article

TILT: Transform Invariant Low-Rank Textures

Authors:

Zhengdong Zhang,

Yi MaAuthors Info & Claims

International Journal of Computer Vision, Volume 99, Issue 1

Pages 1 - 24

https://doi.org/10.1007/s11263-012-0515-x

Published: 01 August 2012 Publication History

Abstract

In this paper, we propose a new tool to efficiently extract a class of "low-rank textures" in a 3D scene from user-specified windows in 2D images despite significant corruptions and warping. The low-rank textures capture geometrically meaningful structures in an image, which encompass conventional local features such as edges and corners as well as many kinds of regular, symmetric patterns ubiquitous in urban environments and man-made objects. Our approach to finding these low-rank textures leverages the recent breakthroughs in convex optimization that enable robust recovery of a high-dimensional low-rank matrix despite gross sparse errors. In the case of planar regions with significant affine or projective deformation, our method can accurately recover both the intrinsic low-rank texture and the unknown transformation, and hence both the geometry and appearance of the associated planar region in 3D. Extensive experimental results demonstrate that this new technique works effectively for many regular and near-regular patterns or objects that are approximately low-rank, such as symmetrical patterns, building facades, printed text, and human faces.

References

[1]

Baker, S., & Matthews, I. (2004). Lucas-Kanade 20 years on: A unifying framework. International Journal of Computer Vision, 56(3), 221-255.

Digital Library

[2]

Becker, S., Candès, E., & Grant, M. (2011). Templates for convex cone problems with applications to sparse signal recovery. Mathematical Programming Computation, 3(3), 165-218.

[3]

Bertsekas, D. P. (2004). Nonlinear programming. Belmont: Athena Scientific.

[4]

Cai, J., Candès, E., & Shen, Z. (2010). A singular value thresholding algorithm for matrix completion. SIAM Journal on Optimization, 20(4), 1956-1982.

[5]

Candès, E., & Recht, B. (2009). Exact matrix completion via convex optimization. Foundations of Computational Mathematics, 9(6), 717-772.

[6]

Candès, E., & Tao, T. (2010). The power of convex relaxation: Near-optimal matrix completion. IEEE Transactions on Information Theory, 56(5), 2053-2080.

Digital Library

[7]

Candès, E., Li, X., Ma, Y., & Wright, J. (2011). Robust principal component analysis? Journal of the ACM, 58(3), 11:1-11:37.

Digital Library

[8]

Canny, J. (1986). A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(6), 679-698.

Digital Library

[9]

Chandrasekaran, V., Sanghavi, S., Parrilo, P., & Willsky, A. (2011). Rank-sparsity incoherence for matrix decomposition. SIAM Journal on Optimization, 21(2), 572-596.

[10]

Chang, S., Davis, L., Dunn, S., Eklundh, J., & Rosenfeld, A. (1987). Texture discrimination by projective invariants. Pattern Recognition Letters, 5, 337-342.

Digital Library

[11]

Chen, J. L., & Kundu, A. (1994). Rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden Markov model. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(2), 208-214.

Digital Library

[12]

Cohen, F., Fan, Z., & Patel, M. (1991). Classification of rotated and scaled texture images using Gaussian Markov random field models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 192-202.

Digital Library

[13]

Cromme, L. (1978). Strong uniqueness: A far-reaching criterion for the convergence analysis of iterative procedures. Numerische Mathematik, 29, 179-193.

Digital Library

[14]

Do, M., & Vetterli, M (2002). Rotation invariant texture characterization and retrieval using steerable wavelet-domain hidden Markov models. IEEE Transactions on Multimedia, 4(4), 517-527.

Digital Library

[15]

Eckstein, J., & Bertsekas, D. (1992). On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55, 293-318.

[16]

Frey, B., & Jojic, N. (1999). Transformed component analysis: Joint estimation of spatial transformations and image components. In Proc. of IEEE international conference on computer vision.

[17]

Gabay, D., & Mercier, B. (1976). A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Computers and Mathematics with Applications, 2, 17-40.

[18]

Ganesh, A., Lin, Z., Wright, J., Wu, L., Chen, M., & Ma, Y. (2009). Fast algorithms or recovering a corrupted low-rank matrix. In Proceedings of third international workshop on computational advances in multi-sensor adaptive processing.

[19]

Garding, J., & Lindeberg, T. (1996). Direct computation of shape cues using scale-adapted spatial derivative operators. International Journal of Computer Vision, 17(2), 163-191.

Digital Library

[20]

Glowinski, R., & Marroco, A. (1975). Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualite, d'une classe de problemes de Dirichlet non lineares. Revuew Francaise d'Automatique, Informatique et Recherche Operationelle, 9, 41-76.

[21]

Greenspan, H., Belongie, S., Goodman, R., & Perona, P. (1994). Rotation invariant texture recognition using a steerable pyramid. In Proc. of international conference on pattern recognition.

