# TRACKING SOLUTIONS OF TIME VARYING LINEAR INVERSE PROBLEMS

### Martin Kleinsteuber, Simon Hawe

#### Abstract

The reconstruction of a signal from only a few measurements, deconvolving, or denoising are only a few interesting signal processing applications that can be formulated as linear inverse problems. Commonly, one overcomes the ill-posedness of such problems by finding solutions which best match some prior assumptions. These are often sparsity assumptions as in the theory of Compressive Sensing. In this paper, we propose a method to track solutions of linear inverse problems. We assume that the corresponding solutions vary smoothly over time. A discretized Newton flow allows to incorporate the time varying information for tracking and predicting the subsequent solution. This prediction requires to solve a linear system of equation, which is in general computationally cheaper than solving a new inverse problem. It may also serve as an additional prior that takes the smooth variation of the solutions into account, or, as an initial guess for the preceding reconstruction. We exemplify our approach with the reconstruction of a compressively sampled synthetic video sequence.

#### References

- Baumann, M., Helmke, U., and Manton, J. (2005). Reliable tracking algorithms for principal and minor eigenvector computations. In 44th IEEE Conference on Decision and Control and European Control Conference, pages 7258-7263.
- Becker, S., Bobin, J., and Candès, E. J. (2009). Nesta: a fast and accurate first-order method for sparse recovery. SIAM Journal on Imaging Sciences, 4(1):1-39.
- Benke, G. (1994). Generalized rudin-shapiro systems. Journal of Fourier Analysis and Applications, 1(1):87- 101.
- Bioucas-Dias, J. and Figueiredo, M. (2007). A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration. Image Processing, IEEE Transactions on, 16(12):2992 -3004.
- Candès, E. J. and Romberg, J. (2007). Sparsity and incoherence in compressive sampling. Inverse Problems, 23(3):969-985.
- Chartrand, R. and Staneva, V. (2008). Restricted isometry properties and nonconvex compressive sensing. Inverse Problems, 24(3):1-14.
- Combettes, P. L. and Pesquet, J. C. (2004). Image restoration subject to a total variation constraint. IEEE Transactions on Image Processing, 13(9):1213-1222.
- Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289-1306.
- Donoho, D. L. and Elad, M. (2003). Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100(5):2197-2202.
- Elad, M., Milanfar, P., and Rubinstein, R. (2007). Analysis versus synthesis in signal priors. Inverse Problems, 3(3):947-968.
- Hawe, S., Kleinsteuber, M., and Diepold, K. (2011). Dense disparity maps from sparse disparity measurements. In IEEE 13th International Conference on Computer Vision.
- Lustig, M., Donoho, D., and Pauly, J. M. (2007). Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine, 58(6):1182-1195.
- Nocedal, J. and Wright, S. J. (2006). Numerical Optimization, 2nd Ed. Springer, New York.
- Romberg, J. (2008). Imaging via compressive sampling. IEEE Signal Processing Magazine, 25(2):14-20.
- Rudin, L. I., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Phys. D, 60(1-4):259-268.

#### Paper Citation

#### in Harvard Style

Kleinsteuber M. and Hawe S. (2012). **TRACKING SOLUTIONS OF TIME VARYING LINEAR INVERSE PROBLEMS** . In *Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,* ISBN 978-989-8425-98-0, pages 253-257. DOI: 10.5220/0003864102530257

#### in Bibtex Style

@conference{icpram12,

author={Martin Kleinsteuber and Simon Hawe},

title={TRACKING SOLUTIONS OF TIME VARYING LINEAR INVERSE PROBLEMS},

booktitle={Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

year={2012},

pages={253-257},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003864102530257},

isbn={978-989-8425-98-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,

TI - TRACKING SOLUTIONS OF TIME VARYING LINEAR INVERSE PROBLEMS

SN - 978-989-8425-98-0

AU - Kleinsteuber M.

AU - Hawe S.

PY - 2012

SP - 253

EP - 257

DO - 10.5220/0003864102530257