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The Beltrami flow [13,14] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the Beltrami
flow equation can be arranged in a reaction-diffusion form. This reveals the edge-enhancing properties of the equation and
suggests the application of additive operator split (AOS) methods [4,5] for faste...
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... goal of this numerical experiment is to show that the weakly coupled Beltrami smoothing operator may be replaced by its decoupled approximation without essential loss of accuracy and with a great saving of the computational time. Fig. 4 presents the simulation results for implicit scheme with different values of acceleration factor f = 2 . . . 50. The first row includes the initial image (left picture) and the smoothing simulation results for the Beltrami filtering, using the coupled explicit difference scheme with a full edge indicator matrix (central picture) and ...
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... corresponds exactly with the governing partial differential equation for the Beltrami flow equation fro a gray level image. This means that the mechanism of the edge enhancement is exactly the same. In the proximity of the edge, 1 /g reaches minimum values. The gradient of 1 /g is directed outside the thin pass of the edge. The gradient of the pixel value is also normal to the edge, but its direction coincides with the gradient of 1 /g for larger pixel values, and is opposite to that direction for smaller pixel values. The reactive component of the equation becomes positive for larger pixel values and negative for smaller pixel in the proximity of the edge. So, the large values become even larger, and the small values become even smaller. thereby enhancing and sharpening the edge. Note that the simplified decoupled form of the Beltrami smoothing may be used for numerical computations. Close to the edge this form is justified because the coupling between x and y components of gradient becomes weak. Away from the edge, the decoupled form brings a definite inaccuracy, and the flow is no longer exactly Beltrami. However, we consider that the accuracy is crucial at the proximity of the edge and less important elsewhere. In this case, the decoupled form of the governing equation (40) leads to a considerable saving of the computational time. Furthermore, the decoupled form makes it possible to apply the additive splitting algorithm leading to a semi-implicit linearized difference scheme, and this yields much faster and unconditionally stable computation. For a specific raw or column of pixels, the left-side matrix of the difference equation set is the same for all three color components, and only the right-side vectors differ. Thus, we solve the three sets of equations simultaneously. The goal of this numerical experiment is to show that the weakly coupled Beltrami smoothing operator may be replaced by its decoupled approximation without essential loss of accuracy and with a great saving of the computational time. Fig. 4 presents the simulation results for implicit scheme with different values of acceleration factor f = 2 . . . 50. The first row includes the initial image (left picture) and the smoothing simulation results for the Beltrami filtering, using the coupled explicit difference scheme with a full edge indicator matrix (central picture) and explicit difference scheme with decoupled governing equation with the eigenvalue instead of the mathix (right picture). In the second row the results are plotted for implicit difference scheme with decoupled governing equation and acceleration factor f = 1 , 2 , 5. The third row corresponds to f = 10 , 20 , 50. As we see, for acceleration factors of up to 20, the flow is simulated with a reasonable ...
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... Diffusion equations have been widely used in image processing as a form of noise reduction starting with Perona and Malik in 1990's [25]. Although numerous techniques have been developed for performing diffusion along surfaces [12,2,29,28,22,30], most approaches are nonparametric and requires finite element or finite difference schemes which are known to suffer various numerical instabilities [11]. ...
... Although (21) is intractable to directly compute, we will approximate (22) using the expected Euler characteristic (EC) method [37,35]. The power (22) can be written as ...
We present a unified statistical approach to modeling 3D anatomical objects
extracted from medical images. Due to image acquisition and preprocessing
noises, it is expected the imaging data is noisy. Using the Laplace-Beltrami
(LB) eigenfunctions, we smooth out noisy data and perform statistical analysis.
The method is applied in characterizing the 3D growth pattern of human hyoid
bone between ages 0 and 20 obtained from computed tomography (CT). We detected
a significant age effect on localized parts of the hyoid bone.
... Diffusion equations have been widely used in image processing as a form of noise reduction starting with Perona andMalik since 1990 (Perona andMalik, 1990). Numerous techniques have been developed for surface fairing and mesh regularization (Sochen et al., 1998;Malladi and Ravve, 2002;Tang et al., 1999;Taubin, 2000) and surface data smoothing (Andrade et al., 2001;Chung et al., 2001;Cachia et al., 2003a,b;Chung et al., 2005;Joshi et al., 2009). Isotropic heat diffusion on surfaces has been introduced in brain imaging for subsequent statistical analysis involving the random field theory that assumes an isotropic covariance function as a noise model (Andrade et al., 2001;Chung and Taylor, 2004;Cachia et al., 2003a,b). ...
