2—D Fourier Transform Method for Optical Measurement of 3—D Surface

A emthod of 2-D Fourier transform used in 3-D surface profilometry is proposed,analyzed and compared with 1-D Fourier transform method in theory and practical measuring result.It was proved that the 2-D Fourier transform method has more advantages over 1-D Fourier transform methodd in biggest crook-rate limits,accuracy and sensitivity of measuring.Styudy on measuring object surface details with large crook-rate changing accurately used new higher-power index low-pass filter of spatial frequency domain.A new method of automatic produced reference grating image and error-correcting is proposed.One undeform row of deform grating image is used to extend a complete reference grating image,and some error-correcting method is used to process the result to get accurate surface shape and the deflection of reference surface normal line deviated from the axle of camera.By this new method,one deform rectangle grating image is only used to get the 3-D shape accurately.     

Quantitative characterization of electron micrograph image usingfractal feature

In this investigation, texture analysis was carried out on electron micrograph images. Fractal dimensions and spatial grey level co-occurrence matrices statistics were estimated on each homogeneous region of interest, The fractal model has the advantages that the fractal dimension correlates to the roughness of the surface and is stable over transformations of scale and linear transforms of intensity. It can be calculated using three different methods. The first method estimates fractal dimension based on the average intensity difference of pixel pairs. In the second method, fractal dimension is determined from the Fourier transformed domain. Finally, fractal dimension can be estimated using reticular cell counting approach. Moreover, automatic image segmentation was performed using fractal dimensions, spatial grey level co-occurrence matrices statistics, and grey level thresholding. Each image was segmented into a number of regions corresponding to distinctly different morphologies: heterochromatin, euchromatin, and background. Fractal dimensions and spatial grey level co-occurrence matrices statistics were found to be able to characterize and segment electron micrograph images     

An efficient Fourier method for 3-D radon inversion in exactcone-beam CT reconstruction

The radial derivative of the three-dimensional (3-D) radon transform of an object is an important intermediate result in many analytically exact cone-beam reconstruction algorithms. The authors briefly review Grangeat's (1991) approach for calculating radon derivative data from cone-beam projections and then present a new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3-D radon transform, called direct Fourier inversion (DFI). The method is based directly on the 3-D Fourier slice theorem. From the 3-D radon derivative data, which is assumed to be sampled on a spherical grid, the 3-D Fourier transform of the object is calculated by performing fast Fourier transforms (FFTs) along radial lines in the radon space. Then, an interpolation is performed from the spherical to a Cartesian grid using a 3-D gridding step in the frequency domain. Finally, this 3-D Fourier transform is transformed back to the spatial domain via 3-D inverse FFT. The algorithm is computationally efficient with complexity in the order of N ³ log N. The authors have done reconstructions of simulated 3-D radon derivative data assuming sampling conditions and image quality requirements similar to those in medical computed tomography (CT)     

机载大斜视雷达成像算法研究

基于大斜视合成孔径雷达成像的空间几何模型和回波数据的特点,通过距离向和方位向解线性调频处理和在时域进行距离走动和距离弯曲校正,分别对方位向和距离向进行傅里叶变换将信号压至正确的位置上。该算法从成像的实时性出发,立足于在分辨率达到要求的条件下尽可能降低运算量和复杂性。由于补偿了高次相位项,此算法受斜视角的影响很小,能精确地处理大距离走动和弯曲。点目标仿真和实测数据成像结果证明了算法的可行性。     

Fast splitting alpha-rooting method of image enhancement: tensor representation.

In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the alpha-rooting method of image enhancement can be fulfilled through processing separately a maximum of 3N/2 splitting-signals of an image (N x N), where N is a power of two. In this paper, we propose a fast implementation of the a-rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The implementation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D N-point DFTs instead of two 2-D N x N-point DFTs in the traditional method of alpha-rooting.     

