Isotropic Smoothing Research Articles

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Let Ω i ⊂ R n i \Omega _i\subset \mathbb {R}^{n_i} , i = 1 , … , m i=1,\ldots ,m , be given domains. In this article, we study the low-rank approximation with respect to L 2 ( Ω 1 × ⋯ × Ω m ) L^2(\Omega _1\times \dots \times \Omega _m) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.

Multitrace Impedance Inversion Based on Structure-Oriented Regularization

As an effective approach of reservoir prediction, poststack impedance inversion has been widely used in industry. However, like other inverse problems, poststack impedance inversion is a quintessential ill-conditioned problem. Conventional impedance inversion methods often use regularization techniques such as Tikhonov-type regularization to improve the stability of the inversion solution. Nevertheless, because this type of regularization method constrains the inversion process by applying isotropic smoothness to the impedance, it will result in blurred boundaries and micro-geological structures, thereby reducing the resolution of the inversion result. In order to address this problem, we have introduced a structure-oriented regularization (SOR) method based on the local geological structural direction for impedance inversion. Compared with conventional isotropic smooth constraints, SOR can apply smoothness along the direction of geological structures, so it can effectively protect significant geological information from being blurred. Three steps are required to complete the SOR-based seismic impedance inversion method. To begin with, the structural orientations are estimated from seismic image. Then, the SOR term is constructed and combined with the multitrace seismic data misfit term to formulate the objective function for the inversion of impedance. Finally, the objective function can be easily solved by the conjugate gradient (CG) method. We compare our method with the classic model-based impedance inversion method on synthetic and real seismic data. The inversion results demonstrate the benefit of our method in seismic impedance inversion.

An experimental study for the effects of noise on face recognition algorithms under varying illumination

When the illumination changes, the appearance of facial images will change dramatically. Lighting changes make face recognition a very challenging and difficult job. In addition, the effects of noise on existing face recognition methods have been neglected in the literature, to the best of our knowledge. In this work, we study the effects of noise on existing illumination-invariant face recognition methods. We tested such noise as Gaussian white noise, Poisson noise, salt & pepper noise, speckle noise, etc. In total, 21 methods have been included in this study in this work. We find out that, when noise is added to facial images, Tan and Triggs’ method achieves the best results for both the extended Yale B face database and the CMU-PIE face database. When facial images do not contain noise, isotropic smoothing is preferred because it obtains the highest average recognition rate (96%) for the extended Yale B face database and 16 methods obtain perfect correct recognition rates (100%) for the CMU-PIE face database.

Mini-batch optimized full waveform inversion with geological constrained gradient filtering

High computation cost and generating solutions without geological sense have hindered the wide application of Full Waveform Inversion (FWI). Source encoding technique is a way to dramatically reduce the cost of FWI but subject to fix-spread acquisition setup requirement and slow convergence for the suppression of cross-talk. Traditionally, gradient regularization or preconditioning is applied to mitigate the ill-posedness. An isotropic smoothing filter applied on gradients generally gives non-geological inversion results, and could also introduce artifacts. In this work, we propose to address both the efficiency and ill-posedness of FWI by a geological constrained mini-batch gradient optimization method. The mini-batch gradient descent optimization is adopted to reduce the computation time by choosing a subset of entire shots for each iteration. By jointly applying the structure-oriented smoothing to the mini-batch gradient, the inversion converges faster and gives results with more geological meaning. Stylized Marmousi model is used to show the performance of the proposed method on realistic synthetic model.

An adaptive motion regularization technique to support sliding motion in deformable image registration.

