A Partial Converse to Hadamard’s Theorem on Homeomorphism

A theorem by Hadamard gives a two-part condition under which a map from one Banach space to another is a homeomorphism. The theorem, while often very useful, is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met. Here, under a typically weak additional assumption on the map, we show that Hadamards condition is met if and only if the map is a homeomorphism with a Lipschitz continuous inverse. An application is given concerning the relation between the stability of a nonlinear system and the stability of related linear systems.     

On the construction of invertible filter banks on the 2-sphere

The theories of signal sampling, filter banks, wavelets, and "overcomplete wavelets" are well established for the Euclidean spaces and are widely used in the processing and analysis of images. While recent advances have extended some filtering methods to spherical images, many key challenges remain. In this paper, we develop theoretical conditions for the invertibility of filter banks under continuous spherical convolution. Furthermore, we present an analogue of the Papoulis generalized sampling theorem on the 2-Sphere. We use the theoretical results to establish a general framework for the design of invertible filter banks on the sphere and demonstrate the approach with examples of self-invertible spherical wavelets and steerable pyramids. We conclude by examining the use of a self-invertible spherical steerable pyramid in a denoising experiment and discussing the computational complexity of the filtering framework.     

A proof of the spherical homeomorphism conjecture for surfaces

The human cerebral cortex is topologically equivalent to a sphere when it is viewed as closed at the brain stem. Due to noise and/or resolution issues, magnetic resonance imaging may see "handles" that need to be eliminated to reflect the true spherical topology. Shattuck and Leahy present an algorithm to correct such an image. The basis for their correction strategy is a conjecture, which they call the spherical homeomorphism conjecture, stating that the boundary between the foreground region and the background region is topologically spherical if certain associated foreground and background multigraphs are both graph-theoretic trees. In this paper, we prove the conjecture, and its converse, under the assumption that the foreground/background boundary is a surface.     

Direct method for rotation estimation from spherical images using 3D mesh surfaces with SPHARM representation

Since a large field of view obviously bears important advantages, the use of spherical images is becoming increasingly important in various computer vision and image processing applications. This paper presents a novel rotation estimation approach for spherical images based on 3D mesh representation of gray level intensity. Once the 3D meshes of the underlying spherical images are obtained, the 3D rotation can be estimated directly and efficiently, without feature extraction and matching process. Subsequently, we propose a direct method for 3D object rotation estimation using spherical harmonics representation with SVD decomposition and ICP algorithm for estimation refinement. Experimental results validate our approach and prove its suitability and robustness for rotation estimation. Moreover, it performs well against noisy images, brightness changes, image compression and occlusions. A comparative study of our proposed approach with four similar methods for 3D rotation estimation between spherical images, is realized to prove its effectiveness.     

Simplified approach to probe-corrected spherical near-field scanning

A multipole representation for the response of an arbitrary receiving antenna is derived that allows the formulation of probe-corrected spherical near-field scanning simply in terms of conventional vector spherical harmonics. Both the representation and formulation are free of rotational and translational addition theorems. A sampling theorem derived for Legendre functions is used to evaluate the resulting orthogonality integrals by direct summation in a computer time proportional to (ka)3.     

基于小波域特征提取的360度全景图像质量评价

随着虚拟现实(VR)技术的发展,其中代表性的360度全景图像越来越受到关注.本文针对360度全景图像在传输、压缩、储存过程中容易产生失真的特点,结合研究当下流行的编码技术对其质量影响的客观需求,研究了360 度全景图像的质量评价.设计了使用小波域提取能量波动等特征信息的半参考质量评价方法,用来对经过压缩的360 度全景图像进行质量评价.通过与其他图像质量评价算法结果的比较,验证了此方法具有较高的性能.     

Radiation pattern computation of a spherical lens using Mie series

The radiation pattern of a sphericai layered dielectric lens is computed using eigenfunction expansions. The method is an adaptation of the well-known classical solution for plane wave scattering from a layered sphere but uses the less well-known vector addition theorem for spherical wave functions to represent the off-center sources. The usefulness of this formulation is that the solution is analytical and "exact," and is an efficient tool for investigating spherical lens radiation properties. The method is illustrated for a typical Luneburg lens.     

