Discretization of the radon transform and of its inverse by spline convolutions

Three-dimensional isotropic imaging of live suspension cells enabled by droplet microvortices

Fast, nondestructive three-dimensional (3D) imaging of live suspension cells remains challenging without substrate treatment or fixation, precluding scalable single-cell morphometry with minimal alterations. While optical sectioning techniques achieve 3D live cell imaging, lateral versus depth resolution differences further complicate analysis. We present a scalable microfluidic method capable of 3D fluorescent isotropic imaging of live, nonadherent cells suspended inside picoliter droplets with high-speed single-cell volumetric readout (800 to 1,200 slices in 5 to 8 s) and near-diffraction limit resolution (~216 nm). The platform features a droplet trap array that leverages flow-induced droplet interfacial shear to generate intradroplet microvortices, which rotate single cells on their axis to enable optical projection tomography (OPT)-based imaging. This allows gentle (~1 mPa shear stress) observation of cells encapsulated inside nontoxic isotonic buffer droplets, facilitating scalable OPT acquisition by simultaneous spinning of hundreds of cells. We demonstrate 3D imaging of live myeloid and lymphoid cells in suspension, including K562 cells, as well as naive and activated T cells-small cells prone to movement in their suspended phenotype. Our fully suspended, orientation-independent cell morphometry, driven by isotropic imaging and spherical harmonic analysis, enabled the study of primary T cells across various immunological activation states. This approach unveiled six distinct nuclear content distributions, contrasting with conventional 2D images that typically portray spheroid and bean-like nuclear shapes associated with lymphocytes. Our arrayed-droplet OPT technology is capable of isotropic, single live-cell 3D imaging, with the potential to perform large-scale morphometry of immune cell effector function states while providing compatibility with microfluidic droplet operations.

Dynamic Speed of Sound Adaptive Transmission-Reflection Ultrasound Computed Tomography

Ultrasound computed tomography (USCT) can visualize a target with multiple imaging contrasts, which were demonstrated individually previously. Here, to improve the imaging quality, the dynamic speed of sound (SoS) map derived from the transmission USCT will be adapted for the correction of the acoustic speed variation in the reflection USCT. The variable SoS map was firstly restored via the optimized simultaneous algebraic reconstruction technique with the time of flights selected from the transmitted ultrasonic signals. Then, the multi-stencils fast marching method was used to calculate the delay time from each element to the grids in the imaging field of view. Finally, the delay time in conventional constant-speed-assumed delay and sum (DAS) beamforming would be replaced by the practical computed delay time to achieve higher delay accuracy in the reflection USCT. The results from the numerical, phantom, and in vivo experiments show that our approach enables multi-modality imaging, accurate target localization, and precise boundary detection with the full-view fast imaging performance. The proposed method and its implementation are of great value for accurate, fast, and multi-modality USCT imaging, particularly suitable for highly acoustic heterogeneous medium.

X-ray Cherenkov-luminescence tomography reconstruction with a three-component deep learning algorithm: Swin transformer, convolutional neural network, and locality module

SIGNIFICANCE

X-ray Cherenkov-luminescence tomography (XCLT) produces fast emission data from megavoltage (MV) x-ray scanning, in which the excitation location of molecules within tissue is reconstructed. However standard filtered backprojection (FBP) algorithms for XCLT sinogram reconstruction can suffer from insufficient data due to dose limitations, so there are limits in the reconstruction quality with some artifacts. We report a deep learning algorithm for XCLT with high image quality and improved quantitative accuracy.

AIM

To directly reconstruct the distribution of emission quantum yield for x-ray Cherenkov-luminescence tomography, we proposed a three-component deep learning algorithm that includes a Swin transformer, convolution neural network, and locality module model.