Isotropic smoothing has been conventionally used to regularize deformation vector fields (DVFs) in deformable image registration (DIR). However, the isotropic smoothing method enforces global smoothness and therefore cannot accurately model the complex tissue deformation, such as sliding motion at organ boundaries. To accurately model and estimate sliding tissue motion, an adaptive direction-dependent DVF regularization technique was developed in this study. A DVF is computed and updated iteratively by minimizing the intensity differences between the images. In each iteration, the DVF was smoothed using an adaptive direction-dependent filter which enforces different motion propagation mechanisms along the primary normal and tangential directions of soft tissue local structures. A Gaussian isotropic filter was used along the normal direction while a bilateral filter was used along the tangential direction. To support large sliding motion, an automatic method was developed to delineate sliding surfaces, such as the chest wall and abdominal wall, where large organ sliding motion occurs. Parameters of the DVF regularization were adjusted adaptively based on a distance map to the sliding surfaces. The proposed method was tested on 14 4D-CT datasets at End-Inhalation (EI) and End-Exhalation (EE) phases of a respiratory cycle (10 public lung datasets, 3 upper abdomen datasets and 1 digital phantom dataset). TRE results of the 10 lung datasets were compared to results from six other existing DIR methods. For the three upper abdomen patient datasets, DIR accuracy was evaluated using manually defined landmarks across the lung and the abdomen. For the digital phantom dataset, DIR accuracy was evaluated using the ground truth displacement of a total 40,000 points that were evenly distributed across the phantom. The results showed that the sliding motion was preserved near the surface of chest wall and abdominal wall. The average target registration error (TRE) was reduced by 35.1% using the proposed method in comparison with five other methods on the 10 lung datasets. The sum of squared difference (SSD) after registration using the proposed method was 4.4% and 11.4% smaller than the SSDs obtained using isotropic smoothing and bilateral smoothing respectively. On the digital phantom, the average TRE was reduced by 59.6% near the surface of liver and by 53.7% near the surface of spleen using the proposed method. Contour propagation and Jacobian determinant analysis of DVF suggested an overall improved accuracy using the proposed method. An adaptive direction-dependent DVF regularization method has been developed to model the sliding tissue motion of the thoracic and abdominal organs. The overall motion estimation accuracy has been improved especially near the chest wall and abdominal wall where large organ sliding motion occurs.

Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in [formula omitted

We investigate the rate of convergence of linear sampling numbers of the embedding Hα,β(Td)↪Hγ(Td). Here α governs the mixed smoothness and β the isotropic smoothness in the space Hα,β(Td) of hybrid smoothness, whereas Hγ(Td) denotes the isotropic Sobolev space. If γ>β we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on “energy-norm based sparse grids” for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding Hmixα(Td)↪Hmixγ(Td) and achieve optimality for Smolyak’s algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding Hmixα(Td)↪Lq(Td) for 2<q≤∞ where again Smolyak’s algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation β=γ (including the embedding into L2(Td)).

Regularization Strategies for Discontinuity-Preserving Optical Flow Methods

The aim of this paper is to study several strategies for the preservation of flow discontinuities in variational optical flow methods. We analyze the combination of robust functionals and diffusion tensors in the smoothness assumption. Our study includes the use of tensors based on decreasing functions, which has shown to provide good results. However, it presents several limitations and usually does not perform better than other basic approaches. It typically introduces instabilities in the computed motion fields in the form of independent blobs of vectors with large magnitude. We propose two alternatives to overcome these drawbacks: first, a simple approach that combines the decreasing function with a minimum isotropic smoothing, and second, a method that looks for the best parameter configuration that preserves the important motion contours and avoid instabilities. It relies on the input images and the regularization parameter. It is fully automatic, providing a near-optimal value for many sequences, as shown in the experiments. Both proposals allow to detect the contours of the motion field and produce more stable solutions for a large range of parameters. In the experimental results, we present a detailed study and comparison of the different strategies.

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let $$X_n = \{x^j\}_{j=1}^n$$Xn={xj}j=1n be a set of n points in the d-cube $${\mathbb {I}}^d:=[0,1]^d$$Id:=[0,1]d, and $$\Phi _n = \{\varphi _j\}_{j =1}^n$$źn={źj}j=1n a family of n functions on $${\mathbb {I}}^d$$Id. We consider the approximate recovery of functions f on $${{\mathbb {I}}}^d$$Id from the sampled values $$f(x^1), \ldots , f(x^n)$$f(x1),ź,f(xn), by the linear sampling algorithm $$ L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. $$Ln(Xn,źn,f):=źj=1nf(xj)źj. The error of sampling recovery is measured in the norm of the space $$L_q({\mathbb {I}}^d)$$Lq(Id)-norm or the energy quasi-norm of the isotropic Sobolev space $$W^\gamma _q({\mathbb {I}}^d)$$Wqź(Id) for $$1 0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β of a hybrid of mixed smoothness $$\alpha > 0$$ź>0 and isotropic smoothness $$\beta \in {\mathbb {R}}$$βźR, and spaces $$B^a_{p,\theta }$$Bp,źa of a nonuniform mixed smoothness $$a \in {\mathbb {R}}^d_+$$aźR+d. We constructed asymptotically optimal linear sampling algorithms $$L_n(X_n^*,\Phi _n^*,\cdot )$$Ln(Xnź,źnź,·) on special sparse grids $$X_n^*$$Xnź and a family $$\Phi _n^*$$źnź of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in $$B^{\alpha ,\beta }_{p,\theta }$$Bp,źź,β and $$B^a_{p,\theta }$$Bp,źa. As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.

