The discrete wavelet transform can be found at the heart of many image-processing algorithms. Until now, the transform on general-purpose processors (CPUs) was mostly computed using a separable lifting scheme. As the lifting scheme consists of a small number of operations, it is preferred for processing using single-core CPUs. However, considering a parallel processing using multi-core processors, this scheme is inappropriate due to a large number of steps. On such architectures, the number of steps corresponds to the number of points that represent the exchange of data. Consequently, these points often form a performance bottleneck. Our approach appropriately rearranges calculations inside the transform, and thereby reduces the number of steps. In other words, we propose a new scheme that is friendly to parallel environments. When evaluating on multi-core CPUs, we consistently overcome the original lifting scheme. The evaluation was performed on 61-core Intel Xeon Phi and 8-core Intel Xeon processors.
The DWT splits the input signal into two components, according to a parity of its samples. The components are often referred to as L and H. The transform can be computed by a pair of quadrature mirror filters, referred to as G, followed by subsampling by a factor of 2. Formally, this can be represented by the polyphase matrix , where operators @math and @math denote the even and odd terms of @math . This equation defines one-dimensional convolution scheme. Further, Sweldens showed @cite_3 how the convolution scheme can be decomposed into a sequence of simple steps. These filters are referred to as the lifting steps and the scheme as the lifting scheme. The following paragraph discusses the lifting scheme in detail.
@cite_3
2015370045
This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula SL(n; R(Z, z-I)) = E(n; R(z, z-l))); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. A new transform is proposed that derives the overcomplete discrete wavelet transform (ODWT) subbands from the critically sampled DWT subbands (complete representation). This complete-to-overcomplete DWT (CODWT) has certain advantages in comparison to the conventional approach that performs the inverse DWT to reconstruct the input signal, followed by the a spl grave -trous or the lowband shift algorithm. Specifically, the computation of the input signal is not required. As a result, the minimum number of downsampling operations is performed and the use of upsampling is avoided. The proposed CODWT computes the ODWT subbands by using a set of prediction-filter matrices and filtering-and-downsampling operators applied to the DWT. This formulation demonstrates a clear separation between the single-rate and multirate components of the transform. This can be especially significant when the CODWT is used in resource-constrained environments, such as resolution-scalable image and video codecs. To illustrate the applicability of the proposed transform in these emerging applications, a new scheme for the transform-calculation is proposed, and existing coding techniques that benefit from its usage are surveyed. The analysis of the proposed CODWT in terms of arithmetic complexity and delay reveals significant gains as compared with the conventional approach. In this paper, a new algorithm to efficiently implement the two-dimensional lifting scheme is presented. The 1D lifting-scheme performs in-place processing of the input samples, and hence it provides reduction in memory requirements. However, for image processing (2D), in-place computation is not enough, resulting in a memory-intensive algorithm, since it has to keep the whole image in memory. We propose the use of line-by-line processing algorithm for the lifting scheme, and we address some issues on how to perform synchronization among different buffer levels, so that an implementation can be easily written. Experimental results show that, for a 5-megapixel image, our algorithm requires 200 times less memory and it is more than 3 times faster than the usual one. We present a novel method to implement lifting based wavelet transforms on general graphs. The detail and approximation coefficients computed from this graph transform can be interpreted similarly to their counterparts in standard signal processing process. Our approach is based on partitioning all nodes in the graph into two sets, containing "even" and "odd" nodes, respectively. Then, as in standard lifting, nodes of one parity are used to predict update those of the other. We discuss the even-odd assignment problem on the graph and provide a solution that is well suited to construct the transform. As an example we discuss how our transform could be used in a denoising application. I. I NTRODUCTION This paper demonstrates that convolution with random waveform followed by random time-domain subsampling is a universally efficient compressive sensing strategy. We show that an @math -dimensional signal which is @math -sparse in any fixed orthonormal representation can be recovered from @math samples from its convolution with a pulse whose Fourier transform has unit magnitude and random phase at all frequencies. The time-domain subsampling can be done in one of two ways: in the first, we simply observe @math samples of the random convolution; in the second, we break the random convolution into @math blocks and summarize each with a single randomized sum. We also discuss several imaging applications where convolution with a random pulse allows us to superresolve fine-scale features, allowing us to recover high-resolution signals from low-resolution measurements. The widespread usage of the discrete wavelet transform (DWT) has motivated the development of fast DWT algorithms and their tuning on all sorts of computer systems. Several studies have compared the performance of the most popular schemes, known as filter bank scheme (FBS) and lifting scheme (LS), and have always concluded that LS is the most efficient option. However, there is no such study on streaming processors such as modern Graphics Processing Units (GPUs). Current trends have transformed these devices into powerful stream