domain fractional differentiation function along the x-direction also called forward differencing of u in x-direction is defined as: or uðx; yÞ 1 urFx ðx; yÞ ¼ F ¼ ox ðDxÞr n r o uðx; yÞ ruðx þ 1; yÞ þ uðx þ 2; yÞ. . . 2 ð2Þ Reverse differencing in x-direction is defined as or uðx; yÞ 1 urRx ðx; yÞ ¼ R ¼ ox ðDxÞr n r o uðx; yÞ ruðx 1; yÞ þ uðx 2; yÞ. . . 2 ð3Þ Similarly forward differencing urFy and reverse differencing urRy in y-direction can be formulated. The fractional gradient at the image pixel p(x, y) is defined to be rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 2 n r ð4Þ jr uj ¼ maxð~ urFx ; u~rRx Þ þ maxð~ urFy ; u~rRy Þ
et al., demonstrated the effectiveness of these edge detectors while segmenting the range images which are inherently noisy. When the derivatives are taken at the lower order, the response of the fractional order edge detector is prominent for true edges while suppressing the noisy edges. Prediction of the order can be automated using machine learning techniques available in the literature. For example, fractional order filters have been used for image contrast enhancement with automated fractional order prediction using regularization networks in Khanna and Chandrasekaran [26]. Similar approach can be applied in this case as well. For the sake of brevity the order prediction exercise has been excluded. This paper mainly focuses on the effectiveness of the proposed fractional order gradient function for filling order prioritization.
6 Overview of proposed method Given an image I, the inpainting domain D is manually marked by the user. The boundary of inpainting domain will be denoted as dD. D shrinks at each time step n as algorithm proceeds. The source region is given by U ¼ InD provides a sequence of patches for inpainting. The algorithm is summarized as follows (use Fig. 3): 1.
Locate dD in I, n = 0;
2.
V pixels p, set KðpÞ ¼
3.
Repeat until dD(n) = /
u~rF ¼
urF ;
R;G;B
u~rR ¼
X
i. Compute |rr u(p)| as given in Eq. (4) ii. Compute data term F(p) = |rr u(p)| ? log(1 ? |rr u(p)|) iii. V patch Pp, compute confidence term P KðpÞ ¼
urR
R;G;B
where symbol * represents x or y. Here Dx and Dy are set to unity. 5.2 Motivation Fractional derivative-based edge detectors are shown to be robust against noise [25]. In this paper, Chandrasekaran
0 8p 2 D 1 8p 2 InD
(a) Set n = n ? 1; (b) 8p 2 dDðnÞ;
where X
iv.
q2Pp
KðqÞ
jPp j
; here Pp represents a
square patch, typically of size 9 9 9 pixels. Compute priority values a(p) = K(p) 9 F(p)
(c) Find the patch Pq for which a(q) is maximum (d) Find the best exemplar Cq0 from the region U that minimizes dðCq0 ; Pq Þ where d denotes L2 norm in the CIE color Lab space (L*, a*, b*). (e) Copy appropriate image data from Cq0 to Pq, 8p 2 Pq \ D (f) Update K(p), 8p 2 Pq \ D:
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7 Experimental results and discussion 7.1 Synthetic images
Fig. 3 Notation diagram
As can be seen from the above algorithm, we have chosen a novel function defined on rr u . |.| denotes the L2 norm. rru denotes the fractional derivative field of the image. A single iteration of the proposed algorithm is elucidated as follows: For each pixel p on the inpainting boundary dD, a patch Pp is constructed with p as its center. Inpainting of patches along the boundary requires a patch filling order as demonstrated by Criminisi et al. [4]. This is accomplished by computing a patch filling order priority at every pixel p along the boundary and the one which has a maximum priority is chosen for region filling. It is the product of two elements, viz., a confidence term K(p), and a data term F(p). The confidence term describes how many valid source pixels from the domain U are present in the patch, and the data term describes the strength of the function of fractional derivative at that point. After finding the priority point q, the algorithm searches in U to find a patch that is the most similar to the patch Pq. The Euclidean metrics on CIE Lab color space reasonably emulates the color distance metrics commonly accepted for humans. When the most similar patch Cq0 is found, the appropriate part of this patch is copied to the inpainting area contained in Pq \ D. After filling the patch, the boundary is updated and the process is repeated. Eventually, the whole inpainting domain will be filled. The proposed priority scheme is driven by the fact that the point which returns the maximum value for the function of the gradient must be addressed first as neighbourhood of that point contains significant intensity changes passing through that point. This gradient-based location selection permits continuation of linear structure inherently thereby eliminating the risk of introducing broken structure artifacts and visibility problems at the seams.