APPROACH

A data-to-image model x-ray Cherenkov-luminescence tomography is developed based on a Swin transformer, which is used to extract pixel-level prior information from the sinogram domain. Meanwhile, a convolutional neural network structure is deployed to transform the extracted pixel information from the sinogram domain to the image domain. Finally, a locality module is designed between the encoder and decoder connection structures for delivering features. Its performance was validated with simulation, physical phantom, and in vivo experiments.

RESULTS

This approach can better deal with the limits to data than conventional FBP methods. The method was validated with numerical and physical phantom experiments, with results showing that it improved the reconstruction performance mean square error ( > 94.1 % ), peak signal-to-noise ratio ( > 41.7 % ), and Pearson correlation ( > 19 % ) compared with the FBP algorithm. The Swin-CNN also achieved a 32.1% improvement in PSNR over the deep learning method AUTOMAP.

CONCLUSIONS

This study shows that the three-component deep learning algorithm provides an effective reconstruction method for x-ray Cherenkov-luminescence tomography.

Pixel-driven computation of parallel and fan-beam projections of a digital image based on pixel-representation using a new formula

BACKGROUND

The computation of the projections of a digital image - modelled as a superposition of square pixels - is essential in several algorithms in computed tomography, and also in many machine vision applications. Projections of digital images are computed through the ray-driven approach. Current pixel-driven methods, though simpler, involve interpolation kernels in the projection-domain - not based on the exact Radon transform (RT) of a square.

METHODS

A new analytical formula - for the line-integral of the unit pixel - simpler than that published previously, is derived. The formula allows easy, pixel-driven computation of the RT of a digital image based on the pixel model i.e., Riemann-sum approximation to the line integral. The method naturally allows pixel-driven backprojection, based on the same (pixel) model. The approach is extended to computing projections over divergent (fan-) beams, and its application as a generalized version of the traditional Hough transform, is discussed.

RESULTS

The Radon transform of the unit-pixel match that of a digital square image. The RT, of a mathematical phantom consisting of a superposition of elliptical disks, compares well with that based on analytical formula. A comparative study with the pixel driven approach with interpolation in the projection-domain, and its important variant, is included. The fan-beam projections of the square image and the phantom are presented. The applicability of the RT in estimating the Hough transform over precise lines, is shown.

CONCLUSION

The new formula, a simplified version of that of Deans, is useful in pixel-driven computation of parallel and fan-beam projections based on Riemann-sum approximation, which is exact in the case of the common pixel-based image model. Pixel-driven approach is amenable to parallel, and also region-of-interest computation. The method is useful in CT as well as machine vision applications.

Direct inversion algorithm for focal plane scanning optical projection tomography

To achieve approximately parallel projection geometry, traditional optical projection tomography (OPT) requires the use of low numerical aperture (NA) objectives, which have a long depth-of-field at the expense of poor lateral resolution. Particularly promising methods to improve spatial resolution include ad-hoc post-processing filters that limit the effect of the system's MTF and focal-plane-scanning OPT (FPS-OPT), an alternative acquisition procedure that allows the use of higher NA objectives by limiting the effect of their shallow depth of field yet still assumes parallel projection rays during reconstruction. Here, we provide a detailed derivation that establishes the existence of a direct inversion formula for FPS-OPT. Based on this formula, we propose a point spread function-aware algorithm that is similar in form and complexity to traditional filtered backprojection (FBP). With simulations, we demonstrate that our point-spread-function aware FBP for FPS-OPT leads to more accurate images than both traditional OPT with deconvolution and FPS-OPT with naive FBP reconstruction. We further illustrate the technique on experimental zebrafish data, which shows that our approach reduces out-of-focus blur compared to a direct FBP reconstruction with FPS-OPT.

