Optimal Isotropic Wavelets for Localized Tight Frame Representations

Ward, John Paul; Pad, Pedram; Unser, Michaël

doi:10.1109/lsp.2015.2448233

Cited by 8 publications

(7 citation statements)

References 26 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our preliminary investigations have revealed the practical benefits of such an optimization [22]- [24]. In this work, we extend the framework and present a unified treatment that relies on more sophisticated tools and improves on our previous findings.…”

mentioning

confidence: 62%

“…More precisely, the energy of a function computed over some spatial neighbourhood should be well represented by the wavelet coefficients associated to that neighbourhood and its vicinity. According to [22], if f = m∈Z 2 f m is an L 2 -function from R 2 to R and f m is the restriction of f to the unit square centered at m, then…”

Section: A Measures Of Localizationmentioning

confidence: 99%

“…First of all, it has been shown analytically [22] that the Simoncelli wavelet minimizes the criterion U 2D . Thus, we already know the optimal profile with respect to the measure U 2D .…”

Section: Numerical Optimizationmentioning

confidence: 99%

“…Moreover, it has been theoretically shown that wavelets with better localization are more efficient for decoupling and sparsifying signals [21]. It is worth mentioning that the Simoncelli wavelet performs well in a wide range of applications and is shown to be the most-localized wavelet in a specific sense [22].…”

mentioning

confidence: 98%

See 3 more Smart Citations

Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

Pad

Uhlmann

Unser

2016

IEEE Trans. on Image Process.

Self Cite

View full text Add to dashboard Cite

Abstract-A crucial component of steerable wavelets is the radial profile of the generating function in the frequency domain. In this paper, we present an infinite-dimensional optimization scheme that helps us find the optimal profile for a given criterion over the space of tight frames. We consider two classes of criteria that measure the localization of the wavelet. The first class specifies the spatial localization of the wavelet profile, and the second that of the resulting wavelet coefficients. From these metrics and the proposed algorithm, we construct tight wavelet frames that are optimally localized and provide their analytical expression. In particular, one of the considered criterion helps us finding back the popular Simoncelli wavelet profile. Finally, the investigation of local orientation estimation, image reconstruction from detected contours in the wavelet domain, and denoising indicate that optimizing wavelet localization improves the performance of steerable wavelets, since our new wavelets outperform the traditional ones.

show abstract

mentioning

confidence: 62%

Section: A Measures Of Localizationmentioning

confidence: 99%

“…First of all, it has been shown analytically [22] that the Simoncelli wavelet minimizes the criterion U 2D . Thus, we already know the optimal profile with respect to the measure U 2D .…”

Section: Numerical Optimizationmentioning

confidence: 99%

mentioning

confidence: 98%

See 2 more Smart Citations

Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

Pad

Uhlmann

Unser

2016

IEEE Trans. on Image Process.

Self Cite

View full text Add to dashboard Cite

show abstract

“…To maintain the property of translation-invariance, the undecimated wavelet transform was used. The wavelet framework used is based on isotropicwavelets with optimal bandwidth [33], [61]. The dimension M of the Riesz transform of each scale for the orders 1 to 4 are 3, 6, 10 and 15 respectively.…”

Section: Methodsmentioning

confidence: 99%

3D Solid Texture Classification Using Locally-Oriented Wavelet Transforms

Cid

Müller

Platon

et al. 2017

IEEE Trans. on Image Process.

View full text Add to dashboard Cite

Abstract-Many image acquisition techniques used in biomedical imaging, material analysis, and structural geology are capable of acquiring 3D solid images. Computational analysis of these images is complex but necessary, since it is difficult for humans to visualize and quantify their detailed 3D content. One of the most common methods to analyze 3D data is to characterize the volumetric texture patterns. Texture analysis generally consists of encoding the local organization of image scales and directions, which can be extremely diverse in 3D. Current state-of-the-art techniques face many challenges when working with 3D solid texture, where most approaches are not able to consistently characterize both scale and directional information. 3D Riesz-wavelets can deal with both properties. One key property of Riesz filterbanks is steerability, which can be used to locally align the filters and compare textures with arbitrary (local) orientations. This paper proposes and compares three novel local alignment criteria for higher-order 3D Rieszwavelet transforms. The estimations of local texture orientations are based on higher-order extensions of regularized structure tensors. An experimental evaluation of the proposed methods for the classification of synthetic 3D solid textures with alterations (such as rotations and noise) demonstrated the importance of local directional information for robust and accurate solid texture recognition. These alignment methods achieved an accuracy of 0.95 in the rotated data, three times more than the unaligned Riesz descriptor that achieved 0.32. The accuracy obtained is better than all other techniques that are published and tested on the same database.Index Terms-3D solid texture analysis, 3D texture classification, Riesz-wavelets steerability, aligned higher-order Riesz-wavelet transform, rotation-covariance.

show abstract