Nonseparable wavelet-based cone-beam reconstruction in 3-D rotational angiography

In this paper, we propose a new wavelet-based reconstruction method suited to three-dimensional (3-D) cone-beam (CB) tomography. It is derived from the Feldkamp algorithm and is valid for the same geometrical conditions. The demonstration is done in the framework of nonseparable wavelets and requires ideally radial wavelets. The proposed inversion formula yields to a filtered backprojection algorithm but the filtering step is implemented using quincunx wavelet filters. The proposed algorithm reconstructs slice by slice both the wavelet and approximation coefficients of the 3-D image directly from the CB projection data. The validity of this multiresolution approach is demonstrated on simulations from both mathematical phantoms and 3-D rotational angiography clinical data. The same quality is achieved compared with the standard Feldkamp algorithm, but in addition, the multiresolution decomposition allows to apply directly image processing techniques in the wavelet domain during the inversion process. As an example, a fast low-resolution reconstruction of the 3-D arterial vessels with the progressive addition of details in a region of interest is demonstrated. Other promising applications are the improvement of image quality by denoising techniques and also the reduction of computing time using the space localization of wavelets.     

A Method for Stop-Action Imaging of the Heart Using Gated Computed Tomography

A new method for the reconstruction of limited angle projection data in rotary fan-beam X-ray computed tomography (CT) is presented. Missing views resulting from ECG-gated cardiac CT are estimated, and the standard fan-beam reconstruction algorithm is used to convolve and backproject both measured and estimated views. The estimation of the missing views takes place in three stages: first, the projection data is augmented by incorporating into each missing view the line integrals that do not pass through the heart, and which otherwise would be considered missing due to ECG-gating; second, line integrals corresponding to source positions in the range 180°±fan angle away from missing view angles are reflected; third, those line integrals that remain missing are estimated by interpolation. This method has been applied to ECG-gated cardiac imaging in dogs without requiring extensive interpolation; end-systolic and end-diastolic images were generated with short-interval gating (?cycle) and total scan time (breath holding period) of 12 s. An important advantage of this method over other proposed limited angle reconstruction techniques is that it uses the existing fan-beam convolution-backprojection algorithm for image reconstruction.     

A Class of Multiresolution Directional Filter Banks

In this paper, we introduced a class of directional filter banks (DFBs) having the previously proposed uniform DFB (uDFB) as a special case. Except for the uDFB, each DFB in this class can be used to decompose an image yielding up to 12 directions while maintaining perfect reconstruction and maximal decimation. A multiresolution representation can be obtained by repeating the same decomposition at the lowpass band. The permissible property of the filter banks in cases of being implemented by a tree structure and by direct implementation is discussed. The result shows that only one DFB in the class, called the uniform quincunx DFB (uqDFB), satisfies the permissible property when being implemented directly without using the tree structure. The nonuniform quincunx DFB (nuqDFB) is then constructed from the uqDFB by merging its two lowpass subbands. An alternative structure for constructing the nuqDFB is presented. The new structure, while yielding the same frequency partitioning, allows the DFB to be realized with complexity comparable to that of the separable wavelet filter bank. The connection between the discrete filter bank and the continuous directional wavelet is also established. Numerical experiments on directional feature extractions, image denoising and nonlinear approximation are presented at the end of the paper to demonstrate the potential of the nuqDFB     

Oriented wavelet transform for image compression and denoising.

In this paper, we introduce a new transform for image processing, based on wavelets and the lifting paradigm. The lifting steps of a unidimensional wavelet are applied along a local orientation defined on a quincunx sampling grid. To maximize energy compaction, the orientation minimizing the prediction error is chosen adaptively. A fine-grained multiscale analysis is provided by iterating the decomposition on the low-frequency band. In the context of image compression, the multiresolution orientation map is coded using a quad tree. The rate allocation between the orientation map and wavelet coefficients is jointly optimized in a rate-distortion sense. For image denoising, a Markov model is used to extract the orientations from the noisy image. As long as the map is sufficiently homogeneous, interesting properties of the original wavelet are preserved such as regularity and orthogonality. Perfect reconstruction is ensured by the reversibility of the lifting scheme. The mutual information between the wavelet coefficients is studied and compared to the one observed with a separable wavelet transform. The rate-distortion performance of this new transform is evaluated for image coding using state-of-the-art subband coders. Its performance in a denoising application is also assessed against the performance obtained with other transforms or denoising methods.     

