Nonhomogeneous Scaling Optimization for Realtime Image Resizing

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1 Noname manuscript No. (will be inserted by the editor) Nonhomogeneous Scaling Optimization for Realtime Image Resizing Yong Jin Ligang Liu Qingbiao Wu Received: date / Accepted: date Abstract We present a novel approach for interactive content-aware image resizing. The resizing is performed on warping a triangular mesh over the image, which captures the image saliency information as well as the underlying image features. The warped triangular mesh and the horizontal and vertical scales of all triangles are simultaneously obtained by a quadratic optimization which can be achieved by solving a sparse linear system. Our approach can preserve the shapes of curved features in the resized images. The resizing operation can be performed in an interactive rate which makes the proposed approach practically useful for realtime image resizing. To guarantee foldover free resizing result, we modify the optimization to a standard quadratic programming. A number of experimental results have shown that our approach has obtained pleasing results and outperforms the previous approaches. Keywords Content-aware image resizing Image retargeting Visual saliency Triangulation Feature preserving 1 Introduction Image resizing or retargeting has been becoming a research focus due to the diversity of display devices and the versatility of applications. Under many circumstance, it is desirable to change the aspect ratios of an image to fit the target screen or to satisfy a variety of purposes, such as customized image editing and photo printing. The common solution is uniformly stretching the original image to fit the target aspect ratio, which might Department of Mathematics, Zhejiang University, China {jsnjkl,ligangliu,qbwu}@zju.edu.cn Corresponding author. not to be able to get a satisfied result all the time. The important content in the image might suffer a terrible distortion or become too small to be noticed. An alternative solution is cropping the region which contains the important content. However, the method inevitably discards information of background. Our goal is to design an image resizing method that preserves the aspect ratios of important content of the image, while the less noticeable region of the image may stretch more severely as a compensation. This is typically referred to as content-aware image resizing, which was first introduced in the Seam Carving work of Avidan and Shamir [1]. Seaming carving [1,15] is an efficient method to solve the problem of content-ware image resizing. The seam caving method greedily removes less noticeable seams to fit the target aspect ratio. Grid mesh based methods [20,21,9] achieve resizing the image by warping a quad grid mesh. The mesh is placed over the original image, then an energy function is employed to preserve the aspect ratios of important content during the warping, while fitting the boundaries of the mesh to target image dimensions. Triangular mesh has also been used to deal with the problem of image resizing [6]. The method takes advantage of the flexibility of triangular mesh to catch salient objects in the image and convert the problem to a nonlinear optimization programming. However, this approach needs to estimate the mesh edges in advance which is not reasonable for computing the mesh in some cases. We present a novel approach for content-aware image resizing by using a triangular mesh. Fig. 1 illustrates the process of our algorithm. We first build a triangular mesh over the original image that is consistent with the existing image features which include the sharp edges, feature curves, and image boundaries. Then a

2 2 1.1 Related Work (a) (c) (b) (d) (e) (f) (g) Fig. 1 Algorithm overview. Given an input image (a), a saliency map (b) is computed. In (c), the Canny edges (in white) and the feature line (in orange) are detected. Some feature points (d) are sampled from Canny edges (in white), feature line (in orange), and image boundaries (in yellow), respectively. A triangular mesh (e) is constructed over the feature points by constrained Delaunay triangulation algorithm in which the important triangles are determined from (b) and marked in blue. The triangular mesh is warped to fit the target image dimensions while preserving the salient content, as shown in (f) and the result image (g) is then obtained by texture mapping from (f). global nonhomogeneous scale optimization is performed to warp the triangular mesh. The result image can then be obtained by texture mapping. The contributions of our approach for image resizing are summarized in the following: 1) Quadratic global optimization: We describe a quadratic global optimization for image resizing which can be obtained by solving a sparse linear system in an interactive rate. 2) Feature preserving: Our approach can preserve salient line features as well as curved features efficiently. Our approach can also emphasize important image content whose aspect ratios can be specified by the user. 3) Foldover free resizing: The quadratic global optimization can be modified to a quadratic programming with additional constraints to prevent foldovers in the results. The concept of content-aware image resizing was first proposed in the Seam Carving work of Avidan and Shamir [1]. This paradigm was later extended in a number of ways, including mesh warping methods [21, 20, 9, 6] and synthesis methods [2, 13]. An intensive survey is beyond the scope of this paper [18]. The approach presented here is one kind of mesh warping method. The scheme of constructing triangular mesh closely follows the work of Guo et al. [6], however, the optimization schemes for image resizing of the two methods are totally different. The optimization scheme adopted here is partly inspired by the local/global approach for mesh parametrization presented by Liu et al. [11], which is recently used in sensor network localization [22]. We take advantage of the special nature of images to convert the local/global method to a quadratic global optimization, which can be obtained by efficiently solving a sparse linear system. We will try to relate more relevant benefits and shortcomings between our approach and the approaches already presented in the remainder of the paper, particularly in the sections dealing with experimental results. 2 Preliminaries We build a mesh over the input image, which is used as a controlling mesh. Instead of using quad mesh, which was generally used in content aware image resizing [20, 9], we use triangular mesh as it is more flexible to adapt feature constraints than quad mesh does [6]. The triangular mesh is generated so that its edges capture the image saliency information as well as the underlying image geometric features. The resizing problem is then formulated as a nonhomogeneous warping problem which finds a target mesh with same topology and respecting the salient features. 2.1 Problem and notations Given an input image I p, the resizing operation rescales the x-axis and y-axis and thus changes the image aspect ratio. The rescale factors of the image in the x- and y- directions are denoted by s x and s y, respectively. We generate a triangular mesh M p over I p. Our goal is to compute a new triangular mesh M q, which has the same topology with M p, for the resized image I q. This can be formulated as a problem of warping the original mesh M p to the target mesh M q. Intuitively, the important triangles should be preserved by their aspect ratios in

3 3 both x- and y-directions. The less important triangles can be stretched to a greater extent. Denote T as the triangle set of the mesh. The vertices in M p and M q are denoted by p i = (p x i, py i )T and q i = (qi x, qy i )T (i = 1,, n), respectively, where n is the number of vertices. 2.2 Feature aligned triangular mesh Strong geometric elements such as salient edges or curves are important visual features. They are vital clues for understanding image content, and should be maintained in the results. Therefore, we generate a triangular mesh to align the image features of I p. First, Canny operator [3] is applied to detect sharp edges in the image, see white lines in Fig. 1(c). Feature curves in the image are also detected. The feature lines can be automatically detected by Hough transformation [4], see orange lines in Fig. 1(c). Other feature curves can optionally be specified by the user. Finally, the four boundary lines are also considered as the feature lines. A set of feature points are uniformly sampled from the above features (edges, feature curves, and boundaries), as shown in Fig. 1(d). Then a triangular mesh M p with these feature points as its vertices is constructed using the constrained Delaunay triangulation algorithm [17], see Fig. 1(e). For each triangle t T, we have a scalar measure of its importance or saliency, given by σ t [0, 1]. We employ the method of [7] to compute the saliency map and thus obtain σ t of t by averaging the saliency values of pixels in t. The saliency map may also be provided in semi-automatic mode by the user. Important triangles with saliency values larger than some threshold (0.6) are marked in blue as shown in Fig. 1(e). The set of important triangles is denoted by T. 3 Nonhomogeneous scaling optimization Unlike [6], we do not compute the edge lengths of the target mesh M q. Actually, the edge lengths can not be well estimated in advance. To accomplish the warping from M p to M q, we employ an optimization framework. We now describe our technique for casting the image resizing problem as a nonhomogeneous scaling optimization over the triangular mesh. 3.1 Scale transformation errors We evaluate the quality of the scale transformation between the corresponding triangles using two error functions as follows. Scaling error. We would like to preserve the aspect ratios of salient regions during warping. Ideally, the important triangles should be homogeneously scaled, i.e., preserving their aspect ratios, while the others could be nonhomogeneously scaled. As the image is resized along the x- and/or y-directions, we expect that the scales are also along these directions without any rotations. Therefore, we associate each triangle t p M p by two unknown scaling factors s x t, s y t along x- and y- directions respectively. ( We ) denote this scale transformation by G t = t 0 s x 0 s y. t On the other hand, each triangle t p M p is equipped with a unique affine mapping from t p to t q. The linear portion of the affine mapping is denoted as a 2 2 Jacobian matrix which is constant per triangle. We denote this matrix as J t (q) to express its dependence on q. In particular, the elements of J t (q) are linearly dependent on vertices of triangle t q. We define the scaling error as: E s = t T σ t A t J t (q) G t 2 F (1) where A t is the area of triangle t p and F is the Frobenius norm. Minimization of the scaling error