An empirical formulation relating boundary lengths to resolution in specimens showing ‘non‐ideally fractal’ dimensions

Rigaut, Jean Paul

doi:10.1111/j.1365-2818.1984.tb00461.x

Cited by 88 publications

(67 citation statements)

References 10 publications

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“…It is a common practice to put a Wishart distribution (see definition below) prior, on the concentration matrix in multivariate analysis. Moreover, in the case of a Wishart distribution, a closed form expression for the Laplace transform exists and leads to a Rigaut-type asymptotic fractal law [16] which has been observed in many biological systems [8] (see explanation below).…”

Section: The Mixture Of Wisharts Model and Denoising Kernelmentioning

confidence: 99%

Feature Preserving Image Smoothing Using a Continuous Mixture of Tensors

Subakan

Jian

Vemuri

et al. 2007

2007 IEEE 11th International Conference on Computer Vision

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Many computer vision and image processing tasks require the preservation of local discontinuities, terminations and bifurcations. Denoising with feature preservation is a challenging task and in this paper, we present a novel technique for preserving complex oriented structures such as junctions and corners present in images. This is achieved in a two stage process namely, (1) All image data are pre-processed to extract local orientation information using a steerable Gabor filter bank. The orientation distribution at each lattice point is then represented by a continuous mixture of Gaussians. The continuous mixture representation can be cast as the Laplace transform of the mixing density over the space of positive definite (covariance) matrices. This mixing density is assumed to be a parameterized distribution, namely, a mixture of Wisharts whose Laplace transform is evaluated in a closed form expression called the Rigaut type function, a scalar-valued function of the parameters of the Wishart distribution. Computation of the weights in the mixture Wisharts is formulated as a sparse deconvolution problem. (2) The feature preserving denoising is then achieved via iterative convolution of the given image data with the Rigaut type function. We present experimental results on noisy data, real 2D images and 3D MRI data acquired from plant roots depicting bifurcating roots. Superior performance of our technique is depicted via comparison to the state-of-the-art anisotropic diffusion filter.

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Section: The Mixture Of Wisharts Model and Denoising Kernelmentioning

confidence: 99%

Feature Preserving Image Smoothing Using a Continuous Mixture of Tensors

Subakan

Jian

Vemuri

et al. 2007

2007 IEEE 11th International Conference on Computer Vision

View full text Add to dashboard Cite

show abstract

“…The Wishart distribution γ p,Σ is known to have the closed-form Laplace transform: (4) Let f in (2) be the density function of γ p,Σ with the expected value D= pΣ, we have (5) This is a familiar Rigaut-type asymptotic fractal expression [19] implying a signal decay characterized by a power-law which is the expected asymptotic behavior for the MR signal attenuation in porous media. Note that although this form of a signal attenuation curve had been phenomenologically fitted to the diffusion-weighted MR data before [17], until now, there was no rigorous derivation of the Rigaut-type expression used to explain the MR signal behavior as a function of b-value.…”

Section: Definitionmentioning

confidence: 99%

Multi-fiber Reconstruction from Diffusion MRI Using Mixture of Wisharts and Sparse Deconvolution

Jian

Vemuri

2007

Lecture Notes in Computer Science

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In this paper, we present a novel continuous mixture of diffusion tensors model for the diffusionweighted MR signal attenuation. The relationship between the mixing distribution and the MR signal attenuation is shown to be given by the Laplace transform defined on the space of positive definite diffusion tensors. The mixing distribution when parameterized by a mixture of Wishart distributions (MOW) is shown to possess a closed form expression for its Laplace transform, called the Rigauttype function, which provides an alternative to the Stejskal-Tanner model for the MR signal decay. Our model naturally leads to a deconvolution formulation for multi-fiber reconstruction. This deconvolution formulation requires the solution to an ill-conditioned linear system. We present several deconvolution methods and show that the nonnegative least squares method outperforms all others in achieving accurate and sparse solutions in the presence of noise. The performance of our multi-fiber reconstruction method using the MOW model is demonstrated on both synthetic and real data along with comparisons with state-of-the-art techniques.

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“…This is a familiar Rigaut-type asymptotic fractal expression (Rigaut, 1984) when the argument is taken to be the ADC associated with the expected tensor of the Wishart distributed diffusion tensors. 2 The important point is that this expression implies a signal decay characterized by a power-law in the large-|q|, hence large-b region exhibiting asymptotic behavior.…”

Section: Theoremmentioning

confidence: 99%

A novel tensor distribution model for the diffusion-weighted MR signal

et al. 2007

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Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

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An empirical formulation relating boundary lengths to resolution in specimens showing ‘non‐ideally fractal’ dimensions

Abstract: KEY w O R D s . Fractal dimensions, resolution, formulation, image analysis. S U M M A R Y

Cited by 88 publications

References 10 publications

Feature Preserving Image Smoothing Using a Continuous Mixture of Tensors

Feature Preserving Image Smoothing Using a Continuous Mixture of Tensors

Multi-fiber Reconstruction from Diffusion MRI Using Mixture of Wisharts and Sparse Deconvolution

A novel tensor distribution model for the diffusion-weighted MR signal

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