A unified reconstruction framework for both parallel-beam and variable focal-length fan-beam collimators by a Cormack-type inversion of exponential radon transform

A variety of inversions of exponential Radon transform has been derived based on the circular harmonic transform in Fourier space by several research groups. However, these inversions cannot be directly applied to deal with the reconstruction for fan-beam or variable-focal-length fan-beam collimator geometries in single photon emission computed tomography (SPECT). In this paper, we derived a Cormack-type inversion of the exponential Radon transform by employing the circular harmonic transform directly in the projection space and the image space instead of the Fourier space. Thus, a unified reconstruction framework is established for parallel-, fan-, and variable-focal-length fan-beam collimator geometries. Compared to many existing algorithms, the presented one greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry and significantly reduces the computational burden of the special functions, such as Chebyshev or Bessel functions. By the well-established fast-Fourier transform (FFT), our algorithm is very efficient, as demonstrated by several numerical simulations.     

A derivative-free noncircular fan-beam reconstruction formula

In order to perform fan-beam reconstruction using projection data collected from a noncircular scanning locus, existing noncircular fan-beam formulas require a derivative of the scanning locus with respect to the rotation angle. A derivative-free noncircular fan-beam reconstruction formula that is based on a geometrical explanation of the circular equispatial fan-beam reconstruction formula is obtained here. A mathematical proof is then provided under the conditions that the source-to-origin distance is symmetric with respect to the origin of the reconstruction coordinate system, is differentiable almost everywhere, and does not change too fast with respect to the rotation angle. The derivative-free noncircular fan-beam reconstruction formula is the same as the circular one, except that the source-to-origin distance is a function of the rotation angle. A typical simulation result for the noncircular fan-beam formula is given.     

On the possibility of direct Fourier reconstruction from divergent-beam projections

A great number of tomographic systems, especially those equipped with fast data acquisition techniques, scan the objects investigated by divergent (fan) X-ray beams. Fan-beam projection data require special reconstruction techniques to be implemented. Among reconstruction techniques from parallel projection data, the direct Fourier method (DFM) proved to be one of the most promising ones, especially for high-speed image reconstruction in the high-end 3-D and dynamic tomographic systems. The goal of this work is to answer the topical question: how would direct use of the DFM influence the quality of image reconstruction from the fan-beam data? The formula describing the error caused by such an approximation is derived. The conclusions deduced from the formula are confirmed by computer simulations. The boundary values of data acquisition geometry parameters have been estimated for the case of using the DFM without recalculating the fan-beam data.     

Reconstruction algorithms: Transform methods

Transform methods for image reconstruction from projections are based on analytic inversion formulas. In this tutorial paper, the inversion formula for the case of two-dimensional (2-D) reconstruction from line integrals is manipulated into a number of different forms, each of which may be discretized to obtain different algorithms for reconstruction from sampled data. For the convolution-backprojection algorithm and the direct Fourier algorithm the emphasis is placed on understanding the relationship between the discrete operations specified by the algorithm and the functional operations expressed by the inversion formula. The performance of the Fourier algorithm may be improved, with negligible extra computation, by interleaving two polar sampling grids in Fourier space. The convolution-backprojection formulas are adapted for the fan-beam geometry, and other reconstruction methods are summarized, including the rho-filtered layergram method, and methods involving expansions in angular harmonics. A standard mathematical process leads to a known formula for iterative reconstruction from projections at a finite number of angles. A new iterative reconstruction algorithm is obtained from this formula by introducing one-dimensional (1-D) and 2-D interpolating functions, applied to sampled projections and images, respectively. These interpolating functions are derived by the same Fourier approach which aids in the development and understanding of the more conventional transform methods.     