E s makes the linear part of the affine mapping be as close as possible to the nonhomogeneous scaling transformation G t, that is, all triangles tend to be nonhomogeneously scaled without rotations. Smoothness error. To avoid the discontinuity in the result image, we introduce a smoothness error. By smoothness, we are not referring to smoothness of the warped mesh itself, but rather smoothness of the scale transformations applied to the triangles. In particular, we require the scaling transformations applied within a region of the mesh to be as similar as possible. We formulate this constraint to apply between every two triangles that are adjacent in the mesh: E m = σ st A st G t G s 2 F (2) s,t T, s,t are adjacent where A st = (A s + A t )/2 and σ st = (σ s + σ t )/ Feature preservation error Each feature curve c is sampled and represented by a series of points c = {c 1,, c k }. As shown in Section 2.2, all these points and their edges c i c i+1 (i = 1,, k 1) are vertices and edges of the triangular mesh. Our goal is to preserve the shape of c in the result, that is, to preserve the similarity of the shape. Instead of representing a curve with a list of vertices or the intrinsic

4 4 terms using edge lengths and angles between edges [16] or the curve Lapacian coordinates [19], we use a scaleinvariant intrinsic terms using length ratios and angles of consecutive edges. Considering three consecutive points c p i, cp i+1, cp i+2 of c p in the source mesh M p, the relationship between the two edges is represented as (c p i+2 cp i+1 ) = r ir i (c p i+1 c p i ), where r i is the edge length ratio between c p i+1 cp i+2 and c p i cp i+1, θ i is the angle between the two edges, and ( ) cosθi sinθ R i = i is a rotation matrix of angle sinθ i cosθ i θ i. In particular, a feature line has all angles θ i = 0. To preserve the shape of c q in the target mesh M q, we minimize the following feature preservation error: k 2 E f = (c q i+2 cq i+1 ) r ir i (c q i+1 cq i ) 2 (3) i=1 Minimization of E f tries to preserve the edge ratios and angles of c q with those of c p. That is, the shape of c q is as similar as possible with c p by minimizing E f. 3.3 Total error function and implementation Our objective function E is the weighted sum of the three error functions: M q and the scale factors of all triangles. We denote all the variables as a big vector z. Then the energy function (4) can be simply written as E = Cz 2 (6) where C is a sparse matrix derived from (4). Thus minimization of (6) can be obtained by solving a sparse linear system. We use the direct solver [8] in our implementation. The Cholesky factorization of the coefficient matrix is first computed. Then the solution is obtained quickly. Most of the time is spent on computing the Cholesky factorization, while the time of the solving is negligible. Although we solve for both the mesh vertices of M q and the scale factors of all triangles, we are interested only in M q while the scale factors of triangles play an auxiliary role only. 4 Discussions In this section, we discuss about the generalization of our approach and propose an approach for preventing foldover in the resizing results. E = E s + λe m + µe f (4) where λ and µ are weights. Saliency constraints. The important triangles should better preserve their aspect ratios and less important triangles can be stretched in much extent. Therefore, we assign the expected scale transformation matrix for important triangles as ( ) βs 0 0 G t = 0 βs 0, t T (5) where s 0 = max{s x, s y } and β is a scalar to determine how large would we preserve the size of the important regions in the resized image I q. If we want to enlarge the salient regions in I q, we can set large value of β. Boundary constraints. We have to ensure that the coordinates of points on the rectangular boundary of the original image remain on the rectangular boundary of the new image. We have two types of constraints on the boundary. For the point q i on the left side of M q we have the positional constraint: qi x = 0. For the triangle t with an edge on the left side of M q we have the scale constraint: s y t = s y. The other 3 sets of boundary constraints are similar. Implementation. It is easy to see that the objective function (4) is quadratic with respect to the vertices of 4.1 Generalization of optimization Our approach is also partly inspired from the parameterization work of Liu et al. [11]. As in [11], the auxiliary transformation G t for each triangle can be taken from other family of allowed transformations in addition to the family of nonhomogeneous scale transformations. As a generalization, G t can be taken from the family of affine transformations, similarity transformations, or rotation transformations. We call the corresponding methods as AAAP (as-affine-as-possible), ASAP (as-similar-as-possible), and ARAP (as-rigid-aspossible), respectively. We have compared our approach with these approaches by a number of examples. Fig. 2 shows the result comparisons with the other approaches. In this example, the triangles in the region of two persons are set as important triangles. As ARAP method needs an initialization, we estimate edge lengths of triangles in the target mesh as in [6]. Our approach obtains similar result with that produced by [6] and performs better than other approaches with less distortion and stretch. For example, AAAP, ASAP, and ARAP might introduce rotations for some triangles which are not expected in image resizing.