High-Quality Parallel-Ray X-Ray CT Back Projection Using Optimized Interpolation

We propose a new, cost-efficient method for computing back projections in parallel-ray X-ray CT. Forward and back projections are the basis of almost all X-ray CT reconstruction methods, but computing these accurately is costly. In the special case of parallel-ray geometry, it turns out that reconstruction requires back projection only. One approach to accelerate the back projection is through interpolation: fit a continuous representation to samples of the desired signal, then sample it at the required locations. Instead, we propose applying a prefilter that has the effect of orthogonally projecting the underlying signal onto the space spanned by the interpolator, which can significantly improve the quality of the interpolation. We then build on this idea by using oblique projection, which simplifies the computation while giving effectively the same improvement in quality. Our experiments on analytical phantoms show that this refinement can improve the reconstruction quality for both filtered back projection and iterative reconstruction in the high-quality regime, i.e., with low noise and many measurements.

A Survey of the Use of Iterative Reconstruction Algorithms in Electron Microscopy

One of the key steps in Electron Microscopy is the tomographic reconstruction of a three-dimensional (3D) map of the specimen being studied from a set of two-dimensional (2D) projections acquired at the microscope. This tomographic reconstruction may be performed with different reconstruction algorithms that can be grouped into several large families: direct Fourier inversion methods, back-projection methods, Radon methods, or iterative algorithms. In this review, we focus on the latter family of algorithms, explaining the mathematical rationale behind the different algorithms in this family as they have been introduced in the field of Electron Microscopy. We cover their use in Single Particle Analysis (SPA) as well as in Electron Tomography (ET).

Deep Convolutional Neural Network for Inverse Problems in Imaging

In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise nonlinearity) when the normal operator (H*H, where H* is the adjoint of the forward imaging operator, H) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a 512 × 512 image on the GPU.

A comparison of linear interpolation models for iterative CT reconstruction

PURPOSE

Recent reports indicate that model-based iterative reconstruction methods may improve image quality in computed tomography (CT). One difficulty with these methods is the number of options available to implement them, including the selection of the forward projection model and the penalty term. Currently, the literature is fairly scarce in terms of guidance regarding this selection step, whereas these options impact image quality. Here, the authors investigate the merits of three forward projection models that rely on linear interpolation: the distance-driven method, Joseph's method, and the bilinear method. The authors' selection is motivated by three factors: (1) in CT, linear interpolation is often seen as a suitable trade-off between discretization errors and computational cost, (2) the first two methods are popular with manufacturers, and (3) the third method enables assessing the importance of a key assumption in the other methods.

METHODS

One approach to evaluate forward projection models is to inspect their effect on discretized images, as well as the effect of their transpose on data sets, but significance of such studies is unclear since the matrix and its transpose are always jointly used in iterative reconstruction. Another approach is to investigate the models in the context they are used, i.e., together with statistical weights and a penalty term. Unfortunately, this approach requires the selection of a preferred objective function and does not provide clear information on features that are intrinsic to the model. The authors adopted the following two-stage methodology. First, the authors analyze images that progressively include components of the singular value decomposition of the model in a reconstructed image without statistical weights and penalty term. Next, the authors examine the impact of weights and penalty on observed differences.

RESULTS

Image quality metrics were investigated for 16 different fan-beam imaging scenarios that enabled probing various aspects of all models. The metrics include a surrogate for computational cost, as well as bias, noise, and an estimation task, all at matched resolution. The analysis revealed fundamental differences in terms of both bias and noise. Task-based assessment appears to be required to appreciate the differences in noise; the estimation task the authors selected showed that these differences balance out to yield similar performance. Some scenarios highlighted merits for the distance-driven method in terms of bias but with an increase in computational cost. Three combinations of statistical weights and penalty term showed that the observed differences remain the same, but strong edge-preserving penalty can dramatically reduce the magnitude of these differences.

CONCLUSIONS

In many scenarios, Joseph's method seems to offer an interesting compromise between cost and computational effort. The distance-driven method offers the possibility to reduce bias but with an increase in computational cost. The bilinear method indicated that a key assumption in the other two methods is highly robust. Last, strong edge-preserving penalty can act as a compensator for insufficiencies in the forward projection model, bringing all models to similar levels in the most challenging imaging scenarios. Also, the authors find that their evaluation methodology helps appreciating how model, statistical weights, and penalty term interplay together.