Implementation of adaptive edge-preserving smoothing regularisation technique for simultaneous multi-frequency full waveform inversion

The regularisation method simply involves adding a regularisation term to a misfit function, thereby generally stabilising ill-posed inverse problems. In this paper, a pure-gradient based full waveform inversion is carried out in the frequency domain, and smoothing regularisation is exploited in the form of preconditioning, contrary to conventional regularisations in general inverse theory. For more accurate descriptions of subsurface structures, we select adaptive regularisation rather than isotropic smoothing regularisation. The adaptive smoothing regularisation method is used to both smooth the image and differentiate between layers and boundaries. Dip information is estimated using the Hilbert transform, an adaptive smoothing filter with an edge-preserving property is designed from the estimated dip information, and full waveform inversion is then performed using the adaptive smoothing filter. The adaptive smoothing filter is periodically updated during inversion iterations to adapt to the inverted velocity model. We conducted numerical experiments to test our adaptive smoothing filter on a simple model, the Marmousi model, and the SEG/EAGE salt model, and the results demonstrated that our adaptive smoothing regularisation method yields more reliable and precise information compared with the conventional regularisation method.

Image-guided inversion in steady-state hydraulic tomography

In steady-state hydraulic tomography, the head data recorded during a series of pumping or/and injection tests can be inverted to determine the transmissivity distributions of an aquifer. This inverse problem is usually under-determined and ill-posed. We propose to use structural information inferred from a guiding image to constrain the inversion process. The guiding image can be drawn from soft data sets such as seismic and ground penetrating radar sections or from geological cross-sections inferred from the wells and some geological expertise. The structural information is extracted from the guiding image through some digital image analysis techniques. Then, it is introduced into the inversion process of the head data as a weighted four direction smoothing matrix used in the regularizer. Such smoothing matrix allows applying the smoothing along the structural features. This helps preserving eventual drops in the hydraulic properties. In addition, we apply a procedure called image-guided interpolation. This technique starts with the tomogram obtained from the image-guided inversion and focus this tomogram. These new approaches are applied on four synthetic toy problems. The hydraulic distributions estimated from the image-guided inversion are closer to the true transmissivity model and have higher resolution than those computed from a classical Gauss–Newton method with uniform isotropic smoothing.

Image Decompositions Using Spaces of Variable Smoothness and Integrability

In this paper new variational image decomposition models are proposed which split an image into its geometrical component and oscillating component. An interpolation of total variation regularization and isotropic smoothing is used as the structure energy, while the energy for the oscillating component is a modular in a variable index Besov space. The existence of minimizers of the new structure-texture energies is studied. The proper choices of the bounded variation--Sobolev integrability and the Besov indices aim to reduce oversmoothing of the cartoon component. If there is noise present, the choices help reduce staircasing. Experimental results demonstrate the possible advantages of the variable index models over the constant index models.

The 4D hyperspherical diffusion wavelet: A new method for the detection of localized anatomical variation.

Recently, the HyperSPHARM algorithm was proposed to parameterize multiple disjoint objects in a holistic manner using the 4D hyperspherical harmonics. The HyperSPHARM coefficients are global; they cannot be used to directly infer localized variations in signal. In this paper, we present a unified wavelet framework that links Hyper-SPHARM to the diffusion wavelet transform. Specifically, we will show that the HyperSPHARM basis forms a subset of a wavelet-based multiscale representation of surface-based signals. This wavelet, termed the hyperspherical diffusion wavelet, is a consequence of the equivalence of isotropic heat diffusion smoothing and the diffusion wavelet transform on the hypersphere. Our framework allows for the statistical inference of highly localized anatomical changes, which we demonstrate in the first-ever developmental study on the hyoid bone investigating gender and age effects. We also show that the hyperspherical wavelet successfully picks up group-wise differences that are barely detectable using SPHARM.