processors with enough flexibility to perform intensive and complex floating-point calculations. The opportunities opened up by these platforms, as well as the growing popularity of the DWT within the computer graphics field, make a new performance comparison of great practical interest. Our study indicates that FBS outperforms LS in current-generation GPUs. In our experiments, the actual FBS gains range between 10 percent and 140 percent, depending on the problem size and the type and length of the wavelet filter. Moreover, design trends suggest higher gains in future-generation GPUs. Abstract We present the lifting scheme, a new idea for constructing compactly supported wavelets with compactly supported duals. The lifting scheme uses a simple relationship between all multiresolution analyses with the same scaling function. It isolates the degrees of freedom remaining after fixing the biorthogonality relations. Then one has full control over these degrees of freedom to custom design the wavelet for a particular application. The lifting scheme can also speed up the fast wavelet transform. We illustrate the use of the lifting scheme in the construction of wavelets with interpolating scaling functions. Whereas formal category theory is classically considered within a @math -category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a @math -category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding @math defines @math as the free small cocompletion of @math , in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad @math on a hypervirtual double category @math several hypervirtual double categories @math of @math -algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors @math . The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in @math lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in @math . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". This article proposes a novel image-based method to transfer illumination from a reference face image to a target face image through edge-preserving filters. According to our method, only a single reference image, without any knowledge of the 3D geometry or material information of the target face, is needed. We first decompose the lightness layers of the reference and the target images into large-scale and detail layers through weighted least square (WLS) filter after face alignment. The large-scale layer of the reference image is filtered with the guidance of the target image. Adaptive parameter selection schemes for the edge-preserving filters is proposed in the above two filtering steps. The final relit result is obtained by replacing the large-scale layer of the target image with that of the reference image. We acquire convincing relit result on numerous target and reference face images with different lighting effects and genders. Comparisons with previous work show that our method is less affected by geometry differences and can preserve better the identification structure and skin color of the target face. We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra @math for a functor @math , we define a framework for deriving pseudometrics on @math which measure the behavioral distance of states. A crucial step is the lifting of the functor @math on @math to a functor @math on the category @math of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. If @math has a final coalgebra, every lifting @math yields in a canonical way a behavioral distance which is usually branching-time, i.e., it generalizes bisimilarity. In order to model linear-time metrics (generalizing trace equivalences), we show sufficient conditions for lifting distributive laws and monads. These results enable us to employ the generalized powerset construction. In this paper, we have a close look at the runtime performance of the intra-component transform employed in the reference implementations of the JPEG2000 image coding standard. Typically, wavelet lifting is used to obtain a wavelet decomposition of the source image in a computationally efficient way. However, so far no attention has been paid to the impact of the CPU's memory cache on the overall performance. We propose two simple techniques that dramatically reduce the number of cache misses and cut column filtering runtime by a factor of 10. Theoretical estimates as well as experimental results on a number of hardware platforms show the effectivity of our approach. Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this problem, known as PhaseMax. Standard convex-relaxation-based methods in phase retrieval resort to the idea of “lifting” which makes them computationally inefficient, since the number of unknowns is effectively squared. In contrast, PhaseMax is a novel convex relaxation that does not increase the number of unknowns. Instead it relies on an initial estimate of the true signal which must be externally provided. In this paper, we investigate the required number of measurements for exact recovery of the signal in the large system limit and when the linear measurement matrix is random with iid standard normal entries. If @math denotes the dimension of the unknown complex signal and @math the number of phaseless measurements, then in the large system limit, @math measurements is necessary and sufficient to recover the signal with high probability, where @math is the angle between the initial estimate and the true signal. Our result indicates a sharp phase transition in the asymptotic regime which matches the empirical result in numerical simulations. For an unweighted graph @math , @math is a subgraph if @math , and @math is a Steiner graph if @math , and for any pair of vertices @math , the distance between them in @math (denoted @math ) is at least the distance between them in @math (denoted @math ). In this paper we introduce the notion of distance preserver. A subgraph (resp., Steiner graph) @math of a graph @math is a subgraph (resp., Steiner) @math -preserver of @math if for every pair of vertices @math with @math , @math . We show that any graph (resp., digraph) has a subgraph @math -preserver with at most @math edges (resp., arcs), and there