This paper reports the development of a Python Non-Uniform Fast Fourier Transform (PyNUFFT) package, which accelerates non-Cartesian image reconstruction on heterogeneous platforms. Scientific computing with Python encompasses a mature and integrated environment. The NUFFT algorithm has been extensively used for non-Cartesian image reconstruction but previously there was no native Python NUFFT library. The current PyNUFFT software enables multi-dimensional NUFFT on heterogeneous platforms. The PyNUFFT also provides several solvers, including the conjugate gradient method, @math 1 total-variation regularized ordinary least square (L1TV-OLS) and @math 1 total-variation regularized least absolute deviation (L1TV-LAD). Metaprogramming libraries were employed to accelerate PyNUFFT. The PyNUFFT package has been tested on multi-core CPU and GPU, with acceleration factors of 6.3 - 9.5 @math on a 32 thread CPU platform and 5.4 - 13 @math on the GPU.
Different kernel functions (interpolators) are available in previous NUFFT implementations, including: (1)the min-max interpoaltor @cite_24 , (2) the fast radial basis functions @cite_23 @cite_11 , (3)least square interpolator @cite_30 , (4)least mean-square error interpolato @cite_8 , (5) fast Gaussian summation @cite_3 , (6) Kaiser-Bessel function @cite_19 , (7) linear system transfer function or inverse reconstruction @cite_31 @cite_10 .
@cite_30 @cite_8 @cite_3 @cite_24 @cite_19 @cite_23 @cite_31 @cite_10 @cite_11
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Radially encoded MRI has gained increasing attention due to its motion insensitivity and reduced artifacts. However, because its samples are collected nonuniformly in the k-space, multidimensional (especially 3-D) radially sampled MRI image reconstruction is challenging. The objective of this paper is to develop a reconstruction technique in high dimensions with on-the-fly kernel calculation. It implements general multidimensional nonuniform fast Fourier transform (NUFFT) algorithms and incorporates them into a k-space image reconstruction framework. The method is then applied to reconstruct from the radially encoded k-space data, although the method is applicable to any non-Cartesian patterns. Performance comparisons are made against the conventional Kaiser-Bessel (KB) gridding method for 2-D and 3-D radially encoded computer-simulated phantoms and physically scanned phantoms. The results show that the NUFFT reconstruction method has better accuracy-efficiency tradeoff than the KB gridding method when the kernel weights are calculated on the fly. It is found that for a particular conventional kernel function, using its corresponding deapodization function as a scaling factor in the NUFFT framework has the potential to improve accuracy. In particular, when a cosine scaling factor is used, the NUFFT method is faster than KB gridding method since a closed-form solution is available and is less computationally expensive than the KB kernel (KB griding requires computation of Bessel functions). The NUFFT method has been successfully applied to 2-D and 3-D in vivo studies on small animals. The fast Fourier transform (FFT) is used widely in signal processing for efficient computation of the FT of finite-length signals over a set of uniformly spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e., a nonuniform FT. Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function. Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type error estimates on compact regions of Rn when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter. We show that the output of a (residual) convolutional neural network (CNN) with an appropriate prior over the weights and biases is a Gaussian process (GP) in the limit of infinitely many convolutional filters, extending similar results for dense networks. For a CNN, the equivalent kernel can be computed exactly and, unlike "deep kernels", has very few parameters: only the hyperparameters of the original CNN. Further, we show that this kernel has two properties that allow it to be computed efficiently; the cost of evaluating the kernel for a pair of images is similar to a single forward pass through the original CNN with only one filter per layer. The kernel equivalent to a 32-layer ResNet obtains 0.84 classification error on MNIST, a new record for GPs with a comparable number of parameters. We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data. The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N ) operations rather than O(N 2 ) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid (A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368-1383). In this paper, we observe that one of the standard interpolation or "gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and three- dimensional settings, saving either 10 d N in storage in d dimensions or a factor of about 5-10 in CPUtime (independent of dimension). Abstract Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale. The proposed construction is shown to be similar to the Nystrom extension of the eigenvectors of the symmetric normalized graph Laplacian matrix. Based on this observation, we provide a new interpretation of the Nystrom extension with suggestions for improvement. Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that storing and computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such computation significantly. Key to Fastfood is the observation that Hadamard matrices, when combined with diagonal Gaussian matrices, exhibit properties similar to dense Gaussian random matrices. Yet unlike the latter, Hadamard and diagonal matrices are inexpensive to multiply and store. These two matrices can be used in lieu of Gaussian matrices in Random Kitchen Sinks proposed by Rahimi and Recht (2009) and thereby speeding up the computation for a large range of kernel functions. Specifically, Fastfood requires O(n log d) time and O(n) storage to compute n non-linear basis functions in d dimensions, a significant improvement from O(nd) computation and storage, without sacrificing accuracy. Our method applies to any translation invariant and any dot-product kernel, such as the popular RBF kernels and polynomial kernels. We prove that the approximation is unbiased and has low variance. Experiments show that we achieve similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory. These