用矢量球面波理论和互易原理分析伦伯透镜天线

伦伯透镜是一种球形分层介质天线,考虑到数值计算的复杂性,本文提出用矢量球面波理论和互易原理相结合,分析其性能。该方法首先用矢量球面波理论计算平面波入射到多层介质球的散射场,在此基础上,用互易原理将散射场转变成天线的远区辐射场。基于这种方法,本文着重研究了伦伯透镜天线的焦区场和远区场特性,包括透镜分层总数以及层与层之间存在的空气间隙对实际应用可能造成的影响。最后,对卫星通信中伦伯透镜天线多波束应用的可行性给出了分析结果。     

灰度图像逻辑或运算 CNN 模板的顽健性设计

通过制定灰度图像的逻辑或运算法则,提出一类灰度图像逻辑或运算CNN,它可以在两幅灰度图像的对应像素点上执行逻辑或运算。对GLOGOR CNN的模板进行鲁棒性分析,建立了一个定理,并给出严格的数学证明。只要模板参数满足定理中给出的参数不等式,CNN就能执行逻辑或运算的任务。数值模拟验证了GLOGOR CNN在应用中的有效性及鲁棒性设计定理的可行性。     

光学ABCD定理的普适性、物理意义和它的应用

本文推广了几何光学近轴光线传输的ABCD定理，使之适用于远离光轴的空间光线的传输，无论光学系统是旋转对称的还是非旋转对称的。又注意到经典力学中的粒子运动也存在类似的ABCD定理。进一步研究发现在这宏观的定理背后的物理含义，即允许存在另一个更为普遍的规律“测不准关系”的可能性。此外应用推广的ABCD定理，可导出推广的含ABCD的衍射积分定理，用角程函的导数来表示。以此进行含初级(赛德耳)象差的衍射积分计算是方便的。文中给出含球差与含慧差的解析结果，及仅含球差时的数值计算。     

Numerical Determination of Potential in Inhomogeneous Dielectrics by Earnshaw's Theorem

Earnshaw's theorem, a characterization of potential functions equivalent to Poisson's equation, expresses a relation between the value of the potential at a point and an average of the function over a spherical surface centered at the point. The theorem therefore lends itself to use in numerical computation of the potential. A formulation of the theorem is presented with particular reference to determination of the potential in a region which is inhomogeneously occupied by dielectric media. This provides a rigorous basis for the formulas used to determine the potential at points on a dielectric interface, in that it avoids the ambiguity which arises in the evaluation of the finite-difference approximation to the Laplacian at such points. The use of the formulation is illustrated by examples of computer-generated graphs giving the potential in the presence of irregular dielectric objects.     

Visual attention on the sphere

Human visual system makes an extensive use of visual attention in order to select the most relevant information and speed-up the vision process. Inspired by visual attention, several computer models have been developed and many computer vision applications rely today on such models. However, the actual algorithms are not suitable to omnidirectional images, which contain a significant amount of geometrical distortion. In this paper, we present a novel computational approach that performs in spherical geometry and thus is suitable for omnidirectional images. Following one of the actual models of visual attention, the spherical saliency map is obtained by fusing together intensity, chromatic, and orientation spherical cue conspicuity maps that are themselves obtained through multiscale analysis on the sphere. Finally, the consecutive maxima in the spherical saliency map represent the spots of attention on the sphere. In the experimental part, the proposed method is then compared to the standard one using a synthetic image. Also, we provide examples of spots detection in real omnidirectional scenes which show its advantages. Finally, an experiment illustrates the homogeneity of the detected visual attention in omnidirectional images.     

球基面反射型彩虹全息的设计和制作

球基面反射型彩虹全息的记录参数与平基面相比有所区别,通过计算位相函数得出球基面反射衍射像的物像关系,以此为基础给出球基面彩虹全息的光路参数设计方法,最后进行了实验验证。     

Efficient Implementation for Spherical Flux Computation and Its Application to Vascular Segmentation

Spherical flux is the flux inside a spherical region, and it is very useful in the analysis of tubular structures in magnetic resonance angiography and computed tomographic angiography. The conventional approach is to estimate the spherical flux in the spatial domain. Its running time depends on the sphere radius quadratically, which leads to very slow spherical flux computation when the sphere size is large. This paper proposes a more efficient implementation for spherical flux computation in the Fourier domain. Our implementation is based on the reformulation of the spherical flux calculation using the divergence theorem, spherical step function, and the convolution operation. With this reformulation, most of the calculations are performed in the Fourier domain. We show how to select the frequency subband so that the computation accuracy can be maintained. It is experimentally demonstrated that, using the synthetic and clinical phase contrast magnetic resonance angiographic volumes, our implementation is more computationally efficient than the conventional spatial implementation. The accuracies of our implementation and that of the conventional spatial implementation are comparable. Finally, the proposed implementation can definitely benefit the computation of the multiscale spherical flux with a set of radii because, unlike the conventional spatial implementation, the time complexity of the proposed implementation does not depend on the sphere radius.      