are graphs and digraphs for which any undirected Steiner @math -preserver contains @math edges. However, we show that if one allows a directed Steiner (diSteiner) @math -preserver, then these bounds can be improved. Specifically, we show that for any graph or digraph there exists a diSteiner @math -preserver with @math arcs, and that this result is tight up to a constant factor. We also study @math -preserving distance labeling schemes, that are labeling schemes that guarantee precise calculation of distances between pairs of vertices that are at a distance of at least @math one from another. We show that there exists a @math -preserving labeling scheme with labels of size @math , and that labels of size @math are required for any @math -preserving labeling scheme. This paper presents a new approach for generating reference currents for an active filter and or a static compensator. It is assumed that the compensator is connected to a load that may either be connected in star or in delta. The load can be unbalanced and may also draw harmonic currents. The purpose of the compensating scheme is to balance the load, as well as make the supply side power factor a desired value. The authors use the theory of instantaneous symmetrical components to obtain an algorithm to compute three phase reference currents which, when injected to the power system, produce desired results. They also propose a suitable compensator structure that will track the reference currents in a hysteresis band control scheme. Finally, the feasibility of such a scheme is demonstrated through simulation studies. We propose mS2GD: a method incorporating a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent (S2GD). We consider the problem of minimizing a strongly convex function represented as the sum of an average of a large number of smooth convex functions, and a simple nonsmooth convex regularizer. Our method first performs a deterministic step (computation of the gradient of the objective function at the starting point), followed by a large number of stochastic steps. The process is repeated a few times with the last iterate becoming the new starting point. The novelty of our method is in introduction of mini-batching into the computation of stochastic steps. In each step, instead of choosing a single function, we sample @math functions, compute their gradients, and compute the direction based on this. We analyze the complexity of the method and show that it benefits from two speedup effects. First, we prove that as long as @math is below a certain threshold, we can reach any predefined accuracy with less overall work than without mini-batching. Second, our mini-batching scheme admits a simple parallel implementation, and hence is suitable for further acceleration by parallelization. The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant @math . However, in many settings the differentiable convex function @math is not uniformly smooth---for example, in @math -optimal design where @math and @math , or even the univariate setting with @math . In this paper we develop a notion of “relative smoothness” and relative strong convexity that is determined relative to a user-specified “reference function” @math (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth with respect to a correspondingly fairly simple reference function @math . We extend two standard algorithms---the primal gradient scheme and the dual averaging scheme---to our new setting, with associated computational guarantees. We apply our new a... An increasing number of applications require processing of signals defined on weighted graphs. While wavelets provide a flexible tool for signal processing in the classical setting of regular domains, the existing graph wavelet constructions are less flexible - they are guided solely by the structure of the underlying graph and do not take directly into consideration the particular class of signals to be processed. This paper introduces a machine learning framework for constructing graph wavelets that can sparsely represent a given class of signals. Our construction uses the lifting scheme, and is based on the observation that the recurrent nature of the lifting scheme gives rise to a structure resembling a deep auto-encoder network. Particular properties that the resulting wavelets must satisfy determine the training objective and the structure of the involved neural networks. The training is unsupervised, and is conducted similarly to the greedy pre-training of a stack of auto-encoders. After training is completed, we obtain a linear wavelet transform that can be applied to any graph signal in time and memory linear in the size of the graph. Improved sparsity of our wavelet transform for the test signals is confirmed via experiments both on synthetic and real data. Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j+1 and 2 sup j (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L sup 2 (R sup n ), the vector space of measurable, square-integrable n-dimensional functions. In L sup 2 (R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. > Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis, non-negative matrix factorization, @math -means clustering, and so on, rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity…), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes. The derived generalization bounds behave proportional to @math with respect to the number of samples @math for the considered matrix factorization techniques. We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the @math sensors in the network is classified into @math classes according to some probability distribution @math . Before deployment, a class- @math sensor is assigned @math cryptographic keys that are selected uniformly at random from a common pool of @math keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters @math and @math so that this graph: 1) has no isolated nodes and 2) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors @math growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by and for the same model, therein referred to as the general random intersection graph.