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For the purpose of testing our proposed method, we have first tested the algorithm on a range of synthetic images under controlled, noisy environment. First it was used on the wellknown structure-rich Kanizsa triangle [1] without noise in Fig. 4. We demonstrate the proposed method with fractional gradient-based filling order prioritization over the method based on isophotes using Fig. 5. In this figure, different shapes of boundaries, namely, spherical, triangular and odd shaped boundaries are taken to show that the proposed method performs well in these cases as against the well known method by Criminisi et al. [4], which fails to produce uniform results over varying boundaries for inpainting. 7.2 Real images with the known source image Proposed technique is used to remove text from an image as shown in Figs. 6 and 7. We present visually pleasing results in a more quantitative fashion with the help of metrics, viz., peak signal to noise ratio (PSNR) and structural similarity index measure (SSIM) [27]. The results are tabulated in Table 1. Comparison of the results in terms of PSNR is shown in Fig. 8. 7.3 Real world images with the unknown source image In these class of images we do not have the reference image with which we can quantitatively compare the inpainted results. As the purpose of image completion is to fill
(a)
(c)
(b)
(d)
Fig. 4 Comparison of reconstructions on Kanizsa triangle
Pattern Anal Applic Fig. 5 Results on varying boundary conditions via synthetic images
(a)Image with text
(b) Fig. 6 Removing the text from the given image
(a)
(b) Fig. 7 Removing the text from a surfing image
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Pattern Anal Applic Table 1 PSNR and SSIM results for real-world images with the known source image Image
PSNR (dB)
SSIM between [0, 1]
Input spherical D (Fig. 5)
9.85
0.9468
Inpainted using Criminisi
29.92
0.9871
Inpainted using proposed method Input odd shaped D (Fig. 5)
38.46 10.97
0.9938 0.8572
Inpainted using Criminisi
35.15
0.9896
Inpainted using proposed method
35.77
0.9913
Image with text (Fig. 6)
12.67
0.7678
Inpainted using Criminisi
12.86
0.8940
Inpainted using proposed method
24.89
0.9029
Surfing input (Fig. 7)
27.82
0.9679
Inpainted using Criminisi
37.35
0.9953
Inpainting using proposed method
52.73
0.9967
(a)
(b)
(d)
(c)
(e)
(g)
(f)
(h)
(i)
Fig. 9 Comparison of reconstructions of a bungee jumper
Fig. 8 Comparison of PSNR results for Criminisi and proposed method
(a) damaged areas while satisfying visual perception and reality considerations, it is commonly accepted that the qualitative assessment of the completed images is easily measured by the human perception. For this reason, comparative results are placed side-by-side for our readers’ judgement. Our algorithm is evaluated on complex real-world images and the results were compared with other popular methods in the literature (see Figs. 9, 10). In all the experiments, the patch size is set to 9 9 9.
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(c)
(b)
(d)
Fig. 10 Comparison of reconstructions on a natural image
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(a)
(b)
(a)
(c)
(b)
(d)
Fig. 11 Comparison of restorations on noisy Kanizsa triangle
(c)
(d)
Fig. 13 Results by considering different data terms
7.4 Other experiments
(a)
(c)
(b)
(d)
Fig. 12 Comparison of restorations on noisy bungee jumper
We consider that the simple algorithm we have proposed in this paper has resulted in visually pleasing image completion. For example in Fig. 9, one can easily note the structural coherence, semantically valid and logical completion task far better than other methods.
Proposed method is robust against noise because of inherent robustness of fractional derivatives. We demonstrated this feature in Figs. 11 and 12. Similarly, the reason for selecting the proposed gradient function is demonstrated using perceptual evaluation of various functions of the gradient which act as data terms. Please see results in Fig. 13. The convex function given in Fig. 13c is from the paper by Seregin [28]. As previously stated, the proposed method uses a patch size of 9 9 9. The effects of varying patch sizes is investigated and results of the experiments are shown in Fig. 14. As can be seen, an optimal patch size selection is important for the inpainting process in order to follow the priority filling order determined by the maximum value of the function of the local gradient. This could be a image dependent process with additional computational burden due to trial and error.
8 Conclusion In this paper we introduced a novel, fractional gradient function-based filling order prioritization during the
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References
(a)
(c)
(b)
(d)
Fig. 14 Results by considering different patch sizes
exemplar-based inpainting. We demonstrated the effectiveness of the same using synthetic images, real-world images with ground truth data and real-world images without known source. The results obtained using the proposed method are always better than the well known existing approach by Criminisi. This is shown with the help of PSNR and SSIM metrics. In the other cases, i.e., the cases where the ground truth data is known, it can be seen that the results obtained by the proposed method is visually pleasing and more coherent than the other methods. In addition, introduction of fractional derivative-based function is considered to be beneficial in terms of its robustness against noise. This is amply demonstrated using the set of synthetic and real-world images. Instead of searching for a patch within the same image, a more generic approach could be to extend the source region given as I \ D to a database of images. In this case we will be finding the best exemplar patch by searching over all the images in the database. Quantitative measures for perceptual quality assessment of inpainted images are available in the literature including s-CIE Lab [29]. This could be taken as research in future.
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