improvements, especially in terms of memory usage, make kernel methods more practical for applications that have large training sets and or require real-time prediction. Nonlinear intersymbol interference (ISI) leads to significant error rate in nonlinear communication and digital storage channel. In this paper, therefore, a novel computationally efficient functional link neural network cascaded with Chebyshev orthogonal polynomial is proposed to combat nonlinear ISI. The equalizer has a simple structure in which the nonlinearity is introduced by functional expansion of the input pattern by trigonometric polynomial and Chebyshev orthogonal polynomial. Due to the input pattern and nonlinear approximation enhancement, the proposed structure can approximate arbitrarily nonlinear decision boundaries. It has been utilized for nonlinear channel equalization. The performance of the proposed adaptive nonlinear equalizer is compared with functional link neural network (FLNN) equalizer, multilayer perceptron (MLP) network and radial basis function (RBF) along with conventional normalized least-mean-square algorithms (NLMS) for different linear and nonlinear channel models. The comparison of convergence rate, bit error rate (BER) and steady state error performance, and computational complexity involved for neural network equalizers is provided. Kernel methods yield state-of-the-art performance in certain applications such as image classification and object detection. However, large scale problems require machine learning techniques of at most linear complexity and these are usually limited to linear kernels. This unfortunately rules out gold-standard kernels such as the generalized RBF kernels (e.g. exponential-c 2 ). Recently, Maji and Berg [13] and Vedaldi and Zisserman [20] proposed explicit feature maps to approximate the additive kernels (intersection, c 2 , etc.) by linear ones, thus enabling the use of fast machine learning technique in a non-linear context. An analogous technique was proposed by Rahimi and Recht [14] for the translation invariant RBF kernels. In this paper, we complete the construction and combine the two techniques to obtain explicit feature maps for the generalized RBF kernels. Furthermore, we investigate a learning method using l 1 regularization to encourage sparsity in the final vector representation, and thus reduce its dimension. We evaluate this technique on the VOC 2007 detection challenge, showing when it can improve on fast additive kernels, and the trade-offs in complexity and accuracy. We present a new trigonometric basis function that is capable of perfectly reproducing circles, spheres and ellipsoids while at the same time being interpolatory. Such basis functions have the advantage that they allow to construct shapes through a sequence of control points that lie on their contour (2-D) or surface (3-D) which facilitates user-interaction, especially in 3-D. Our piecewise exponential basis function has finite support, which enables local control for shape modification. We derive and prove all the necessary properties of the kernel to represent shapes that can be smoothly deformed and show how idealized shapes such as ellipses and spheres can be constructed. This paper demonstrates that the computational effort required to develop numerical solutions to continuous-state dynamic programs can be reduced significantly when cubic piecewise polynomial functions, rather than tensor product linear interpolants, are used to approximate the value function. Tensor product cubic splines, represented in either piecewise polynomial or B-spline form, and multivariate Hermite polynomials are considered. Computational savings are possible because of the improved accuracy of higher-order functions and because the smoothness of higher-order functions allows efficient quasi-Newton methods to be used to compute optimal decisions. The use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of Hermite polynomials for the test problem and easier to program. In comparison to linear interpolation, use of splines in piecewise polynomial form reduced the CPU time to obtain results of equivalent accuracy by a factor of 250–330 for a stochastic 4-dimensional water supply reservoir problem with a smooth objective function, and factors ranging from 25–400 for a sequence of 2-, 3-, 4-, and 5-dimensional problems. As a result, a problem that required two hours to solve with linear interpolation was solved in a less than a minute with spline interpolation with no loss of accuracy. The Hermite radial basis functions (HRBF) implicits reconstruct an implicit function which interpolates or approximates scattered multivariate Hermite data (i.e. unstructured points and their corresponding normals). Experiments suggest that HRBF implicits allow the reconstruction of surfaces rich in details and behave better than previous related methods under coarse and or non-uniform samplings, even in the presence of close sheets. HRBF implicits theory unifies a recently introduced class of surface reconstruction methods based on radial basis functions (RBF), which incorporate normals directly in their problem formulation. Such class has the advantage of not depending on manufactured offset-points to ensure existence of a non-trivial implicit surface RBF interpolant. In fact, we show that HRBF implicits constitute a particular case of Hermite–Birkhoff interpolation with radial basis functions, whose main results we present here. This framework not only allows us to show connections between the present method and others but also enable us to enhance the flexibility of our method by ensuring well-posedness of an interesting combined interpolation regularization approach. Standard video frame interpolation methods first estimate optical flow between input frames and then synthesize an intermediate frame guided by motion. Recent approaches merge these two steps into a single convolution process by convolving input frames with spatially adaptive kernels that account for motion and re-sampling simultaneously. These methods require large kernels to handle large motion, which limits the number of pixels whose kernels can be estimated at once due to the large memory demand. To address this problem, this paper formulates frame interpolation as local separable