We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel method is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, the method is applied to characterize the localized growth pattern of mandible surfaces obtained in CT images between ages 0 and 20 by regressing the length of displacement vectors with respect to a surface template.
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... Since the lengths of displacement defined on mesh vertices are expected to be noisy due to errors associated with image acquisition and image preprocessing, it is necessary to smooth out the noise and increase the signal-to-noise ra- tio [7]. Many previous surface data smoothing approaches have used heat diffusion type of smoothing to reduce surface noise [3, 7, 18, 20, 27, 31, 32]. Instead, we propose to use the Laplace-Beltrami (LB) eigenfunctions in parametrically representing the surface data. ...
We present a new sparse shape modeling framework on the Laplace-Beltrami (LB) eigenfunctions. Traditionally, the LB-eigenfunctions are used as a basis for intrinsically representing surface shapes by forming a Fourier series expansion. To reduce high frequency noise, only the first few terms are used in the expansion and higher frequency terms are simply thrown away. However, some lower frequency terms may not necessarily contribute significantly in reconstructing the surfaces. Motivated by this idea, we propose to filter out only the significant eigenfunctions by imposing l1-penalty. The new sparse framework can further avoid additional surface-based smoothing often used in the field. The proposed approach is applied in investigating the influence of age (38-79 years) and gender on amygdala and hippocampus shapes in the normal population. In addition, we show how the emotional response is related to the anatomy of the subcortical structures.
... Diffusion equations have been widely used in image processing as a form of noise reduction starting with Perona and Malik in 1990 [21] . Although numerous techniques have been developed for surface fairing and mesh regularization [20,22,23,24,25,26] based on heat diffusion. Most diffusion smoothing approaches mainly use finite element or finite difference schemes which is known to suffer numerical instability if the forward Euler scheme is used. ...
We present a new subcortical structure shape modeling framework using heat kernel smoothing constructed with the Laplace-Beltrami eigenfunctions. Cotan discretization is used to numerically obtain the eigenfunctions of the Laplace-Beltrami operator along the surfaces of subcortical structures. The eigenfunctions are then used to construct the heat kernel and used in smoothing out measurements noise along the surface. The proposed framework is applied in investigating the influence of age (38-79 years) and gender on amygdala and hippocampus shape. We detected a significant age effect on hippocampus in accordance with the previous studies. In addition, we also detected a significant gender effect on amygdala. Since we did not find any such differences in the traditional volumetric methods, our results demonstrate the benefit of the current framework over traditional volumetric methods.
... One technique to overcome this computing cost, the Additive Operator Splitting (AOS), has been introduced by Weickert et al. [19] for the nonlinear diffusion flow and later applied by Goldenberg et al. [5] and Kuhne et al. [8] to implement a fast version of the geodesic contour model. A similar work has been done by Malladi and Ravve [10], where the authors applied the AOS method to anisotropic diffusion of gray level, and vector-valued imagery with the so-called Beltrami flow [18]. ...
... One technique to overcome this computing cost, the Additive Operator Splitting (AOS), has been introduced by Weickert et al. [19] for the nonlinear diffusion flow and later applied by Goldenberg et al. [5] and Kuhne et al. [8] to implement a fast version of the geodesic contour model. A similar work has been done by Malladi and Ravve [10], where the authors applied the AOS method to anisotropic diffusion of gray level, and vector-valued imagery with the so-called ...
... But its main drawback is (very) slow convergence and the number of iteration required. We are going to derive a semi-implicit scheme similarly to that used for the Beltrami flow in [10]. ...
We present a new fast approach for segmentation of ob-ject with missing boundaries based on the Subjective Sur-face [15] combined with the Fast-Marching algorithm [17]. The Subjective-Surface [15] deals with constructing percep-tually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result, and its main drawback is very slow convergence and a huge number of iterations required. In this paper, we address those two prob-lems. We first show that the governing equation for the sub-jective surface flow can be re-arranged in an AOS implemen-tation, providing a near real time solution to the shape com-pletion problem in ¢ ¡ and £ ¡ . Then we devise a new ini-tialization paradigm based on the Fast-Marching algorithm [17]. We compare the original method with our new algo-rithm on several examples of real 3D medical images, thus revealing the improvement achieved.
... Let us consider the image function and be the intensity gradient magnitude image where is the gradient operator and indicates the image function at any given time " . " Malladi and Ravve [26] The first term on the right-hand side contributes to the reaction while the second term contributes to the nonlinear diffusion . The parameter controls relative contribution of reaction and diffusion terms. ...