Methodological improvements in voxel‐based analysis of diffusion tensor images: Applications to study the impact of apolipoprotein E on white matter integrity

PurposeTo identify regional differences in apparent diffusion coefficient (ADC) and fractional anisotropy (FA) using customized preprocessing before voxel‐based analysis (VBA) in 14 normal subjects with the specific genes that decrease (apolipoprotein [APO] E ε2) and that increase (APOE ε4) the risk of Alzheimer's disease.Materials and MethodsDiffusion tensor images (DTI) acquired at 1.5 Tesla were denoised with a total variation tensor regularization algorithm before affine and nonlinear registration to generate a common reference frame for the image volumes of all subjects. Anisotropic and isotropic smoothing with varying kernel sizes was applied to the aligned data before VBA to determine regional differences between cohorts segregated by allele status.ResultsVBA on the denoised tensor data identified regions of reduced FA in APOE ε4 compared with the APOE ε2 healthy older carriers. The most consistent results were obtained using the denoised tensor and anisotropic smoothing before statistical testing. In contrast, isotropic smoothing identified regional differences for small filter sizes alone, emphasizing that this method introduces bias in FA values for higher kernel sizes.ConclusionVoxel‐based DTI analysis can be performed on low signal to noise ratio images to detect subtle regional differences in cohorts using the proposed preprocessing techniques. J. Magn. Reson. Imaging 2014;39:387–397. © 2013 Wiley Periodicals, Inc.

Fast Geodesic Active Fields for Image Registration Based on Splitting and Augmented Lagrangian Approaches

In this paper, we present an efficient numerical scheme for the recently introduced geodesic active fields (GAF) framework for geometric image registration. This framework considers the registration task as a weighted minimal surface problem. Hence, the data-term and the regularization-term are combined through multiplication in a single, parametrization invariant and geometric cost functional. The multiplicative coupling provides an intrinsic, spatially varying and data-dependent tuning of the regularization strength, and the parametrization invariance allows working with images of nonflat geometry, generally defined on any smoothly parametrizable manifold. The resulting energy-minimizing flow, however, has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach; data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Our approach is more flexible than standard Gaussian regularization, since one can interpolate freely between isotropic Gaussian and anisotropic TV-like smoothing. In this paper, we compare the geodesic active fields method with the popular Demons method and three more recent state-of-the-art algorithms: NL-optical flow, MRF image registration, and landmark-enhanced large displacement optical flow. Thus, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality. Over the range of example applications, it also consistently produces results not far from more dedicated state-of-the-art methods, illustrating the flexibility of the proposed framework.

On the construction of sparse tensor product spaces

Let Ω1 ⊂ Rn1 and Ω2 ⊂ Rn2 be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness r1 and r2, respectively. In this paper, we study the optimal construction of sparse tensor products made from these spaces. In particular, we derive the resulting cost complexities to approximate functions with anisotropic and isotropic smoothness on the tensor product domain Ω1×Ω2. Numerical results validate our theoretical findings.

On the analysis of anisotropic smoothness

We investigate anisotropic function spaces defined over the multi-level ellipsoid covers of Rn, where the ellipsoids can quickly change shape from point to point and from level to level. We explicitly define an anisotropic modulus of smoothness (already used implicitly in Dahmen et al. (2010) [4]) and investigate its properties. We show anisotropic variants of classic inequalities such as the Marchaud, Nikolskii and Ul’yanov, relationships with isotropic smoothness and applications to anisotropic Besov space embedding.

Higher Degree Total Variation (HDTV) Regularization for Image Recovery

We introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation (TV) scheme. Motivated by novel reinterpretations of the classical TV regularizer, we derive two families of functionals involving higher degree partial image derivatives; we term these families as isotropic and anisotropic higher degree TV (HDTV) penalties, respectively. The isotropic penalty is the L(1) - L(2) mixed norm of the directional image derivatives, while the anisotropic penalty is the separable L(1) norm of directional derivatives. These functionals inherit the desirable properties of standard TV schemes such as invariance to rotations and translations, preservation of discontinuities, and convexity. The use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel, thus encouraging isotropic smoothing. In contrast, the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities, while continuing to smooth along the linelike features; this scheme thus enhances the linelike image characteristics analogous to standard TV. We also introduce efficient majorize-minimize algorithms to solve the resulting optimization problems. The numerical comparison of the proposed scheme with classical TV penalty, current second-degree methods, and wavelet algorithms clearly demonstrate the performance improvement. Specifically, the proposed algorithms minimize the staircase and ringing artifacts that are common with TV and wavelet schemes, while better preserving the singularities. We also observe that anisotropic HDTV penalty provides consistently improved reconstructions compared with the isotropic HDTV penalty.

