Average Error Based Adaptive Regularization Control For The Gradient Constancy Variational Stereo Model

Authors

  • Izhar Ali Amur Department of General Faculty, Shaheed Benazir Bhutto University, Shaheed Benazir Abad, Nawabshah, Pakistan
  • Khuda Bux Amur Department Mathematics and Statistics, QUEST Nawabshah, SBA, Pakistan
  • Muzaffar Bashir Arain Department Mathematics and Statistics, QUEST Nawabshah, SBA, Pakistan
  • Mohsin Ali Amur Department of General Faculty, Shaheed Benazir Bhutto University, Shaheed Benazir Abad, Nawabshah, Pakistan
  • K.N Memon Department Mathematics and Statistics, QUEST Nawabshah, SBA, Pakistan

DOI:

https://doi.org/10.21015/vtm.v11i1.1492

Abstract

The study of the stereo vision problems is most crucial and challenging task in image processing and computer vision. The Stereo Vision problems address the investigation of the correspondence between the two images of same scene (stereo pairs) captured from two different views. Generally, these problems are inverse and ill posed. To deal with such problems the energy-based regularization techniques are considered as an efficient and most successful approaches. The adaptive finite element method is used here as discretization method for the partial differential equation obtained from the optimization of the designed energy functional. Such type of the regularization generally depends on the smoothness parameters, and their suitable choice. The choice of the smoothing parameter in an adaptive way and specifically their choice as a scalar function over the whole computational domain is an interesting idea. In this work, a variational model based on the gradient constancy assumption is proposed, moreover a post optimization method (mesh refinement strategy) is designed which is based on a priori estimate called average absolute disparity error estimate. The post optimization is based on an adaptive intelligent algorithm which is efficient in identifying the less regular regions of the computed disparity image and reduces the value of the parameter to refine the grid. Consequently, the smoothness appears in the solution which is the main goal.

References

Amur, K. [2012], ‘Some regularization strategies for an ill-posed denoising problem’, International Journal

of Tomography and Statistics 19(1), 46–59.

Amur, K. [2013], ‘A posteriori control of regularization for complementary image motion problem’,

Sindh University Research Journal-SURJ (Science Series) 45(3).

AMUR, K., CHANDIO, M. and ALI, E. [2013], ‘A novel mesh adaptation technique for the computation

of stereo depth using texture less simple stereo pairs’, Sindh University Research Journal-SURJ (Science

Series) 45(4).

Aubert, G., Kornprobst, P. and Aubert, G. [2006], Mathematical problems in image processing: partial

differential equations and the calculus of variations, Vol. 147, Springer.

Belhachmi, Z. and Hecht, F. [2011], ‘Control of the effects of regularization on variational optic flow

computations’, Journal of Mathematical Imaging and Vision 40, 1–19.

Ben-Ari, R. and Sochen, N. [2007], Variational stereo vision with sharp discontinuities and occlusion

handling, in ‘2007 IEEE 11th International Conference on Computer Vision’, IEEE, pp. 1–7.

Brox, T., Bruhn, A., Papenberg, N. and Weickert, J. [2004], High accuracy optical flow estimation based

on a theory for warping, in ‘Computer Vision-ECCV 2004: 8th European Conference on Computer

Vision, Prague, Czech Republic, May 11-14, 2004. Proceedings, Part IV 8’, Springer, pp. 25–36.

Bruhn, A., Weickert, J. and Schnörr, C. [2005], ‘Lucas/kanade meets horn/schunck: Combining local

and global optic flow methods’, International journal of computer vision 61, 211–231.

Chambolle, A. [2004], ‘An algorithm for total variation minimization and applications’, Journal of Mathematical

imaging and vision 20, 89–97.

Grimson, W. E. L. [1985], ‘Computational experiments with a feature based stereo algorithm’, IEEE

Transactions on pattern analysis and machine intelligence (1), 17–34.

Hamid, M. S., Abd Manap, N., Hamzah, R. A. and Kadmin, A. F. [2022], ‘Stereo matching algorithm

based on deep learning: A survey’, Journal of King Saud University-Computer and Information Sciences

(5), 1663–1673.

Hamzah, R. A., Ibrahim, H. and Hassan, A. H. A. [2017], ‘Stereo matching algorithm based on per pixel

difference adjustment, iterative guided filter and graph segmentation’, Journal of Visual Communication

and Image Representation 42, 145–160.