Iterative Reconstruction?Reprojection: An Algorithm for Limited Data Cardiac-Computed Tomography

Cardiac X-ray computed tomography (CT) has been limited due to scanning times which are considerably longer (1 s) than required to resolve the beating heart (0.1 s). The otherwise attractive convolution-backprojection algorithm is not suited for CT image reconstruction from measurements comprising an incomplete set of projection data. In this paper, an iterative reconstruction-reprojection (IRR) algorithm is proposed for limited projection data CT image reconstruction. At each iteration, the missing views are estimated based on reprojection, which is a software substitute for the scanning process. The standard fan-beam convolution-backprojection algorithm is then used for image reconstruction. The proposed IRR algorithm enables the use of convolution-backprojection in limited angle of view and in limited field of view CT cases. The potential of this method for cardiac CT reconstruction is demonstrated using computer simulated data.     

光CT的扇束投影重排方法及投影几何分析

在工业过程监控中,由于被测物理过程的发生时间短,因此光CT装置将不可能象医学CT那样有充足的时间进行"静态"成像,而是必须进行实时性较高的"动态"成像(或称为流动成像).其制约是光线扫描的投影视角和投影数量都不可能太多.根据这一情况,在医学CT中常用的一些图像重建算法如反投影法、滤波反投影法和投影重排方法都必须加以改进才能引用.为此,本文研究扇束投影重排方法在光学流动成像中的应用可行性,分析其计算方法和步骤,并确定光线扇束结构中的光源个数、扇形张角和探测器个数之间的约束关系,同时也分析了内插精度对图像质量的影响.在数值模拟实验中,将投影重排方法与反投影法的重建图象进行了比较,进一步评判本文方法的图象质量和成像实时性等方面的性能.     

Fourier-based forward and back-projectors in iterative fan-beam tomographic image reconstruction

Fourier-based forward and back-projection methods can reduce computation in iterative tomographic image reconstruction. Recently, an optimized nonuniform fast Fourier transform (NUFFT) approach was shown to yield accurate parallel-beam projections. In this paper, we extend the NUFFT approach to describe an O (N2 log N) projector/backprojector pair for fan-beam transmission tomography. Simulations and experiments with real CT data show that fan-beam Fourier-based forward and back-projection methods can reduce computation for iterative reconstruction while still providing accuracy comparable to their O (N3) space-based counterparts.     

基于总变分正则化算法的散射计图像重构

目前业务化运行的星载散射计分辨率一般为25 km,在分辨率需求较高的应用中(如极地海冰监测、热带雨林监测、近岸风场研究等)受到了限制。散射计图像重构技术可以在不改变系统硬件的前提下,通过数据处理方法的改进,提高分辨率。现有的散射计图像重构方法(SIR)是基于图像处理领域中较早期的乘性代数重建技术(MART)。该文针对星载扇形旋转扫描散射计,将一种新的图像重构方法总变分正则化 (total variation regularization) 算法应用于散射计图像重构,并通过仿真实验说明,新算法可以在增强分辨率的同时减少噪声,提高重构图像的质量。     

Data consistency based translational motion artifact reduction in fan-beam CT

A basic assumption in the classic computed tomography (CT) theory is that an object remains stationary in an entire scan. In biomedical CT/micro-CT, this assumption is often violated. To produce high-resolution images, such as for our recently proposed clinical micro-CT (CMCT) prototype, it is desirable to develop a precise motion estimation and image reconstruction scheme. In this paper, we first extend the Helgason-Ludwig consistency condition (HLCC) from parallel-beam to fan-beam geometry when an object is subject to a translation. Then, we propose a novel method to estimate the motion parameters only from sinograms based on the HLCC. To reconstruct the moving object, we formulate two generalized fan-beam reconstruction methods, which are in filtered backprojection and backprojection filtering formats, respectively. Furthermore, we present numerical simulation results to show that our approach is accurate and robust.     

Videographic tomography. II. Reconstruction with fan-beam projection data

For pt.I see ibid., vol.9, p.366-75 (1990). Presents approaches for the realization of fast electro-optical reconstruction systems for fan-beam projection data. Two of the proposed systems were implemented. Requiring only 30 ms for the reconstruction of one image, the systems are two orders of magnitude faster than the special-purpose hardware in commercial CT reconstruction systems. The speed of the reconstruction systems would allow real-time interactive CT imaging.     