Near-field imaging for conducting objects

A spherical back-projection algorithm is proposed to reconstruct near-field microwave images of conducting objects. In this imaging scheme, the object is mounted on a rotating pedestal and its backscattered near field is measured at discrete-frequency steps. Implementation of the method consists of first estimating the spherical projection (or range profile) at each aspect by inverse Fourier transforming the range-corrected scattered near field, and then coherently spherically back-projecting the range profiles into the image plane. The images are interpreted and predicted based on the procedure of the spherical back-projection method, and numerical and experimental results are presented to verify the effectiveness of the proposed algorithm     

GPU-friendly rendering for illumination adjustable images

An illumination adjustable image (IAI) contains a large number of pre-recorded reference images under various lighting directions. It describes the appearance of a scene illuminated under various lighting directions. Synthesized images about the scene illuminated by complicated lighting conditions are generated from those reference images. This paper presents practical and real-time rendering methods for IAIs based on spherical Gaussian kernel functions (SGKFs). The lighting property of an IAI is represented by a few number of lightmaps. With those lightmaps, we can use consumer-level graphics processing units (GPUs) to perform the rendering process. The rendering methods for directional light source, point light source and slide projector are discussed. Compared with the conventional spherical harmonic (SH) approach, the proposed SGKF approach offers similar distortion performance but it consumes less graphics memory and has a faster rendering speed.     

基于共形对应的球面图像的计算全息图

研究了基于共形对应的球面图像的计算全息图(CGH)的生成和重现.通常,当用平面波照射全息图时,所重现的图像一般显示在平面上.共形对应在计算机图形学中被广泛应用,它可以将平面图像和任意曲面对应.相对于简单的坐标变换,共形对应关系具有很多优点,如它可以保持变换前后图像之间的几何形状不变性等.将共形对应引入计算机全息图的生成过程中,利用平面与球面的共形对应关系,生成显示于球面的二维图像的计算全息图,并由所生成全息图得到原始图像,从而实现整个全息记录和重现过程的计算机模拟.二维图像重现于球面实际上了产生三维显示,因此上述方法在从计算全息图重现三维图像方面起到重要的作用.     

球状分层微带天线并矢格林函数的有效处理

采用提取格林函数渐进项的方法,成功克服了分层介质中球并矢格林函数计算无穷级数收敛慢的缺陷,提取出的级数形式的格林函数用球汉克尔函数的加法定理转换成闭合形式的格林函数,通过化简,使之与自由空间的格林函数有着相同的形式。因此,可采取相同的方式处理格林函数的奇异性,然后,采用以RWG为基函数的矩量法计算了圆形贴片与方形贴片微带天线的输入特性,并与文献结果、FEKO仿真结果比较,验证了方法的有效性和正确性。     

高密度盘式全息存储及其热固定的实验研究

利用空间-角度复用方式与球面参考波相结合的全息复用存储技术，在盘状LiNbO3：Fe晶体单一扇形区域内成功存储10000幅高分辨率图像，存储面密度达33．7bits／μm^2。在此基础上，将分批热固定的方法应用于盘式体全息存储，分4批次存储了5000幅高分辨率图像，从而将存储面密度进一步提高到50bits／μm^2。会聚的球面参考波及优化的光路配置减小了参考光的反射光斑尺寸，同时也减少了有害的杂散光从晶体表面反射进入成像光路，保证存储图像能得到良好的再现。     

Progressive wavelet correlation using Fourier methods

This paper derives a multiresolution analysis technique for performing correlations on wavelet representations of images. The technique maps the images into the wavelet-frequency domain to take advantage of high-speed correlation in the frequency domain. It builds on Vaidyanathan's (1993) wavelet correlation theorem, which shows that subsamples of correlations of two signals can be obtained from a sum of correlations of subbands of wavelet representations of those signals. Our algorithm produces the correlations at lowest resolution by applying the convolution theorem to subband correlations. A new multiresolution technique fills in the missing correlation data by incrementally inverting the wavelet transform and refining the Fourier transform. When applied to JPEG representations of data, the lowest resolution correlations can he performed directly on the JPEG images to produce 1/64th of the correlation points. Each of three incremental steps quadruple the number of correlation points, and the process can be halted at any point if the intermediate results indicate that the correlation will not find a match