Motion Tracking of Discontinuous Sea Ice

The purpose of this paper is twofold: 1) to develop a high-resolution sea ice motion tracking system at the geospatial mesoscale (1-100 km <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) and 2) to propose an algorithm that measures motion at close proximity to discontinuous regions. Here, we present a motion tracking system that computes differential motion at 400 m resolution and validate the accuracy/precision of this system via four studies. The first study measures the accuracy against displacements measured from in situ Global Positioning System (GPS) buoys deployed at the Sea-ice Experiment: Dynamic Nature of the Arctic (SEDNA) and the Surface HEat Budget of the Arctic Ocean (SHEBA) experiments. The estimates are found to be statistically comparable with GPS, with an average error of 361.9 and 600.6 m for the experiments, respectively. The second study compares the estimated displacements to those measured by the RADARSAT Geophysical Processing System. A precision error of 75.7 m is found between the two motion tracking systems. The third study uses intensity warping of randomly sampled measurements to evaluate discontinuous motion tracking. A one-tailed Wilcoxon signed rank test is used to validate these measurements at α = 0.01. Results from this paper prove that anisotropic smoothing produces significantly smaller errors at discontinuous locations (W = 4240 and p <; 0.001) over conventional isotropic smoothing. The fourth study compares displacements measured by anisotropic smoothing against manual measurements. This paper demonstrates an average reduction of the estimation error by 50 m with the use of anisotropic smoothing over the conventional isotropic smoothing.

A method for anisotropic spatial smoothing of functional magnetic resonance images using distance transformation of a structural image

Spatial smoothing using isotropic Gaussian kernels to remove noise reduces spatial resolution and increases the partial volume effect of functional magnetic resonance images (fMRI), thereby reducing localization power. To minimize these limitations, we propose a novel anisotropic smoothing method for fMRI data. To extract an anisotropic tensor for each voxel of the functional data, we derived an intensity gradient using the distance transformation of the segmented gray matter of the fMRI-coregistered T1-weighted image. The intensity gradient was then used to determine the anisotropic smoothing kernel at each voxel of the fMRI data. Performance evaluations on both real and simulated data showed that the proposed method had 10% higher statistical power and about 20% higher gray matter localization compared to isotropic smoothing and robustness to the registration errors (up to 4 mm translations and 4° rotations) between T1 structural images and fMRI data. The proposed method also showed higher performance than the anisotropic smoothing with diffusion gradients derived from the fMRI intensity data.

Ramp-Preserving Denoising for Conductivity Image Reconstruction in Magnetic Resonance Electrical Impedance Tomography

In magnetic resonance electrical impedance tomography, among several conductivity image reconstruction algorithms, the harmonic B(z) algorithm has been successfully applied to B(z) data from phantoms and animals. The algorithm is, however, sensitive to measurement noise in B(z) data. Especially, in in vivo animal and human experiments where injection current amplitudes are limited within a few milliampere at most, measured B(z) data tend to have a low SNR. In addition, magnetic resonance (MR) signal void in outer layers of bones and gas-filled organs, for example, produces salt-pepper noise in the MR phase and, consequently, B(z) images. The B(z) images typically present areas of sloped transitions, which can be assimilated to ramps. Conductivity contrasts change ramp slopes in B(z) images and it is critical to preserve positions of those ramps to correctly recover edges in conductivity images. In this paper, we propose a ramp-preserving denoising method utilizing a structure tensor. Using an eigenvalue analysis, we identified local regions of salt-pepper noise. Outside the identified local regions, we applied an anisotropic smoothing to reduce noise while preserving their ramp structures. Inside the local regions of salt-pepper noise, we used an isotropic smoothing. After validating the proposed denoising method through numerical simulations, we applied it to in vivo animal imaging experiments. Both numerical simulation and experimental results show significant improvements in the quality of reconstructed conductivity images.