[22]

Haley, G., & Manjunath, B. (1999). Rotation-invariant texture classification using a complete space-frequency model. IEEE Transactions on Image Processing, 8, 255-269.

Digital Library

[23]

Harris, C., & Stephens, M. (1988). A combined corner and edge detector. In Proc. of 4th alvey vision conference.

[24]

He, B. (2009). Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities. Computational Optimization and Applications, 42(2).

[25]

He, B., Tao, M., & Yuan, X. (2011). Alternating direction method with Gaussian back substitution for separable convex programming. SIAM Journal on Optimization, under-revision.

[26]

Jittorntrum, K., & Osborne, M. (1980). Strong uniqueness and second order convergence in nonlinear discrete approximation. Numerische Mathematik, 34, 439-455.

Digital Library

[27]

Kondepudy, R., & Healey, G. (1994). Using moment invariants to analyze 3-D color textures. In Proc. of IEEE international conference on image processing.

[28]

Kosecka, J., & Zhang, W. (2005). Extraction, matching, and pose recovery based on dominant rectangular structures. CVGIP. Image Understanding, 100(3), 274-293.

Digital Library

[29]

Lee, S., & Liu, Y. (2010). Skewed rotation symmetry group detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(9), 1659-1672.

Digital Library

[30]

Leung, T., & Malik, J. (2001). Representing and recognizing the visual appearance of materials using three-dimensional textons. International Journal of Computer Vision, 43(1), 29-44.

Digital Library

[31]

Levina, E., & Bickel, P. J. (2006). Texture synthesis and non-parametric resampling of random fields. Annals of Statistics, 34(4), 1751-1773.

[32]

Lin, Z., Chen, M., Wu, L., & Ma, Y. (2009). The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices (UIUC Technical Report UILU-ENG-09-2215).

[33]

Liu, R., Lin, Z., Wei, S., & Su, Z. (2011). Solving principal component pursuit in linear time via l₁ filtering (Preprint).

[34]

Liu, Y., & Collins, R. (2001). Skewed symmetry groups. In Proc. of IEEE conference on computer vision and pattern recognition.

[35]

Liu, Y., Collins, R., & Tsin, Y. (2004). A computational model for periodic pattern perception based on Frieze and wallpaper groups. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(3), 354-371.

Digital Library

[36]

Liu, Y., Hel-Or, H., Kaplan, C. S., & Gool, L. V. (2010). Computational symmetry in computer vision and computer graphics. Foundations and Trends in Computer Graphics and Vision, 5(1-2), 1-195.

[37]

Lowe, D. G. (2004). Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60, 91-110.

Digital Library

[38]

Ma, Y., Soatto, S., Kosecka, J., & Sastry, S. (2004). An invitation to 3D vision. Berlin: Springer.

[39]

Madiraju, S., & Liu, C. (1994). Rotation invariant texture classification using covariance. In Proc. of IEEE international conference on image processing.

[40]

Mikolajczyk, K., & Schmid, C. (2004). Scale and affine invariant interest point descriptors. International Journal of Computer Vision, 60(1), 63-86.

[41]

Mikolajczyk, K., & Schmid, C. (2005). A performance evaluation of local descriptors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27, 1615-1630.

Digital Library

[42]

Mobahi, H., Zhou, Z., Yang, A., & Ma, Y. (2011). Holistic 3D reconstruction of urban structures from low-rank textures. In Third international IEEE workshop on 3D representation and recognition.

[43]

Morel, J. M., & Yu, G. (2009). ASIFT: A new framework for fully affine invariant image comparison. SIAM Journal on Imaging Sciences, 2(2), 438-469.

Digital Library

[44]

Park, M., Lee, S., Chen, P., Kashyap, S., Butt, A., & Liu, Y. (2008). Performance evaluation of state-of-the-art discrete symmetry detection algorithms. In Proc. of IEEE conference on computer vision and pattern recognition.

[45]

Park, M., Brocklehurst, K., Collins, R., & Liu, Y. (2010). Translation-symmetry-based perceptual grouping with applications to urban scenes. In Asian conference on computer vision.

[46]

Peng, Y., Ganesh, A., Wright, J., Xu, W., & Ma, Y. (2010). RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images. In Proc. of IEEE conference on computer vision and pattern recognition.

[47]

Peng, Y., Ganesh, A., Wright, J., Xu, W., & Ma, Y. (2011). RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images. IEEE Transactions on Pattern Analysis and Machine Intelligence, to appear.

[48]

Recht, B., Fazel, M., & Parillo, P. (2010). Guaranteed minimum rank solution of matrix equations via nuclear norm minimization. SIAM Review, 52(3), 471-501.

