Total P-variation Research Articles

Pig-DTpV: A prior information guided directional TpV algorithm for orthogonal translation computed laminography

The local scanning orthogonal translation computed laminography (OTCL) has great potential for tiny fault detection of laminated structure thin-plate parts. However, it generates limited-angle and truncated projection data, which result in aliasing and truncation artifacts in the reconstructed images. The directional total variation (DTV) algorithm has been demonstrated to achieve highly accurate reconstructed images in limited-angle computed tomography (CT). However, its application in local scanning OTCL has not been explored. Based on this algorithm, we introduce the lp norm to better suppress artifacts, and prior information to further constrain the reconstructed image. Thus, we propose a prior information guided directional total p-variation (DTpV) algorithm (Pig-DTpV). The Pig-DTpV model is a constrained non-convex optimization model. The constraint term are the six DTpV terms, whereas the objective term is the data fidelity term. Then, we use the iterative reweighting strategy and the Chambolle–Pock (CP) algorithm to solve the model. The Pig-DTpV reconstruction algorithm’s performance is compared with other algorithms such as simultaneous algebraic reconstruction technique (SART), TV, reweighted anisotropic-TV (RwATV), and DTV in simulation and real data experiments. The experiment results demonstrate that the Pig-DTpV algorithm can reduce truncation and aliasing artifacts and enhance the quality of reconstructed images.

Dilations for operator-valued quantum measures

This paper concerns the dilations of Banach space operator-valued quantum measures. While the recently developed general dilation theory leads to a projection (idempotent) valued dilation for any quantum measure over the projection lattice of a von Neumann algebra that does not contain a Type I2 direct summand, such a dilation does not necessarily guarantee the preservation of countable additivity of the quantum measure. So it remains an open question whether every countably additive B(X)-valued quantum measure can be dilated to a countably additive projection-valued measure. The main purpose of this paper is to prove that such a dilation exists if one of the following two conditions is satisfied: (i) either the underlying Banach space X is ℓp for some 1<p<2 or X has the Schur property, (ii) the quantum measure has bounded p-variation (i.e., the total p-variation is finite) for some 1≤p<∞. These are achieved by establishing a non-commutative version of a minimal dilation theory on the elementary dilation space equipped with an appropriate dilation norm. In particular, we introduce a p-variation norm on the elementary dilation space and prove that every operator-valued quantum measure with bounded p-variation has a projection-valued quantum measure dilation that preserves the boundedness of the p-variation.

Sparse Angle CBCT Reconstruction Based on Guided Image Filtering.

Cone-beam Computerized Tomography (CBCT) has the advantages of high ray utilization and detection efficiency, short scan time, high spatial and isotropic resolution. However, the X-rays emitted by CBCT examination are harmful to the human body, so reducing the radiation dose without damaging the reconstruction quality is the key to the reconstruction of CBCT. In this paper, we propose a sparse angle CBCT reconstruction algorithm based on Guided Image FilteringGIF, which combines the classic Simultaneous Algebra Reconstruction Technique(SART) and the Total p-Variation (TpV) minimization. Due to the good edge-preserving ability of SART and noise suppression ability of TpV minimization, the proposed method can suppress noise and artifacts while preserving edge and texture information in reconstructed images. Experimental results based on simulated and real-measured CBCT datasets show the advantages of the proposed method.

A simple and fast ASD-POCS algorithm for image reconstruction.