5 5 (a) (b) (c) (d) (e) (f) Fig. 2 Compare with other approaches. (a) Original image. (b) Result of Guo et al. [6]. (c) Result of ARAP method. (d) Result of ASAP method. (e) Result of AAAP method. (f) Results of our approach. The two persons are salient regions. The method of [6] needs to estimate the edge lengths of triangles in advance and is slow due to the nonlinear optimization. AAAP, ASAP, and ARAP might introduce rotations in the triangles, thus produce more stretches in the result. 4.2 Folderover free optimization The global scaling optimization approach works remarkable with high efficiency, because only one sparse linear system need to be solved. However, we are not able to guarantee to avoid the foldovers in the result mesh, especially in the cases of extreme aspect ratio stretches. We introduce additional linear inequality constraints in our optimization framework to prevent foldovers. Acute triangle constraints. Each edge e q = q i q j in mesh M q should preserve its direction with its counterpart e p = p i p j in mesh M p. That is, (p x j p x i )(q x j q x i ) δ E, (p y j py i )(qy j qy i ) δ E, e q = q i q j M q (7) where δ E is a positive threshold. Obtuse triangle constraints. The inequality constraints (7) ensure that all acute triangles of the mesh do not flip. But for obtuse triangles, the direction constraints are not enough to avoid foldover. Denote the set of obtuse triangles in M p by T o. For an obtuse triangle t = (p i p j p k ) T o with obtuse edge p i p j, as shown in Fig. 3. Assuming that the triangle t is transformed to the target triangle (q i q j q k ). It is seen that the three edges of target triangle (q i q j q k ) all satisfy the direction constraints (7). However, the triangle still flips. Denote p k = ηt 1p i + η t 2p j as the perpendicular foot of p k onto the edge p i p j. Then the obtuse triangle constraints are defined as: (η t 1p x i + η t 2p x j p x k)(η t 1q x i + η t 2q x j q x k) δ O, (η t 1p y i + ηt 2p y j py k )(ηt 1q y i + ηt 2q y j qy k ) δ O, t = (p i p j p k ) T o (8) where δ O is a positive threshold. It can be easily proved that the obtuse triangles will not flip under inequality constraints (7) and (8) simultaneously. Quadratic programming. We have derived inequality constraints to prevent triangle foldovers. Note that the energy function in (4) is a convex quadratic form. Fig. 3 Obtuse triangle constraints. Now we have the problem of minimizing (6) with linear inequality constraints (7) and (8). This is a standard quadratic program whose global optimal solution can be obtained. 5 Experimental results All the examples presented in this paper were made on a PC with Duo CPU 1.8GHz and 2GB memory. We employed Intel MKL [8] to solve the sparse linear system in our optimization framework (Section 3) and adopted CVX MATLAB toolbox [5] to obtain the solution to the quadratic programming on foldover free resizing (Section 4.2). The averaged length of the edges in triangulations is within pixels. It takes about ms to solve the global scaling optimization and takes about 2-4 seconds to solve foldover free optimization for an image with resolution of In our system, when the user drag to resize the image, only the scaling optimization is performed so that the user can see the resized images in realtime. After the user finish the dragging operation, the foldover free optimization is then performed. There are a few of parameters in our approach. The weight parameters λ and µ in (4) measure the importance of transformation smoothness and feature preservation constraints respectively. In our implementation, we set λ = 0.5 and µ = 5. The parameter β in (6) is set as value of The two positive thresholds δ E in (7) and δ O in (8) are both set as 0.01.