Automatic detection of brain contours in MRI data sets

A software procedure is presented for fully automated detection of brain contours from single-echo 3-D MRI data, developed initially for scans with coronal orientation. The procedure detects structures in a head data volume in a hierarchical fashion. Automatic detection starts with a histogram-based thresholding step, whenever necessary preceded by an image intensity correction procedure. This step is followed by a morphological procedure which refines the binary threshold mask images. Anatomical knowledge, essential for the discrimination between desired and undesired structures, is implemented in this step through a sequence of conventional and novel morphological operations, using 2-D and 3-D operations. A final step of the procedure performs overlap tests on candidate brain regions of interest in neighboring slice images to propagate coherent 2-D brain masks through the third dimension. Results are presented for test runs of the procedure on 23 coronal whole-brain data sets, and one sagittal whole-brain data set. Finally, the potential of the technique for generalization to other problems is discussed, as well as limitations of the technique     

激光焊接裂纹磁光成像频域特征分析

为了研究旋转磁场激励下激光焊接裂纹磁光成像在频域内的特征, 采用对激光焊接裂纹磁光图像进行2维离散傅里叶变换的方法, 进行了理论分析和实验验证, 取得了旋转磁场不同励磁强度下激光焊接裂纹的频谱图数据。结合裂纹磁光图的空域特征, 对所获裂纹频谱图灰度值为255的点进行统计分析。结果表明, 激光焊接裂纹磁光图像的频域特征和空域特征有一定的对应关系; 在一个变化周期内(885帧磁光图), 对应频谱图上会出现先变小再变大、再变小再变大或相反的变化过程, 最终回到初始状态。这一结果验证了旋转磁场下裂纹磁光成像规律的正确性, 对激光焊接缺陷的无损检测是有帮助的。     

A new texture generation method based on pseudo-DCT coefficients.

In this paper, a new method for generating different texture images is presented. This method involves a simple transform from a certain one-dimensional (1-D) signal to an expected two-dimensional (2-D) image. Unlike traditional methods, the input signal is generated by a simple 1-D function in our work instead of a sample texture. We first transform the 1-D input signal into frequency domain using fast Fourier transform. Based on the sufficient analysis in 2-D discrete cosine transform (DCT) domain, where each of the coefficients expresses a texture feature in a certain direction, the 2-D pseudo-DCT coefficients are then constructed by appropriately rearranging the Fourier coefficients in terms of their frequency components. Finally, the corresponding texture image can be produced by 2-D inverse DCT algorithm. We applied the proposed method to generate several stochastic textures (i.e., cloud, illumination, and sand), and several structural texture images. Experimental results indicate the good performance of the proposed method.     

MULTI-SPECTRAL AND HYPERSPECTRAL IMAGE FUSION USING 3-D WAVELET TRANSFORM

Image fusion is performed between one band of multi-spectral image and two bands of hyperspectral image to produce fused image with the same spatial resolution as source multi-spectral image and the same spectral resolution as source hyperspeetral image. According to the characteristics and 3-Dimensional （3-D） feature analysis of multi-spectral and hyperspectral image data volume, the new fusion approach using 3-D wavelet based method is proposed. This approach is composed of four major procedures： Spatial and spectral resampling, 3-D wavelet transform, wavelet coefficient integration and 3-D inverse wavelet transform. Especially, a novel method, Ratio Image Based Spectral Resampling （RIBSR）method, is proposed to accomplish data resampling in spectral domain by utilizing the property of ratio image. And a new fusion rule, Average and Substitution （A＆S） rule, is employed as the fusion rule to accomplish wavelet coefficient integration. Experimental results illustrate that the fusion approach using 3-D wavelet transform can utilize both spatial and spectral characteristics of source images more adequately and produce fused image with higher quality and fewer artifacts than fusion approach using 2-D wavelet transform. It is also revealed that RIBSR method is capable of interpolating the missing data more effectively and correctly, and A＆S rule can integrate coefficients of source images in 3-D wavelet domain to preserve both spatial and spectral features of source images more properly.     

Frequency domain volume rendering by the wavelet X-ray transform

We describe a wavelet based X-ray rendering method in the frequency domain with a smaller time complexity than wavelet splatting. Standard Fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed. We review the implementation of the fast wavelet transform in the frequency domain. The wavelet X-ray transform is derived, and the corresponding Fourier-wavelet volume rendering algorithm (FWVR) is introduced, FWVR uses Haar or B-spline wavelets and linear or cubic spline interpolation. Various combinations are tested and compared with wavelet splatting (WS). We use medical MR and CT scan data, as well as a 3-D analytical phantom to assess the accuracy, time complexity, and memory cost of both FWVR and WS. The differences between both methods are enumerated.     