The adaptive steepest descent projection onto convex set (ASD-POCS) algorithm is a promising algorithm for constrained total variation (TV) type norm minimization models in computed tomography (CT) image reconstruction using sparse and/or noisy data. However, in ASD-POCS algorithm, the existing gradient expression of the TV-type norm appears too complicated in the implementation code and reduces image reconstruction speed. To address this issue, this work aims to develop and test a simple and fast ASD-POCS algorithm. Since the original algorithm is not derived thoroughly, we first obtain a simple matrix-form expression by thorough derivation via matrix representations. Next, we derive the simple matrix expressions of the gradients of TV, adaptive weighted TV (awTV), total p-variation (TpV), high order TV (HOTV) norms by term combinations and matrix representations. The deep analysis is then performed to identify the hidden relations of these terms. The TV reconstruction experiments by use of sparse-view projections via the Shepp-Logan, FORBILD and a real CT image phantoms show that the simplified ASD-POCS (S-ASD-POCS) using the simple matrix-form expression of TV gradient achieve the same reconstruction accuracy relative to ASD-POCS, whereas it enables to speed up the whole ASD process 1.8-2.7 time fast. The derived simple matrix expressions of the gradients of these TV-type norms may simplify the implementation of the ASD-POCS algorithm and speed up the ASD process. Additionally, a general gradient expression suitable to all the sparse transform-based optimization models is demonstrated so that the ASD-POCS algorithm may be tailored to extended image reconstruction fields with accelerated computational speed.

Locally Adaptive Total p-Variation Regularization for Non-Rigid Image Registration With Sliding Motion.

Due to the complicated thoracic movements which contain both sliding motion occurring at lung surfaces and smooth motion within individual organs, respiratory estimation is still an intrinsically challenging task. In this paper, we propose a novel regularization term called locally adaptive total p-variation (LaTpV) and embed it into a parametric registration framework to accurately recover lung motion. LaTpV originates from a modified Lp-norm constraint (1 < p < 2), where a prior distribution of p modeled by the Dirac-shaped function is constructed to specifically assign different values to voxels. LaTpV adaptively balances the smoothness and discontinuity of the displacement field to encourage an expected sliding interface. Additionally, we also analytically deduce the gradient of the cost function with respect to transformation parameters. To validate the performance of LaTpV, we not only test it on two mono-modal databases including synthetic images and pulmonary computed tomography (CT) images, but also on a more difficult thoracic CT and positron emission tomography (PET) dataset for the first time. For all experiments, both the quantitative and qualitative results indicate that LaTpV significantly surpasses some existing regularizers such as bending energy and parametric total variation. The proposed LaTpV based registration scheme might be more superior for sliding motion correction and more potential for clinical applications such as the diagnosis of pleural mesothelioma and the adjustment of radiotherapy plans.

Seismic Acoustic Impedance Inversion Using Mixed Second-Order Fractional ATpV Regularization

Seismic acoustic impedance (AI) inversion is widely used in geophysics as AI can indicate rock characteristics and facilitate stratigraphic analysis. However, traditional AI inversion suffers from a multi-solution problem. To overcome this barrier, anisotropic total p-variation (ATpV) regularization has been applied in inversion as it can improve the accuracy by Lp quasi-norm. Nevertheless, this regularization results in the staircase effect and the scattering effect. To reduce these two effects, we introduced the mixed second-order variations and the fractional difference in AI inversion based on ATpV and proposed a novel AI inversion method using mixed second-order fractional anisotropic total p-variation (MS_FATpV) regularization. Moreover, the alternating direction method of multipliers (ADMM) algorithm is used to build the inversion framework. Numerical experiments demonstrate that the fractional difference and the mixed second-order variant can reduce the staircase and scattering effects. The proposed method reduces the multiplicity and improves the accuracy than some state-of-the-art methods based on anisotropic total variation (ATV).

Compressed Sensing MRI via Extended Anisotropic and Isotropic Total Variation

Compressed sensing (CS) is a technique to reconstruct images from undersampling data, reducing the scanning time of magnetic resonance imaging (MRI). It utilizes the sparsity of images in some transform domains. Total variation (TV) has been applied to enforce sparsity. However, traditional TV based on the l1-norm is not the most direct way to induce sparsity, and it cannot offer a sufficiently sparse representation. Since the lp-norm (0&lt; p &lt; 1) promotes the sparsity better than that of the l1-norm, we propose two extended TV algorithms based on the lp-norm: anisotropic and isotropic total p-variation (TpV). Then we introduce them to the MRI reconstruction model. We apply the Bregman iteration technique to handle the proposed optimization problem. During the iteration, the p-shrinkage operator is employed to resolve the nonconvex problem caused by the lp-norm. Experimental results illustrate that our algorithms could offer the higher SNR and lower relative error compared with traditional TV algorithms and high-degree TV (HDTV) algorithm in MRI reconstruction problem.