6 6 The scalar parameter β in (5) emphasizes how important the salient regions are compared the image. Setting large value of β encourages salient regions to be large in the result image. Fig. 4 shows various results of setting different values of β in an example. Larger β enlarges the salient region (fish) in the resized image (Fig. 4(b)) while smaller β makes the fish shrinkage in the result (Fig. 4(f)). Although we have set constraints (6) on the auxiliary matrix G t of important triangles, there s no guarantee that their Jacobian matrix J t (q) in (1) has exactly the form of (6) because of the compensation to other parts of the energy function. Actually, the effect of preservation of salient triangles will be determined by J t (q). To ensure the J t (q) of important triangles to be close enough to their corresponding G t, we can specify large saliency values σ t of the important triangles t T. Fig. 5 shows such an example in which the salient regions are well preserved in the results by setting large value of σ t. Fig. 6 shows a few examples demonstrating the effectiveness of the feature preservation of our approach. Shapes of feature lines, curve, and circle are well preserved in the result images. Fig. 7 shows an example of foldover free resizing. In this example, both the adult and the kid are salient regions in the image. When the user decrease the width of the image, there are sever self intersections in the warped mesh as shown in the middle row. The result image has discontinuity in the region between the two persons. The foldover can be removed by using our approach as shown in the right column. A more extreme resizing example is shown in Fig. 8. In this example, the wolf is stretched a lot to avoid foldovers in the result. In Fig. 9 we compare with some state-of-the-art approaches for content-aware image resizing, including seam carving method (SC) [1], optimized scale-andstretch method (OSS) [20], and energy based method (EBD) [9]. In the first example, the Brasserie is the most salient object in the image. the aspect ratio of the Brasserie is preserved well while OSS preserves the aspect ratio but generates smaller Brasserie. The Brasseries are largely stretched by SC and EBD methods. In the second example, the girl is the most salient content, which is encouraged both in SC and our method. SC removes some less salient wall, while leaving some jaggy brick contours. The bricks in our result is aligned uniformly both horizontally and vertically. By contrast, the girl is shrinked in OSS and EBD, which is less satisfactory. In the third row, the riders become too small to be recognized in OSS and squashed in EBD. Both SC and our method succeed to produce similar satisfactory re- Fig. 5 Preserving the saliency region strictly by setting large saliency values. The horses are salient regions in the image. Left: original image; middle: resizing result by setting saliency value as 1.0 for the horse regions where the horses shrink in sizes; right: resizing result by setting saliency value as 10.0 for the horse regions where the horses have the same sizes with those in the original image. sults. Moreover, the riders appear in the location which is closer to the original position in our method than in SC. In the Greek wine example of the fourth row, the table and wines, which are referred as the most important content of the image in our method, are encouraged and totally preserved. Both tables and wines in the other three cases do not maintain their proper aspect ratio. Furthermore, legs of the chairs in SC even obtain a little distortion. In the fifth row, the sky between building are recognized as the most salient region due to the sharp contrast with the surrounding environment. The sky distorts obviously in SC. Though they are preserved better in OSS and EBD, a few roof lines or walls of the buildings still obtain relatively small distortion due to the fact that being lack of preservation of straight lines. In our method which encourages straight lines, the two features mentioned above are completely preserved. In the last result of sixth row, there exits multiple features, i.e., the penguins in the image. The second penguin from left shrinks obviously in SC. The other three methods obtain similar results, while the penguins are slightly more magnified in our method. Table 1 shows the timing statistics and some other information of examples shown in Fig. 9 using our method. We only show the running times of the scaling optimization in the table. The running time is dependent on the mesh resolution and independent on the image resolution. Therefore, our approach can resize large image in an interactive rate as well.

7 7 (a) (b) (c) (d) (e) (f) Fig. 4 Resized images with different values of β. The fish is the salient region in the images. (a) Original image; (b) β = 1.5; (c) β = 1.25; (d) β = 1.0; (e) β = 0.75; (b) β = 0.5. Fig. 6 Resizing results with preserving features like lines (upper row), curve (middle row), and circle (lower row). From left column to right column: the original images, the resizing results without preserving features, the specified features with sampling points shown in orange, the resizing results with preserving features. Table 1 Timing statistics of the examples in Fig. 9 using our method. Example Image Res. Mesh Res. β Time(ms) V/1436T V/2664T V/850T V/1536T V/2560T V/1758T Conclusion We have presented a novel framework for content-aware image resizing by a global nonhomogeneous scaling optimization. The optimization is performed by solving a sparse linear system and thus is able to be achieved with high efficiency. The resizing images can be obtained in an interactive rate. With some extra computational cost, the global scaling optimization can be changed to a quadratic programming in order to avoid foldover strictly in the results. A number of experimental results have shown the applicability and effectiveness of our approach. It is much interesting to develop and study some qualitative measures for content-aware image resizing [10, 14].