一种基于提升小波的多传感器图像融合

提出了一种基于提升方案的五株形小波的多传感器图像融合方法,即在提升五株形小波多尺度分解的基础上,对每一幅图像进行多尺度分解,按照不同的融合规则和融合算子去构造融合图像对应的各小波系数,再通过逆变换重构融合后的图像.该方法被成功地用于多聚焦图像的融合处理.试验结果表明该融合方法十分有效,获得的融合图像更符合人眼的视觉特性、更有利于进行目标监视和侦察.     

Fast predictions of variance images for fan-beam transmission tomography with quadratic regularization

Accurate predictions of image variances can be useful for reconstruction algorithm analysis and for the design of regularization methods. Computing the predicted variance at every pixel using matrix-based approximations [1] is impractical. Even most recently adopted methods that are based on local discrete Fourier approximations are impractical since they would require a forward and backprojection and two fast Fourier transform (FFT) calculations for every pixel, particularly for shift-variant systems like fan-beam tomography. This paper describes new "analytical" approaches to predicting the approximate variance maps of 2-D images that are reconstructed by penalized-likelihood estimation with quadratic regularization in fan-beam geometries. The simplest of the proposed analytical approaches requires computation equivalent to one backprojection and some summations, so it is computationally practical even for the data sizes in X-ray computed tomography (CT). Simulation results show that it gives accurate predictions of the variance maps. The parallel-beam geometry is a simple special case of the fan-beam analysis. The analysis is also applicable to 2-D positron emission tomography (PET).     

Design of signal-adapted multidimensional lifting scheme for lossy coding

This paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform. Our approach is stated in the general case, and is illustrated for the 2-D separable and for the quincunx images. Results are shown for the JPEG2000 database and for satellite images acquired on a quincunx sampling grid. The design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases. Our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case. It is based on a two-step lifting scheme and joins the lifting theory with Wiener's optimization. The prediction step is designed in order to minimize the variance of the signal, and the update step is designed in order to minimize a reconstruction error. Application for lossy compression shows the performances of the method.     

CT扇形束滤波反投影图像重建算法优化

基于等距扇形束滤波反投影(FBP)算法推导了一种新的算法——求导-希尔伯特反投影(DHB)算法,研究了DHB算法在频域对投影的滤波特性。通过理论分析和实验验证,指出由于DHB滤波函数在高频段对于锐截止特性的改善,很大程度上消除了重建图像的抖动现象。并且算法中去掉了反投影算子中的距离加权运算,使计算速度进一步提高。     

Gradient-based multiresolution image fusion

A novel approach to multiresolution signal-level image fusion is presented for accurately transferring visual information from any number of input image signals, into a single fused image without loss of information or the introduction of distortion. The proposed system uses a "fuse-then-decompose" technique realized through a novel, fusion/decomposition system architecture. In particular, information fusion is performed on a multiresolution gradient map representation domain of image signal information. At each resolution, input images are represented as gradient maps and combined to produce new, fused gradient maps. Fused gradient map signals are processed, using gradient filters derived from high-pass quadrature mirror filters to yield a fused multiresolution pyramid representation. The fused output image is obtained by applying, on the fused pyramid, a reconstruction process that is analogous to that of conventional discrete wavelet transform. This new gradient fusion significantly reduces the amount of distortion artefacts and the loss of contrast information usually observed in fused images obtained from conventional multiresolution fusion schemes. This is because fusion in the gradient map domain significantly improves the reliability of the feature selection and information fusion processes. Fusion performance is evaluated through informal visual inspection and subjective psychometric preference tests, as well as objective fusion performance measurements. Results clearly demonstrate the superiority of this new approach when compared to conventional fusion systems.     

A simple method to build oversampled filter banks and tight frames.

This paper presents conditions under which the sampling lattice for a filter bank can be replaced without loss of perfect reconstruction. This is the generalization of common knowledge that removing up/downsampling will not lose perfect reconstruction. The results provide a simple way of building oversampled filter banks. If the original filter banks are orthogonal, these oversampled banks construct tight frames of l2 (Z(n)) when iterated. As an example, a quincunx lattice is used to replace the rectangular one of the standard wavelet transform. This replacement leads to a tight frame that has a higher sampling in both time and frequency. The frame transform is nearly shift invariant and has intermediate scales. An application of the transform to image fusion is also presented.