Fast gridding projectors for analytical and iterative tomographic reconstruction of differential phase contrast data

This paper introduces new gridding projectors designed to efficiently perform analytical and iterative tomographic reconstruction, when the forward model is represented by the derivative of the Radon transform. This inverse problem is tightly connected with an emerging X-ray tube- and synchrotron-based imaging technique: differential phase contrast based on a grating interferometer. This study shows, that the proposed projectors, compared to space-based implementations of the same operators, yield high quality analytical and iterative reconstructions, while improving the computational efficiency by few orders of magnitude.

Fast 3D reconstruction method for differential phase contrast X-ray CT

We present a fast algorithm for fully 3D regularized X-ray tomography reconstruction for both traditional and differential phase contrast measurements. In many applications, it is critical to reduce the X-ray dose while producing high-quality reconstructions. Regularization is an excellent way to do this, but in the differential phase contrast case it is usually applied in a slice-by-slice manner. We propose using fully 3D regularization to improve the dose/quality trade-off beyond what is possible using slice-by-slice regularization. To make this computationally feasible, we show that the two computational bottlenecks of our iterative optimization process can be expressed as discrete convolutions; the resulting algorithms for computation of the X-ray adjoint and normal operator are fast and simple alternatives to regridding. We validate this algorithm on an analytical phantom as well as conventional CT and differential phase contrast measurements from two real objects. Compared to the slice-by-slice approach, our algorithm provides a more accurate reconstruction of the analytical phantom and better qualitative appearance on one of the two real datasets.

Accelerated statistical reconstruction for C-arm cone-beam CT using Nesterov's method

PURPOSE

To accelerate model-based iterative reconstruction (IR) methods for C-arm cone-beam CT (CBCT), thereby combining the benefits of improved image quality and/or reduced radiation dose with reconstruction times on the order of minutes rather than hours.

METHODS

The ordered-subsets, separable quadratic surrogates (OS-SQS) algorithm for solving the penalized-likelihood (PL) objective was modified to include Nesterov's method, which utilizes "momentum" from image updates of previous iterations to better inform the current iteration and provide significantly faster convergence. Reconstruction performance of an anthropomorphic head phantom was assessed on a benchtop CBCT system, followed by CBCT on a mobile C-arm, which provided typical levels of incomplete data, including lateral truncation. Additionally, a cadaveric torso that presented realistic soft-tissue and bony anatomy was imaged on the C-arm, and different projectors were assessed for reconstruction speed.

RESULTS

Nesterov's method provided equivalent image quality to OS-SQS while reducing the reconstruction time by an order of magnitude (10.0 ×) by reducing the number of iterations required for convergence. The faster projectors were shown to produce similar levels of convergence as more accurate projectors and reduced the reconstruction time by another 5.3 ×. Despite the slower convergence of IR with truncated C-arm CBCT, comparison of PL reconstruction methods implemented on graphics processing units showed that reconstruction time was reduced from 106 min for the conventional OS-SQS method to as little as 2.0 min with Nesterov's method for a volumetric reconstruction of the head. In body imaging, reconstruction of the larger cadaveric torso was reduced from 159 min down to 3.3 min with Nesterov's method.

CONCLUSIONS

The acceleration achieved through Nesterov's method combined with ordered subsets reduced IR times down to a few minutes. This improved compatibility with clinical workflow better enables broader adoption of IR in CBCT-guided procedures, with corresponding benefits in overcoming conventional limits of image quality at lower dose.