8 8 (a) (b) (c) (d) (e) Fig. 9 Comparisons with some state-of-the-art resizing methods: (a) Original image; (b) seam carving method (SC) [1]; (c) optimized scale-and-stretch method (OSS) [20]; (d) energy based method (EBD) [9]; (e) our method.

9 9 Fig. 7 Example of foldover free resizing. Left: original image. Middle: the resizing result without applying foldover free optimization. There are foldovers in the middle region (between the adult and the kid) of the image. Right: the resizing image without foldover using our foldover free optimization. Fig. 8 An example of extreme resizing. Acknowledgements Thanks to the many Flickr.com users whose images were used in this paper. This work is supported by the National Natural Science Foundation of China( , ), Technology Department of Zhejiang Province(No. 2008C ), and the 973 National Key Basic Research Foundation of China (No. 2009CB320801). 3. J. Canny, A computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(6): (1986) 4. L. Fernandes and M. Oliveira. Real-time line detection through improved hough transform voting schme. Pattern Recognition, 41(1): (2008) 5. M. Grant, S. Boyd, and Y.Ye, CVX: Matlab software for disciplined convex programming. boyd/cvx. 6. Y. Guo, F. Liu, J. Shi, Z. Zhou, and M. Gleicher, Image retargeting using mesh parametrization, IEEE Transactions on Multimedia, 11(5): (2009) 7. J. Harel, C. Koch, and P. Perona, Graph-based visual saliency, In Proc. Neural Information Processing Systems (NIPS) 19: (2006) 8. Intel math kernel libarary Z. Karni, D. Freedman, and C. Gotsman, Energy-Based content-aware image deformation. Computer Graphics Forum(Proc. Symposium on Geometry Processing), 28(5): (2009) 10. H. Kim and S. Kumara, New quality metrics for digital image resizing, Proc. SPIE, Vol. 6696, (2007) 11. L. Liu, L. Zhang, Y. Xu, C. Gotsman, and S. J. Gortler, A local/global approach to mesh parameterization, Computer Graphics Forum(Proc. Symposium on Geometry Processing), 27(5): (2008) 12. U. Pinkall, K. Polthier, Computing discrete minimal surface their conjugates. Experimental Mathematics, 2(1):15-36(1993) 13. Y. Pritch, E. Kav-Venaki, S. Peleg, Shift-map image editing, Proc. of ICCV (2009) 14. A. Reibman and S. Suthaharan, A no-reference spatial aliasing measure for digital image resizing, Proc. ICIP: (2008) 15. M. Rubinstein, A. Shamir, and S. Avidan, Improved seam carving for video retargeting. ACM Transactions on Graphics(Proc. SIGGRAPH), 27(3):1-9(2008) 16. T.W. Sederberg, P. Gao, G. Wang, H. Mu, 2D shape blending: an intrinsic solution to the vertex path problem, Proc. SIGGRAPH, 15-18(1993) 17. R. Seidel, Constrained delaunay triangulations and voronoi diagrams with obstacles. Technical Report, Inst. for Information Processing, Graz, Austria (1988) 18. A. Shamir and O. Sorkine, Visual media retargeting, In SIG- GRAPH Asia Course (2009) 19. O. Sorkine, Y. Lipman, D. Cohen-Or, M. Alexa, C. Rossl, H.-P. Seidel, Laplacian surface editing. Proc. Symposium on Geometry Processing, (2004). 20. Y.-S. Wang, C.L. Tai, O. Sorkine, and T.-Y. Lee, Optimized scale-and-stretch for image resizing, ACM Transactions on Graphics(Proc. SIGGRAPH Asia),27(5):1-8(2008) 21. L. Wolf, M. Guttmann, and D. Cohen-Or, Nonhomogeneous content-driven video-retargeting, Proc. ICCV, 1-6(2007) 22. L. Zhang, L. Liu, C. Gotsman, and S. J. Gortler. An asrigid-as-possible approach to sensor network localization. ACM Transactions on Sensor Networks, 6(4):1-12(2010) References 1. S. Avidan and A. Shamir, Seam carving for content-aware image resizing, ACM Transactions on Graphics(Proc. SIG- GRAPH), 26(3): (2007) 2. C. Barnes, E. Shechtman, A. Finkelstein, and D. Goldman, PatchMatch: a randomized correspondence algorithm for structural image editing, ACM Transactions on Graphics (Proc. SIGGRAPH), 28(3) (2009)