Statistical analysis of genetic changes within cell nuclei that are far from the primary tumor would help determine whether such changes have occurred prior to tumor invasion. To determine whether the gene amplification in cells is morphologically and/or genetically related to the primary tumor requires quantitative evaluation of a large number of cell nuclei from continuous meaningful structures such as milk-ducts, tumors, etc., located relatively far from the primary tumor. To address this issue, we have designed an integrated image analysis software system for high-throughput segmentation of nuclei. Filters such as Beltrami flow-based reaction-diffusion, directional diffusion, etc., were used to pre-process the images resulting in a better segmentation. The accurate shape of the segmented nucleus was recovered using an iterative "shrink-wrap" operation. The study of two cases of ductal carcinoma in situ in breast tissue supports the biological observation regarding the existence of a preferential intraductal invasion, and therefore a common origin, between the primary tumor and the gene amplification in the cell-nuclei lining the ductal structures in the breast.
... Our goal is to build a fast and reliable method to solve the Beltrami flow equations and it is based on the AOS technique. This work was first reported in Malladi and Ravve [21], where the main technique is discussed and applied to gray-level and vector-valued imagery. In this paper, we first discuss the method as applied to 2D gray scale imagery and extend it to 3D and a related PDE, the governing equation for the subjective surface computation [6]. ...
In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme that is fashioned after additive operator split (AOS) [1], [2] for Beltrami and the subjective surface computation. The Beltrami flow [3], [4], [5] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface [6] deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. In this paper, we first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition" the viewpoint surface using the Fast-Marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved.
... The coordinates p(Ω) are expected to be noisy. To filter out mesh noisy, we perform diffusion-based surface smoothing [1,6,19,25,26] [blind] on mesh vertices. Let ∆ be the spherical Laplacian defined as ...
... Unlike previous literature [1,6,19,25,26] [blind] that solve surface diffusion numerically using the finite difference scheme, Theorem 1 provides a new framework for performing surface diffusion by computing a series expansion. The advantage of this new framework is the ability to explicitly model surface diffusion using the Karhunen-Loeve expansion of a random field [17]. ...
We present a computational framework for analyzing brain hemispheric asymmetry without any kind of image flipping. In order to perform brain asymmetry analysis, it is necessary to flip 3D magnetic resonance images (MRI) and establish the hemispheric correspondence by register-ing the original image to the flipped image. The difference between the original and the flipped images is then used as a measure of cerebral asymmetry. Instead of physically flip-ping MRI and performing image registration, we construct the global algebraic representation of cortical surface using spherical harmonics. Then using the inherent angular sym-metry present in the spherical harmonics, image flipping is done by changing the sign of the asymmetric part in the rep-resentation. The surface registration between hemispheres and different subjects is done algebraically within the repre-sentation itself without any time consuming numerical op-timization. The methodology has been applied in localiz-ing the abnormal cortical asymmetry pattern of in a group of autistic subjects using the logistic discriminant analy-sis. Since the logistic discriminant analysis avoids the tra-ditional hypothesis testing paradigm, the complicated mul-tiple comparison issue that plagues the brain imaging com-munity has been avoided.
We present a unified heat kernel smoothing framework for modeling 3D anatomical surface data extracted from medical images. Due to image acquisition and preprocessing noises, it is expected the medical imaging data is noisy. The surface data of the anatomical structures is regressed using the weighted linear combination of Laplace-Beltrami (LB) eigenfunctions to smooth out noisy data and perform statistical analysis. The method is applied in characterizing the 3D growth pattern of human hyoid bone between ages 0 and 20 obtained from CT images. We detected a significant age effect on localized parts of the hyoid bone.
Heat diffusion has been widely used in brain imaging for surface fairing, mesh regularization and cortical data smoothing. Motivated by diffusion wavelets and convolutional neural networks on graphs, we present a new fast and accurate numerical scheme to solve heat diffusion on surface meshes. This is achieved by approximating the heat kernel convolution using high degree orthogonal polynomials in the spectral domain. We also derive the closed-form expression of the spectral decomposition of the Laplace-Beltrami operator and use it to solve heat diffusion on a manifold for the first time. The proposed fast polynomial approximation scheme avoids solving for the eigenfunctions of the Laplace-Beltrami operator, which is computationally costly for large mesh size, and the numerical instability associated with the finite element method based diffusion solvers. The proposed method is applied in localizing the male and female differences in cortical sulcal and gyral graph patterns obtained from MRI in an innovative way. The MATLAB code is available at
http://www.stat.wisc.edu/~mchung/chebyshev
.