Differential morphology and image processing

Image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain. We provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems. This includes ideas on using partial differential or difference equations (PDEs) to model distance propagation or nonlinear multiscale processes in images. We briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics. We also review some nonlinear PDEs that model the evolution of multiscale morphological operators and use morphological derivatives. Among the new ideas presented, we develop some general 2-D max/min-sum difference equations that model the space dynamics of 2-D morphological systems (including the distance computations) and some nonlinear signal transforms, called slope transforms, that can analyze these systems in a transform domain in ways conceptually similar to the application of Fourier transforms to linear systems. Thus, distance transforms are shown to be bandpass slope filters. We view the analysis of the multiscale morphological PDEs and of the eikonal PDE solved via weighted distance transforms as a unified area in nonlinear image processing, which we call differential morphology, and briefly discuss its potential applications to image processing and computer vision.     

Fan filters, the 3-D Radon transform, and image sequence analysis

This paper develops a theory for the application of fan filters to moving objects. In contrast to previous treatments of the subject based on the 3-D Fourier transform, simplicity and insight are achieved by using the 3-D Radon transform. With this point of view, the Radon transform decomposes the image sequence into a set of plane waves that are parameterized by a two-component slowness vector. Fan filtering is equivalent to a multiplication in the Radon transform domain by a slowness response function, followed by an inverse Radon transform. The plane wave representation of a moving object involves only a restricted set of slownesses such that the inner product of the plane wave slowness vector and the moving object velocity vector is equal to one. All of the complexity in the application of fan filters to image sequences results from the velocity-slowness mapping not being one-to-one; therefore, the filter response cannot be independently specified at all velocities. A key contribution of this paper is to elucidate both the power and the limitations of fan filtering in this new application. A potential application of 3-D fan filters is in the detection of moving targets in clutter and noise. For example, an appropriately designed fan filter can reject perfectly all moving objects whose speed, irrespective of heading, is less than a specified cut-off speed, with only minor attenuation of significantly faster objects. A simple geometric construction determines the response of the filter for speeds greater than the cut-off speed.     

基于小波非线性外推的图像增强方法

针对图像增强的问题,提出了一种新的图像增强方法．该方法将小波变换与频域非线性外推算法相结合,首先利用小波变换提升图像的高频成分,再利用频域非线性外推算法增加图像的高频分量,最终实现图像的增强．实验表明,该新方法有效地提高了图像的视觉效果．     

Fast Splitting$alpha$-Rooting Method of Image Enhancement: Tensor Representation

In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the$alpha$-rooting method of image enhancement can be fulfilled through processing separately a maximum of$3N/2$splitting-signals of an image$(Ntimes N)$, where$N$is a power of two. In this paper, we propose a fast implementation of the$alpha$-rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The implementation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D$N$-point DFTs instead of two 2-D$Ntimes N$-point DFTs in the traditional method of$alpha$-rooting.     

On the Computation of Optical Flow using the 3-D Gabor Transform

The motion of brightness patterns in an image sequence (optical flow) is most intuitively considered as a spatiotemporal phenomenon. It has been shown, however, that motion has a characteristic signature in the spatiotemporal-frequency (Fourier) domain. This fact can be exploited for the computation of optical flow. However, for cases which involve a number of regions in a sequence with different motions, as in scenes with one or more objects moving against a stationary or moving background, the global nature of the Fourier transform makes it unsuitable for this task. The signatures of the different motions cannot be resolved in the Fourier domain, nor associated with their respective regions in the image sequence. Local frequency representations provide a means to address this problem. In this paper, we consider the application of a 3-D version of the widely used Gabor transform to the computation of optical flow.     