A new in-line X-ray phase-contrast computed tomography reconstruction algorithm based on adaptive-weighted anisotropic TpV regularization for insufficient data.

In-line X-ray phase-contrast computed tomography (IL-PCCT) is a valuable tool for revealing the internal detailed structures in weakly absorbing objects (e.g. biological soft tissues), and has a great potential to become clinically applicable. However, the long scanning time for IL-PCCT will result in a high radiation dose to biological samples, and thus impede the wider use of IL-PCCT in clinical and biomedical imaging. To alleviate this problem, a new iterative CT reconstruction algorithm is presented that aims to decrease the radiation dose by reducing the projection views, while maintaining the high quality of reconstructed images. The proposed algorithm combines the adaptive-weighted anisotropic total p-variation (AwaTpV, 0 < p < 1) regularization technique with projection onto convex sets (POCS) strategy. Noteworthy, the AwaTpV regularization term not only contains the horizontal and vertical image gradients but also adds the diagonal image gradients in order to enforce the directional continuity in the gradient domain. To evaluate the effectiveness and ability of the proposed algorithm, experiments with a numerical phantom and synchrotron IL-PCCT were performed, respectively. The results demonstrated that the proposed algorithm had the ability to significantly reduce the artefacts caused by insufficient data and effectively preserved the edge details under noise-free and noisy conditions, and thus could be used as an effective approach to decrease the radiation dose for IL-PCCT.

Constrained Higher Degree Total p-Variation Minimization for MRI Reconstruction From Undersampled K-Space Data

Constrained Higher Degree Total p-Variation Minimization for MRI Reconstruction From Undersampled K-Space Data

Image reconstruction for cone-beam computed tomography using total p-variation plus Kullback–Leibler data divergence**Project supported by the National Natural Science Foundation of China (Grant Nos. 61372172 and 61601518).

Accurate reconstruction from a reduced data set is highly essential for computed tomography in fast and/or low dose imaging applications. Conventional total variation (TV)-based algorithms apply the L1 norm-based penalties, which are not as efficient as Lp () quasi-norm-based penalties. TV with a p-th power-based norm can serve as a feasible alternative of the conventional TV, which is referred to as total p-variation (TpV). This paper proposes a TpV-based reconstruction model and develops an efficient algorithm. The total p-variation and Kullback–Leibler (KL) data divergence, which has better noise suppression capability compared with the often-used quadratic term, are combined to build the reconstruction model. The proposed algorithm is derived by the alternating direction method (ADM) which offers a stable, efficient, and easily coded implementation. We apply the proposed method in the reconstructions from very few views of projections (7 views evenly acquired within 180°). The images reconstructed by the new method show clearer edges and higher numerical accuracy than the conventional TV method. Both the simulations and real CT data experiments indicate that the proposed method may be promising for practical applications.

Efficient TpV minimization for circular, cone-beam computed tomography reconstruction via non-convex optimization

An efficient iterative algorithm, based on recent work in non-convex optimization and generalized p-shrinkage mappings, is proposed for volume image reconstruction from circular cone-beam scans. Conventional total variation regularization makes use of L1 norm of gradient magnitude images (GMI). However, this paper utilizes a generalized penalty function, induced by p-shrinkage, of GMI which is proven to be a better measurement of its sparsity. The reconstruction model is formed using generalized total p-variation (TpV) minimization, which differs with the state of the art methods, with the constraint that the estimated projection data is within a specified tolerance of the available data and that the values of the volume image are non-negative. Theoretically, the proximal mapping for penalty functions induced by p-shrinkage has an exact and closed-form expression; thus, the constrained optimization can be stably and efficiently solved by the alternating direction minimization (ADM) scheme. Each sub-problem decoupled by variable splitting is minimized by explicit and easy-to-implement formulas developed by ADM. The proposed algorithm is efficiently implemented using a graphics processing unit and is referred to as “TpV-ADM.” This method is robust and accurate even for very few view reconstruction datasets. Verifications and comparisons performed using various datasets (including ideal, noisy, and real projections) illustrate that the proposed method is effective and promising.