Horn, B. K. and Schunck, B. G. [1981], ‘Determining optical flow’, Artificial intelligence 17(1-3), 185–203.

Ji, P., Li, J., Li, H. and Liu, X. [2021], ‘Superpixel alpha-expansion and normal adjustment for stereo

matching’, Journal of Visual Communication and Image Representation 79, 103238.

Kichenassamy, S. [1997], ‘The perona–malik paradox’, SIAM Journal on Applied Mathematics 57(5), 1328–

Klaus, A., Sormann, M. and Karner, K. [2006], Segment-based stereo matching using belief propagation

and a self-adapting dissimilarity measure, in ‘18th International Conference on Pattern Recognition

(ICPR’06)’, Vol. 3, IEEE, pp. 15–18.

Kolmogorov, V. and Zabih, R. [2002], Multi-camera scene reconstruction via graph cuts, in ‘Computer

Vision—ECCV 2002: 7th European Conference on Computer Vision Copenhagen, Denmark, May 28–

, 2002 Proceedings, Part III 7’, Springer, pp. 82–96.

Lei, C., Selzer, J. and Yang, Y.-H. [2006], Region-tree based stereo using dynamic programming optimization,

in ‘2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition

(CVPR’06)’, Vol. 2, IEEE, pp. 2378–2385.

Li, G. and Zucker, S. W. [2006], Differential geometric consistency extends stereo to curved surfaces,

in ‘Computer Vision–ECCV 2006: 9th European Conference on Computer Vision, Graz, Austria, May

-13, 2006, Proceedings, Part III 9’, Springer, pp. 44–57.

Lin, X., Wang, J. and Lin, C. [2020], Research on 3d reconstruction in binocular stereo vision based on

feature point matching method, in ‘2020 IEEE 3rd International Conference on Information Systems

and Computer Aided Education (ICISCAE)’, IEEE, pp. 551–556.

Marr, D. and Poggio, T. [1976], ‘Cooperative computation of stereo disparity: A cooperative algorithm

is derived for extracting disparity information from stereo image pairs.’, Science 194(4262), 283–287.

Mozerov, M. G. and Van De Weijer, J. [2015], ‘Accurate stereo matching by two-step energy minimization’,

IEEE Transactions on Image Processing 24(3), 1153–1163.

Scharstein, D. and Szeliski, R. [2002], ‘A taxonomy and evaluation of dense two-frame stereo correspondence

algorithms’, International journal of computer vision 47, 7–42.

Slesareva, N., Bruhn, A. and Weickert, J. [2005], Optic flow goes stereo: A variational method for

estimating discontinuity-preserving dense disparity maps, in ‘Pattern Recognition: 27th DAGM Symposium,

Vienna, Austria, August 31-September 2, 2005. Proceedings 27’, Springer, pp. 33–40.

Tihonov, A. N. [1963], ‘Solution of incorrectly formulated problems and the regularization method’,

Soviet Math. 4, 1035–1038.

Tikhonov, A. N., Goncharsky, A., Stepanov, V. V. and Yagola, A. G. [1995], Numerical methods for the

solution of ill-posed problems, Vol. 328, Springer Science & Business Media.

Xiang, X., Zhang, R., Zhai, M., Xiao, D. and Bai, E. [2018], ‘Scene flow estimation based on adaptive

anisotropic total variation flow-driven method’, Mathematical Problems in Engineering 2018.

Yan, T., Gan, Y., Xia, Z. and Zhao, Q. [2019], ‘Segment-based disparity refinement with occlusion handling

for stereo matching’, IEEE Transactions on Image Processing 28(8), 3885–3897.

Zimmer, H., Bruhn, A., Valgaerts, L., Breuß, M., Weickert, J., Rosenhahn, B. and Seidel, H.-P. [2008],

Pde-based anisotropic disparity-driven stereo vision., in ‘VMV’, pp. 263–272.

Downloads

Published

2023-06-06

How to Cite

Amur, I. A., Amur, K. B., Arain, M. B., Amur, M. A., & Memon, K. (2023). Average Error Based Adaptive Regularization Control For The Gradient Constancy Variational Stereo Model. VFAST Transactions on Mathematics, 11(1), 156–169. https://doi.org/10.21015/vtm.v11i1.1492