Linear Image Reconstruction from a Sparse Set of α-Scale Space Features by Means of Inner Products of Sobolev Type

Duits, Remco; Janssen, Bart; Kanters, Frans; Florack, Luc

doi:10.1007/11577812_9

Cited by 4 publications

(7 citation statements)

References 10 publications

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“…Inspired by the general works on image reconstruction [47,41,38], we have shown [35,14,11,31,32] that up to second order differential structure at these points allow a visually appealing reconstruction of the image at scale s = 0 (via orthogonal projection). This suggests that top-points are visually descriptive feature points.…”

Section: Input: Tagged Mr Imagesmentioning

confidence: 96%

A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

et al. 2012

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The investigation and quantification of cardiac motion is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider a new method to track cardiac motion from magnetic resonance (MR) tagged images. Tracking is achieved by following the spatial maxima in scale-space of the MR images over time. Reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev norm expressed in covariant derivatives. These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. Furthermore, we propose a multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in one nonsingular analytic kernel operator in order to decompose the optic flow vector field in a divergence-free and a rotation-free part. Finally, we combine both the multi-scale Helmholtz decomposition and our vector field reconstruction (based on covariant derivatives) in a single algorithm and show the practical benefit of this approach by an experiment on real cardiac images.

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Section: Input: Tagged Mr Imagesmentioning

confidence: 96%

A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

et al. 2012

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“…As this gives quite unsatisfactory reconstruction results in the static case (unless one has a very rich set of anchor points), this is probably not a good choice for the dynamic case either. To stay within the linear framework Kanters et al, Duits et al, and Janssen et al [4,5,6,7] have considered more general Hilbert spaces, notably Sobolev spaces of a certain type. These include L 2 (IR n ) as a limiting case of a vanishing coupling constant.…”

Section: Recommendationsmentioning

confidence: 99%

“…the non-regularized case obtained by setting = 0, cf. [2,3,4,5,6,7], and subsequently apply the above formal replacement rules for f , g, φ i , and Φ ij , so as to obtain the -regularized reconstruction of interest. Since the Gram matrix Φ ij is known in closed form for the case of standard reconstruction and Gaussian derivative filters φ i of scale s i , say, the -regularized reconstruction scheme is likewise in closed form, and boils down to scale replacements s i → s i + in the unregularized Gram matrix, Φ 0 , and s i + 2 in the free expansion filters, φ i , recall Eq.…”

Section: A2 Theorymentioning

confidence: 99%

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Towards a New Paradigm for Motion Extraction

Florack

Janssen

Kanters

et al. 2006

Lecture Notes in Computer Science

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Abstract.A new multiscale paradigm is proposed for motion extraction. It exploits the fact that certain geometrically meaningful, isolated points in scale space provide unambiguous motion evidence, and the fact that such evidence narrows down the space of admissible motion fields. The paradigm combines the strengths of multiscale and variational frameworks.Besides spatial velocity, two additional degrees of freedom are taken into account, viz. a temporal and an isotropic spatial scale component. The first one is conventionally set to unity ("temporal gauge"), but may in general account for creation or annihilation of structures over time. The second one pertains to changes in the image that can be attributed to a sharpening or blurring of structures over time.This paper serves to introduce the new generalized motion paradigm de-emphasizing performance issues. We focus on the conceptual idea and provide recommendations for future directions.

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“…Lillhom and Nielsen [8] described an entropy-based variational reconstruction method. Inspired by Lillholm et al [8,9], Kanters et al showed that images can be reconstructed based on the information content of SPs, with only a few hundred points in Gaussian scale space [10].…”

Section: Introductionmentioning

confidence: 99%

Image Reconstruction from Multiscale Singular Points Based on the Dual-Tree Complex Wavelet Transform

Liu

Tremblais

Carré

et al. 2021

Security and Communication Networks

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The representation of an image with several multiscale singular points has been the main concern in image processing. Based on the dual-tree complex wavelet transform (DT-CWT), a new image reconstruction (IR) algorithm from multiscale singular points is proposed. First, the image was transformed by DT-CWT, which provided multiresolution wavelet analysis. Then, accurate multiscale singular points for IR were detected in the DT-CWT domain due to the shift invariance and directional selectivity properties of DT-CWT. Finally, the images were reconstructed from the phases and magnitudes of the multiscale singular points by alternating orthogonal projections between the CT-DWT space and its affine space. Theoretical analysis and experimental results show that the proposed IR algorithm is feasible, efficient, and offers a certain degree of denoising. Furthermore, the proposed IR algorithm outperforms other classical IR algorithms in terms of performance metrics such as peak signal-to-noise ratio, mean squared error, and structural similarity.

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Linear Image Reconstruction from a Sparse Set of α-Scale Space Features by Means of Inner Products of Sobolev Type

Cited by 4 publications

References 10 publications

A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

Towards a New Paradigm for Motion Extraction

Image Reconstruction from Multiscale Singular Points Based on the Dual-Tree Complex Wavelet Transform

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