Multiscale bilateral filtering for improving image quality in digital breast tomosynthesis.

Detection of subtle microcalcifications in digital breast tomosynthesis (DBT) is a challenging task because of the large, noisy DBT volume. It is important to enhance the contrast-to-noise ratio (CNR) of microcalcifications in DBT reconstruction. Most regularization methods depend on local gradient and may treat the ill-defined margins or subtle spiculations of masses and subtle microcalcifications as noise because of their small gradient. The authors developed a new multiscale bilateral filtering (MSBF) regularization method for the simultaneous algebraic reconstruction technique (SART) to improve the CNR of microcalcifications without compromising the quality of masses. The MSBF exploits a multiscale structure of DBT images to suppress noise and selectively enhance high frequency structures. At the end of each SART iteration, every DBT slice is decomposed into several frequency bands via Laplacian pyramid decomposition. No regularization is applied to the low frequency bands so that subtle edges of masses and structured background are preserved. Bilateral filtering is applied to the high frequency bands to enhance microcalcifications while suppressing noise. The regularized DBT images are used for updating in the next SART iteration. The new MSBF method was compared with the nonconvex total p-variation (TpV) method for noise regularization with SART. A GE GEN2 prototype DBT system was used for acquisition of projections at 21 angles in 3° increments over a ± 30° range. The reconstruction image quality with no regularization (NR) and that with the two regularization methods were compared using the DBT scans of a heterogeneous breast phantom and several human subjects with masses and microcalcifications. The CNR and the full width at half maximum (FWHM) of the line profiles of microcalcifications and across the spiculations within their in-focus DBT slices were used as image quality measures. The MSBF method reduced contouring artifacts and enhanced the CNR of microcalcifications compared to the TpV method, thus preserving the image quality of the structured background. The MSBF method achieved the highest CNR of microcalcifications among the three methods. The FWHM of the microcalcifications and mass spiculations resulting from the MSBF method was comparable to that without regularization, and superior to that of the TpV method. The SART regularized by the multiscale bilateral filtering method enhanced the CNR of microcalcifications and preserved the sharpness of microcalcifications and spiculated masses. The MSBF method provided better image quality of the structured background and was superior to TpV and NR for enhancing microcalcifications while preserving the appearance of mass margins.

Constrained TpV Minimization for Enhanced Exploitation of Gradient Sparsity: Application to CT Image Reconstruction.

Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the n}{}ell_{1} norm of the image gradient magnitude, and reducing the n}{}ell_{1} norm is known to encourage sparsity in its argument. Recently, there has been interest in employing nonconvex n}{}ell_{p} quasinorms with n}{}0<p<1 for sparsity exploiting image reconstruction, which is potentially more effective than n}{}ell_{1} because nonconvex n}{}ell_{p} is closer to n}{}ell_{0}—a direct measure of sparsity. This paper develops algorithms for constrained minimization of the total n}{}p-variation n}{}({rm T}p{rm V}), n}{}ell_{p} of the image gradient. Use of the algorithms is illustrated in the context of breast CT—an imaging modality that is still in the research phase and for which constraints on X-ray dose are extremely tight. The n}{}{rm T}p{rm V}-based image reconstruction algorithms are demonstrated on computer simulated data for exploiting gradient magnitude sparsity to reduce the projection view angle sampling. The proposed algorithms are applied to projection data from a realistic breast CT simulation, where the total X-ray dose is equivalent to two-view digital mammography. Following the simulation survey, the algorithms are then demonstrated on a clinical breast CT data set.

