Sparse representation utilizing tight frame for phase retrieval

Overcomplete representation is attracting interest in image restoration due to its potential to generate sparse representations of signals. However, the problem of seeking sparse representation must be unstable in the presence of noise. Restricted Isometry Property (RIP), playing a crucial role in providing stable sparse representation, has been ignored in the existing sparse models as it is hard to integrate into the conventional sparse models as a regularizer. In this paper, we propose a stable sparse model with non-tight frame (SSM-NTF) via applying the corresponding frame condition to approximate RIP. Our SSM-NTF model takes into account the advantage of the traditional sparse model, and meanwhile contains RIP and closed-form expression of sparse coefficients which ensure stable recovery. Moreover, benefitting from the pair-wise of the non-tight frame (the original frame and its dual frame), our SSM-NTF model combines a synthesis sparse system and an analysis sparse system. By enforcing the frame bounds and applying a second-order truncated series to approximate the inverse frame operator, we formulate a dictionary pair (frame pair) learning model along with a two-phase iterative algorithm. Extensive experimental results on image restoration tasks such as denoising, super resolution and inpainting show that our proposed SSM-NTF achieves superior recovery performance in terms of both subjective and objective quality.

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Incoherent Dictionary Learning Method Based on Unit Norm Tight Frame and Manifold Optimization for Sparse Representation

Mathematical Problems in Engineering ◽

10.1155/2016/9407503 ◽

2016 ◽

Vol 2016 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

HongZhong Tang ◽

Xiaogang Zhang ◽

Hua Chen ◽

Ling Zhu ◽

Xiang Wang ◽

...

Keyword(s):

Sparse Representation ◽

Optimization Problem ◽

Tight Frame ◽

Gram Matrix ◽

Mutual Coherence ◽

Learned Dictionary ◽

Manifold Optimization ◽

Audio Data ◽

Update Process ◽

Unit Norm

Optimizing the mutual coherence of a learned dictionary plays an important role in sparse representation and compressed sensing. In this paper, a efficient framework is developed to learn an incoherent dictionary for sparse representation. In particular, the coherence of a previous dictionary (or Gram matrix) is reduced sequentially by finding a new dictionary (or Gram matrix), which is closest to the reference unit norm tight frame of the previous dictionary (or Gram matrix). The optimization problem can be solved by restricting the tightness and coherence alternately at each iteration of the algorithm. The significant and different aspect of our proposed framework is that the learned dictionary can approximate an equiangular tight frame. Furthermore, manifold optimization is used to avoid the degeneracy of sparse representation while only reducing the coherence of the learned dictionary. This can be performed after the dictionary update process rather than during the dictionary update process. Experiments on synthetic and real audio data show that our proposed methods give notable improvements in lower coherence, have faster running times, and are extremely robust compared to several existing methods.

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Quantitative analysis by reconstruction of HREM focal series

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100171432 ◽

1994 ◽

Vol 52 ◽

pp. 740-741

Author(s):

W. Coene ◽

A. Thust ◽

M. Op de Beeck ◽

D. Van Dyck

Keyword(s):

Phase Retrieval ◽

Image Plane ◽

Initial Guess ◽

Recursive Least Squares ◽

Objective Lens ◽

Electron Wave ◽

Electron Sources ◽

Original Algorithm ◽

Coherent Field ◽

Direct Interpretation

Compared to conventional electron sources, the use of a highly coherent field-emission gun (FEG) in TEM improves the information resolution considerably. A direct interpretation of this extra information, however, is hampered since amplitude and phase of the electron wave are scrambled in a complicated way upon transfer from the specimen exit plane through the objective lens towards the image plane. In order to make the additional high-resolution information interpretable, a phase retrieval procedure is applied, which yields the aberration-corrected electron wave from a focal series of HRTEM images (Coene et al, 1992).Kirkland (1984) tackled non-linear image reconstruction using a recursive least-squares formalism in which the electron wave is modified stepwise towards the solution which optimally matches the contrast features in the experimental through-focus series. The original algorithm suffers from two major drawbacks : first, the result depends strongly on the quality of the initial guess of the first step, second, the processing time is impractically high.

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Limitations to the Spacing Effect

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie) ◽

10.1027/1618-3169.52.4.257 ◽

2005 ◽

Vol 52 (4) ◽

pp. 257-263 ◽

Cited By ~ 22

Author(s):

Peter P. J. L. Verkoeijen ◽

Remy M. J. P. Rikers ◽

Henk G. Schmidt

Keyword(s):

Free Recall ◽

Phase Retrieval ◽

Incidental Learning ◽

Factor Model ◽

Recall Performance ◽

Spacing Effect ◽

Word List ◽

Intentional Learning ◽

Study Phase ◽

Learning Condition

Abstract. The spacing effect refers to the finding that memory for repeated items improves when the interrepetition interval increases. To explain the spacing effect in free-recall tasks, a two-factor model has been put forward that combines mechanisms of contextual variability and study-phase retrieval (e.g., Raaijmakers, 2003 ; Verkoeijen, Rikers, & Schmidt, 2004 ). An important, yet untested, implication of this model is that free recall of repetitions should follow an inverted u-shaped relationship with interrepetition spacing. To demonstrate the suggested relationship an experiment was conducted. Participants studied a word list, consisting of items repeated at different interrepetition intervals, either under incidental or under intentional learning instructions. Subsequently, participants received a free-recall test. The results revealed an inverted u-shaped relationship between free recall and interrepetition spacing in both the incidental-learning condition and the intentional-learning condition. Moreover, for intentionally learned repetitions, the maximum free-recall performance was located at a longer interrepetition interval than for incidentally learned repetitions. These findings are interpreted in terms of the two-factor model of spacing effects in free-recall tasks.

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