convolution over input frames using pairs of 1D kernels. Compared to regular 2D kernels, the 1D kernels require significantly fewer parameters to be estimated. Our method develops a deep fully convolutional neural network that takes two input frames and estimates pairs of 1D kernels for all pixels simultaneously. Since our method is able to estimate kernels and synthesizes the whole video frame at once, it allows for the incorporation of perceptual loss to train the neural network to produce visually pleasing frames. This deep neural network is trained end-to-end using widely available video data without any human annotation. Both qualitative and quantitative experiments show that our method provides a practical solution to high-quality video frame interpolation. Standard video frame interpolation methods first estimate optical flow between input frames and then synthesize an intermediate frame guided by motion. Recent approaches merge these two steps into a single convolution process by convolving input frames with spatially adaptive kernels that account for motion and re-sampling simultaneously. These methods require large kernels to handle large motion, which limits the number of pixels whose kernels can be estimated at once due to the large memory demand. To address this problem, this paper formulates frame interpolation as local separable convolution over input frames using pairs of 1D kernels. Compared to regular 2D kernels, the 1D kernels require significantly fewer parameters to be estimated. Our method develops a deep fully convolutional neural network that takes two input frames and estimates pairs of 1D kernels for all pixels simultaneously. Since our method is able to estimate kernels and synthesizes the whole video frame at once, it allows for the incorporation of perceptual loss to train the neural network to produce visually pleasing frames. This deep neural network is trained end-to-end using widely available video data without any human annotation. Both qualitative and quantitative experiments show that our method provides a practical solution to high-quality video frame interpolation. Interpolation between pairs of values, typically vectors, is a fundamental operation in many computer graphics applications. In some cases simple linear interpolation yields meaningful results without requiring domain knowledge. However, interpolation between pairs of distributions or pairs of functions often demands more care because features may exhibit translational motion between exemplars. This property is not captured by linear interpolation. This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending. The functions can be non-uniformly sampled, high-dimensional, and defined on non-Euclidean manifolds, e.g., spheres and tori. Our method decomposes distributions or functions into sums of radial basis functions (RBFs). We solve a mass transport problem to pair the RBFs and apply partial transport to obtain the interpolated function. We describe practical methods for computing the RBF decomposition and solving the transport problem. We demonstrate the interpolation approach on synthetic examples, BRDFs, color distributions, environment maps, stipple patterns, and value functions. Abstract: We examine the performance profile of Convolutional Neural Network training on the current generation of NVIDIA Graphics Processing Units. We introduce two new Fast Fourier Transform convolution implementations: one based on NVIDIA's cuFFT library, and another based on a Facebook authored FFT implementation, fbfft, that provides significant speedups over cuFFT (over 1.5x) for whole CNNs. Both of these convolution implementations are available in open source, and are faster than NVIDIA's cuDNN implementation for many common convolutional layers (up to 23.5x for some synthetic kernel configurations). We discuss different performance regimes of convolutions, comparing areas where straightforward time domain convolutions outperform Fourier frequency domain convolutions. Details on algorithmic applications of NVIDIA GPU hardware specifics in the implementation of fbfft are also provided. We examine the performance profile of Convolutional Neural Network training on the current generation of NVIDIA Graphics Processing Units. We introduce two new Fast Fourier Transform convolution implementations: one based on NVIDIA's cuFFT library, and another based on a Facebook authored FFT implementation, fbfft, that provides significant speedups over cuFFT (over 1.5x) for whole CNNs. Both of these convolution implementations are available in open source, and are faster than NVIDIA's cuDNN implementation for many common convolutional layers (up to 23.5x for some synthetic kernel configurations). We discuss different performance regimes of convolutions, comparing areas where straightforward time domain convolutions outperform Fourier frequency domain convolutions. Details on algorithmic applications of NVIDIA GPU hardware specifics in the implementation of fbfft are also provided. Abstract The integral ∫0Leiνφ(s,t)f(s)dswith a highly oscillatory kernel (large ν, ν is up to 2000) is considered. This integral is accurately evaluated with an improved trapezoidal rule and effectively transcribed using local Fourier basis and adaptive multiscale local Fourier basis. The representation of the oscillatory kernel in these bases is sparse. The coefficients after the application of local Fourier transform are smoothed. Sometimes this enables us to obtain further compression with wavelets. We consider the problem of constructing a globally smooth analytic function that represents a surface implicitly by way of its zero set, given sample points with surface normal vectors. The contributions of the paper include a novel means of regularising multi-scale compactly supported basis functions that leads to the desirable interpolation properties previously only associated with fully supported bases. We also provide a regularisation framework for simpler and more direct treatment of surface normals, along with a corresponding generalisation of the representer theorem lying at the core of kernel-based machine learning methods. We demonstrate the techniques on 3D problems of up to 14 million data points, as well as 4D time series data and four-dimensional interpolation between three-dimensional shapes.