Estimates of the total [formula omitted]-variation of bivariate functions

We obtain sharp estimates of the Hardy–Vitali type total p-variation of a function of two variables in terms of its mixed modulus of continuity in Lp([0,1]2).

Selective‐diffusion regularization for enhancement of microcalcifications in digital breast tomosynthesis reconstruction

Digital breast tomosynthesis (DBT) has been shown to improve mass detection. Detection of microcalcifications is more challenging because of the large breast volume to be searched for subtle signals. The simultaneous algebraic reconstruction technique (SART) was found to provide good image quality for DBT, but the image noise is amplified with an increasing number of iterations. In this study, the authors developed a selective-diffusion (SD) method for noise regularization with SART to improve the contrast-to-noise ratio (CNR) of microcalcifications in the DBT slices for human or machine detection. The SD method regularizes SART reconstruction during updating with each projection view. Potential microcalcifications are differentiated from the noisy background by estimating the local gradient information. Different degrees of regularization are applied to the signal or noise classes, such that the microcalcifications will be enhanced while the noise is suppressed. The new SD method was compared to several current methods, including the quadratic Laplacian (QL) method, the total variation (TV) method, and the nonconvex total p-variation (TpV) method for noise regularization with SART. A GE GEN2 prototype DBT system with a stationary digital detector was used for the acquisition of DBT scans at 21 angles in 3 degrees increments over a +/-30 degrees range. The reconstruction image quality without regularization and that with the different regularization methods were compared using the DBT scans of an American College of Radiology phantom and a human subject. The CNR and the full width at half maximum (FWHM) of the line profiles of microcalcifications within the in-focus DBT slices were used as image quality measures. For the comparison of large microcalcifications in the DBT data of the subject, the SD method resulted in comparable CNR to the nonconvex TpV method. Both of them performed better than the other two methods. For subtle microcalcifications, the SD method was superior to other methods in terms of CNR. In both the subject and phantom DBT data, for large microcalcifications, the FWHM of the SD method was comparable to that without regularization, which was wider than that of the TV type methods. For subtle microcalcifications, the SD method had comparable FWHM values to the TV type methods. All three regularization methods were superior to the QL method in terms of FWHM. The SART regularized by the selective-diffusion method enhanced the CNR and preserved the sharpness of microcalcifications. In comparison with three existing regularization methods, the selective-diffusion regularization was superior to the other methods for subtle microcalcifications.

Enhanced imaging of microcalcifications in digital breast tomosynthesis through improved image‐reconstruction algorithms

The authors develop a practical, iterative algorithm for image-reconstruction in undersampled tomographic systems, such as digital breast tomosynthesis (DBT). The algorithm controls image regularity by minimizing the image total p variation (TpV), a function that reduces to the total variation when p = 1.0 or the image roughness when p = 2.0. Constraints on the image, such as image positivity and estimated projection-data tolerance, are enforced by projection onto convex sets. The fact that the tomographic system is undersampled translates to the mathematical property that many widely varied resultant volumes may correspond to a given data tolerance. Thus the application of image regularity serves two purposes: (1) Reduction in the number of resultant volumes out of those allowed by fixing the data tolerance, finding the minimum image TpV for fixed data tolerance, and (2) traditional regularization, sacrificing data fidelity for higher image regularity. The present algorithm allows for this dual role of image regularity in undersampled tomography. The proposed image-reconstruction algorithm is applied to three clinical DBT data sets. The DBT cases include one with microcalcifications and two with masses. Results indicate that there may be a substantial advantage in using the present image-reconstruction algorithm for microcalcification imaging.

On functions of bounded p-variation

We obtain estimates of the total p -variation ( 1 < p < ∞ ) and other related functionals for a periodic function f ∈ L p [ 0 , 1 ] in terms of its L p -modulus of continuity ω ( f ; δ ) p . These estimates are sharp for any rate of the decay of ω ( f ; δ ) p . Moreover, the constant coefficients in them depend on parameters in an optimal way.