基于小波变换与稀疏傅里叶变换相结合的光场重构方法

随着计算机图形学和计算机视觉技术的发展,光场开始进入人们的视线并被迅速应用于各个领域.然而光场的获取需要大量的图像,具有数据量大,获取成本高等特点,因此学者们越来越关注如何利用少量的光场数据获取整个光场这一问题,并且做出了大量的工作.针对上述问题,本文将小波变换与稀疏傅里叶变换相结合,利用光场在角度域的稀疏性提出一种新的光场重构方法.首先,利用小波变换多分辨率分析的特点,通过小波变换将原始图像分解为多个不同频率的子图像;然后分别对每个子图像通过傅里叶切片定理恢复其频率位置,从而可以分别得到它们的二维角度谱;最后将每个子图像的二维角度谱合并,进行小波逆变换获得整个光场.本文方法利用小波变换将原图像分解为多个不同频率的子图像分别同时处理,不仅降低了算法的复杂度,大大减少了算法的运行时间,为光场的广泛应用提供了条件,而且相比于单独运用稀疏傅里叶算法重构,本方法有效地抑制了窗口效应,使重构结果更加准确.此外,本文方法将高频信息和低频信息分开重构,可以有效地改善并网恢复中小频率丢失的问题,进一步改进重构结果.最后通过仿真验证了算法的有效性.     

Two-dimensional matched filtering for motion estimation

In this work, we describe a frequency domain technique for the estimation of multiple superimposed motions in an image sequence. The least-squares optimum approach involves the computation of the three-dimensional (3-D) Fourier transform of the sequence, followed by the detection of one or more planes in this domain with high energy concentration. We present a more efficient algorithm, based on the properties of the Radon transform and the two-dimensional (2-D) fast Fourier transform, which can sacrifice little performance for significant computational savings. We accomplish the motion detection and estimation by designing appropriate matched filters. The performance is demonstrated on two image sequences.     

3-D discrete analytical ridgelet transform.

In this paper, we propose an implementation of the 3-D Ridgelet transform: the 3-D discrete analytical Ridgelet transform (3-D DART). This transform uses the Fourier strategy for the computation of the associated 3-D discrete Radon transform. The innovative step is the definition of a discrete 3-D transform with the discrete analytical geometry theory by the construction of 3-D discrete analytical lines in the Fourier domain. We propose two types of 3-D discrete lines: 3-D discrete radial lines going through the origin defined from their orthogonal projections and 3-D planes covered with 2-D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3-D DART adapted to a specific application. Indeed, the 3-D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3-D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3-D DART and its extension to the Local-DART (with smooth windowing) to the denoising of 3-D image and color video. These experimental results show that the simple thresholding of the 3-D DART coefficients is efficient.     

High-resolution radar imaging using 2-D linear prediction

An algorithm for radar imaging is described. The algorithm is based on two-dimensional (2-D) linear prediction of 2-D Cartesian frequency spectra. It is shown that the algorithm provides much better resolution than the ISAR image obtained using a 2-D inverse Fourier transform. The algorithm is especially useful for imaging targets using small-bandwidth RCS data over limited aspect angle regions     

Fast fully 3-D image reconstruction in PET using planograms

We present a method of performing fast and accurate three-dimensional (3-D) backprojection using only Fourier transform operations for line-integral data acquired by planar detector arrays in positron emission tomography. This approach is a 3-D extension of the two-dimensional (2-D) linogram technique of Edholm. By using a special choice of parameters to index a line of response (LOR) for a pair of planar detectors, rather than the conventional parameters used to index a LOR for a circular tomograph, all the LORs passing through a point in the field of view (FOV) lie on a 2-D plane in the four-dimensional (4-D) data space. Thus, backprojection of all the LORs passing through a point in the FOV corresponds to integration of a 2-D plane through the 4-D "planogram." The key step is that the integration along a set of parallel 2-D planes through the planogram, that is, backprojection of a plane of points, can be replaced by a 2-D section through the origin of the 4-D Fourier transform of the data. Backprojection can be performed as a sequence of Fourier transform operations, for faster implementation. In addition, we derive the central-section theorem for planogram format data, and also derive a reconstruction filter for both backprojection-filtering and filtered-backprojection reconstruction algorithms. With software-based Fourier transform calculations we provide preliminary comparisons of planogram backprojection to standard 3-D backprojection and demonstrate a reduction in computation time by a factor of approximately 15.