Digital Library

[49]

Schindler, G., Krishnamurthy, P., Lublinerman, R., Liu, Y., & Dellaert, F. (2008). Detecting and matching repeated patterns for automatic geo-tagging in urban environments. In Proc. of IEEE conference on computer vision and pattern recognition.

[50]

Sundaramoorthi, G., Petersen, P., Varadarajan, V. S., & Soatto, S. (2009). On the set of images modulo viewpoint and contrast changes. In Proc. of IEEE conference on computer vision and pattern recognition.

[51]

Toh, K., & Yun, S. (2010). An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pacific Journal of Optimization, 6, 615-640.

[52]

Winder, S., & Brown, M. (2007). Learning local image descriptor. In Proc. of IEEE conference on computer vision and pattern recognition.

[53]

Wu, W. R., & Wei, S. C. (1996). Rotation and gray-scale transform-invariant texture classification using spiral resampling, subband decomposition and hidden Markov model. IEEE Transactions on Image Processing, 5, 1423-1434.

Digital Library

[54]

Yang, A., Huang, K., Rao, S., & Ma, Y. (2005). Symmetry-based 3- D reconstruction from perspective images. Computer Vision and Image Understanding, 99(2), 210-240.

Digital Library

[55]

Yuan, X., & Tao, M. (2011). Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM Journal on Optimization, 21(1), 57-81.

[56]

Zhang, Z., Liang, X., & Ma, Y. (2011a). Unwrapping low-rank textures on generalized cylindrical surfaces. In Proc. of IEEE international conference on computer vision.

[57]

Zhang, Z., Matsushita, Y., & Ma, Y. (2011b). Camera calibration with lens distortion from low-rank textures. In Proc. of IEEE conference on computer vision and pattern recognition.

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https://dl.acm.org/doi/10.1007/s10851-024-01216-8
Ryou DYouwang KOh T(2024)Multi-stage adaptive rank statistic pruning for lightweight human 3D mesh recovery modelThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-023-02798-x40:2(535-543)Online publication date: 1-Feb-2024
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TILT: Transform Invariant Low-Rank Textures
1. Computing methodologies
  1. Artificial intelligence
    1. Computer vision
  2. Computer graphics

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Published In

cover image International Journal of Computer Vision

International Journal of Computer Vision Volume 99, Issue 1

August 2012

121 pages

ISSN:0920-5691

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Copyright © Copyright © 2012 Springer Science+Business Media, LLC.

Publisher

Kluwer Academic Publishers

United States

Publication History

Published: 01 August 2012

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Cited By

Li KWen YChan R(2024)Cartoon–Texture Image Decomposition Using Least Squares and Low-Rank RegularizationJournal of Mathematical Imaging and Vision10.1007/s10851-024-01216-867:1Online publication date: 30-Dec-2024
https://dl.acm.org/doi/10.1007/s10851-024-01216-8
Ryou DYouwang KOh T(2024)Multi-stage adaptive rank statistic pruning for lightweight human 3D mesh recovery modelThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-023-02798-x40:2(535-543)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s00371-023-02798-x
Skoryukina NTropin DShemiakina JArlazarov V(2023)Document Localization and Classification As Stages of a Document Recognition SystemPattern Recognition and Image Analysis10.1134/S105466182304043033:4(699-716)Online publication date: 1-Dec-2023
https://dl.acm.org/doi/10.1134/S1054661823040430
Tao YZhang YChen X(2022)Element-Arrangement Context Network for Facade ParsingJournal of Computer Science and Technology10.1007/s11390-022-2189-337:3(652-665)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s11390-022-2189-3
Liu XHao CSu ZQi ZFu SLi YHan H(2022)Image inpainting algorithm based on tensor decomposition and weighted nuclear normMultimedia Tools and Applications10.1007/s11042-022-12635-382:3(3433-3458)Online publication date: 5-Jul-2022
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Yu CChang YLi YZhao XYan LShen HZhuang YSmith JYang YCesar PMetze FPrabhakaran B(2021)Unsupervised Image Deraining: Optimization Model Driven Deep CNNProceedings of the 29th ACM International Conference on Multimedia10.1145/3474085.3475441(2634-2642)Online publication date: 17-Oct-2021
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Qiu DBai MNg MZhang X(2021)Robust Low Transformed Multi-Rank Tensor Methods for Image AlignmentJournal of Scientific Computing10.1007/s10915-021-01437-887:1Online publication date: 1-Apr-2021
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Liu WXu JLiu YWu Y(2020)A Robust Progressive Text Line Segmentation Framework with Markov Line DescriptorsProceedings of the 2020 4th International Conference on Video and Image Processing10.1145/3447450.3447482(199-212)Online publication date: 25-Dec-2020
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Zhang XWang DZhou ZMa Y(2020)Robust Low-Rank Tensor Recovery with Rectification and AlignmentIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2019.292904343:1(238-255)Online publication date: 3-Dec-2020
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