A Forward Regridding Method With Minimal Oversampling for Accurate and Efficient Iterative Tomographic Algorithms

Reconstruction of underconstrained tomographic data sets remains a major challenge. Standard analytical techniques frequently lead to unsatisfactory results due to insufficient information. Several iterative algorithms, which can easily integrate a priori knowledge, have been developed to tackle this problem during the last few decades. Most of these iterative algorithms are based on an implementation of the Radon transform that acts as forward projector. This operator and its adjoint, the backprojector, are typically called few times per iteration and represent the computational bottleneck of the reconstruction process. Here, we present a Fourier-based forward projector, founded on the regridding method with minimal oversampling. We show that this implementation of the Radon transform significantly outperforms in efficiency other state-of-the-art operators with O(N2log2N) complexity. Despite its reduced computational cost, this regridding method provides comparable accuracy to more sophisticated projectors and can, therefore, be exploited in iterative algorithms to substantially decrease the time required for the reconstruction of underconstrained tomographic data sets without loss in the quality of the results.

Spline Driven: High Accuracy Projectors for Tomographic Reconstruction From Few Projections

Tomographic iterative reconstruction methods need a very thorough modeling of data. This point becomes critical when the number of available projections is limited. At the core of this issue is the projector design, i.e., the numerical model relating the representation of the object of interest to the projections on the detector. Voxel driven and ray driven projection models are widely used for their short execution time in spite of their coarse approximations. Distance driven model has an improved accuracy but makes strong approximations to project voxel basis functions. Cubic voxel basis functions are anisotropic, accurately modeling their projection is, therefore, computationally expensive. Both smoother and more isotropic basis functions better represent the continuous functions and provide simpler projectors. These considerations have led to the development of spherically symmetric volume elements, called blobs. Set apart their isotropy, blobs are often considered too computationally expensive in practice. In this paper, we consider using separable B-splines as basis functions to represent the object, and we propose to approximate the projection of these basis functions by a 2D separable model. When the degree of the B-splines increases, their isotropy improves and projections can be computed regardless of their orientation. The degree and the sampling of the B-splines can be chosen according to a tradeoff between approximation quality and computational complexity. We quantitatively measure the good accuracy of our model and compare it with other projectors, such as the distance-driven and the model proposed by Long et al. From the numerical experiments, we demonstrate that our projector with an improved accuracy better preserves the quality of the reconstruction as the number of projections decreases. Our projector with cubic B-splines requires about twice as many operations as a model based on voxel basis functions. Higher accuracy projectors can be used to improve the resolution of the existing systems, or to reduce the number of projections required to reach a given resolution, potentially reducing the dose absorbed by the patient.

Multiscale segmentation of the skull in MR images for MRI-based attenuation correction of combined MR/PET

BACKGROUND AND OBJECTIVE

Combined magnetic resonance/positron emission tomography (MR/PET) is a relatively new, hybrid imaging modality. MR-based attenuation correction often requires segmentation of the bone on MR images. In this study, we present an automatic segmentation method for the skull on MR images for attenuation correction in brain MR/PET applications.

MATERIALS AND METHODS

Our method transforms T1-weighted MR images to the Radon domain and then detects the features of the skull image. In the Radon domain we use a bilateral filter to construct a multiscale image series. For the repeated convolution we increase the spatial smoothing in each scale and make the width of the spatial and range Gaussian function doubled in each scale. Two filters with different kernels along the vertical direction are applied along the scales from the coarse to fine levels. The results from a coarse scale give a mask for the next fine scale and supervise the segmentation in the next fine scale. The use of the multiscale bilateral filtering scheme is to improve the robustness of the method for noise MR images. After combining the two filtered sinograms, the reciprocal binary sinogram of the skull is obtained for the reconstruction of the skull image.

RESULTS

This method has been tested with brain phantom data, simulated brain data, and real MRI data. For real MRI data the Dice overlap ratios are 92.2%±1.9% between our segmentation and manual segmentation.

CONCLUSIONS

The multiscale segmentation method is robust and accurate and can be used for MRI-based attenuation correction in combined MR/PET.

Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography

Discretization of continuous (analog) convolution operators by direct sampling of the convolution kernel and use of fast Fourier transforms is highly efficient. However, it assumes the input and output signals are band-limited, a condition rarely met in practice, where signals have finite support or abrupt edges and sampling is nonideal. Here, we propose to approximate signals in analog, shift-invariant function spaces, which do not need to be band-limited, resulting in discrete coefficients for which we derive discrete convolution kernels that accurately model the analog convolution operator while taking into account nonideal sampling devices (such as finite fill-factor cameras). This approach retains the efficiency of direct sampling but not its limiting assumption. We propose fast forward and inverse algorithms that handle finite-length, periodic, and mirror-symmetric signals with rational sampling rates. We provide explicit convolution kernels for computing coherent wave propagation in the context of digital holography. When compared to band-limited methods in simulations, our method leads to fewer reconstruction artifacts when signals have sharp edges or when using nonideal sampling devices.

A box spline calculus for the discretization of computed tomography reconstruction problems

B-splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of box splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported box splines (tensor-product B-splines, separable or not) in any number of dimensions. The proposed box spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an N-direction box spline is generally a (nonuniform) polynomial spline of degree N-1. The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a box spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms.

Simulation tools for two-dimensional experiments in x-ray computed tomography using the FORBILD head phantom

Mathematical phantoms are essential for the development and early stage evaluation of image reconstruction algorithms in x-ray computed tomography (CT). This note offers tools for computer simulations using a two-dimensional (2D) phantom that models the central axial slice through the FORBILD head phantom. Introduced in 1999, in response to a need for a more robust test, the FORBILD head phantom is now seen by many as the gold standard. However, the simple Shepp-Logan phantom is still heavily used by researchers working on 2D image reconstruction. Universal acceptance of the FORBILD head phantom may have been prevented by its significantly higher complexity: software that allows computer simulations with the Shepp-Logan phantom is not readily applicable to the FORBILD head phantom. The tools offered here address this problem. They are designed for use with Matlab®, as well as open-source variants, such as FreeMat and Octave, which are all widely used in both academia and industry. To get started, the interested user can simply copy and paste the codes from this PDF document into Matlab® M-files.

A wavelet multiscale denoising algorithm for magnetic resonance (MR) images

Based on the Radon transform, a wavelet multiscale denoising method is proposed for MR images. The approach explicitly accounts for the Rician nature of MR data. Based on noise statistics we apply the Radon transform to the original MR images and use the Gaussian noise model to process the MR sinogram image. A translation invariant wavelet transform is employed to decompose the MR 'sinogram' into multiscales in order to effectively denoise the images. Based on the nature of Rician noise we estimate noise variance in different scales. For the final denoised sinogram we apply the inverse Radon transform in order to reconstruct the original MR images. Phantom, simulation brain MR images, and human brain MR images were used to validate our method. The experiment results show the superiority of the proposed scheme over the traditional methods. Our method can reduce Rician noise while preserving the key image details and features. The wavelet denoising method can have wide applications in MRI as well as other imaging modalities.

Dynamic PET reconstruction using wavelet regularization with adapted basis functions

Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an l(1) -regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the "optimal" E-spline parameters (poles and zeros) and we investigate their effect on reconstruction quality. We demonstrate the usefulness of spatio-temporal regularization and the superior performance of E-spline wavelets over conventional Battle-LemariE wavelets in a series of experiments: the 1-D fitting of TACs, and the tomographic reconstruction of both simulated and clinical data. We find that the E-spline wavelets outperform the conventional wavelets in terms of the reconstructed signal-to-noise ratio (SNR) and the sparsity of the wavelet coefficients. Based on our simulations, we conclude that replacing the conventional wavelets with E-spline wavelets leads to equal reconstruction quality for a 40% reduction of detected coincidences, meaning an improved image quality for the same number of counts or equivalently a reduced exposure to the patient for the same image quality.