We review a class of methods that can be collected under the name nonlinear transform coding (NTC), which over the past few years have become competitive with the best linear transform codecs for images, and have superseded them in terms of rate--distortion performance under established perceptual quality metrics such as MS-SSIM. We assess the empirical rate--distortion performance of NTC with the help of simple example sources, for which the optimal performance of a vector quantizer is easier to estimate than with natural data sources. To this end, we introduce a novel variant of entropy-constrained vector quantization. We provide an analysis of various forms of stochastic optimization techniques for NTC models; review architectures of transforms based on artificial neural networks, as well as learned entropy models; and provide a direct comparison of a number of methods to parameterize the rate--distortion trade-off of nonlinear transforms, introducing a simplified one.
Insufficiency of linear coding for the network coding problem was first proved by providing an instance which is solvable only by nonlinear network coding (Dougherty et al., 2005).Based on the work of Effros, et al., 2015, this specific network coding instance can be modeled as a groupcast index coding (GIC)instance with 74 messages and 80 users (where a message can be requested by multiple users). This proves the insufficiency of linear coding for the GIC problem. Using the systematic approach proposed by Maleki et al., 2014, the aforementioned GIC instance can be cast into a unicast index co…
In this paper, we propose a new graph-based transform and illustrate its potential application to signal compression. Our approach relies on the careful design of a graph that optimizes the overall rate-distortion performance through an effective graph-based transform. We introduce a novel graph estimation algorithm, which uncovers the connectivities between the graph signal values by taking into consideration the coding of both the signal and the graph topology in rate-distortion terms. In particular, we introduce a novel coding solution for the graph by treating the edge weights as another g…
The k-pair problem in network coding theory asks to send k messages simultaneously between k source-target pairs over a directed acyclic graph. In a previous paper [ICALP 2009, Part I, pages 622--633] the present authors showed that if a classical k-pair problem is solvable by means of a linear coding scheme, then the quantum k-pair problem over the same graph is also solvable, provided that classical communication can be sent for free between any pair of nodes of the graph. Here we address the main case that remained open in our previous work, namely whether nonlinear classical network coding…
We propose deterministic sampling strategies for compressive imaging based on Delsarte-Goethals frames. We show that these sampling strategies result in multi-scale measurements which can be related to the 2D Haar wavelet transform. We demonstrate the effectiveness of our proposed strategies through numerical experiments.
Beijing University of Posts and Telecommunications（北京邮电大学）；Peng Cheng Laboratory（鹏城实验室）
In this paper, we propose a class of high-efficiency deep joint source-channel coding methods that can closely adapt to the source distribution under the nonlinear transform, it can be collected under the name nonlinear transform source-channel coding (NTSCC). In the considered model, the transmitter first learns a nonlinear analysis transform to map the source data into latent space, then transmits the latent representation to the receiver via deep joint source-channel coding. Our model incorporates the nonlinear transform as a strong prior to effectively extract the source semantic features …
I present a method for lossy transform coding of digital audio that uses the Weyl symbol calculus for constructing the encoding and decoding transformation. The method establishes a direct connection between a time-frequency representation of the signal dependent threshold of masked noise and the encode/decode pair. The formalism also offers a time-frequency measure of perceptual entropy.
Revisiting Shannon's Source Coding Theorem with Distributional Uncertainty under the Nonlinear Expectation Theory
National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, CAS（中国科学院数学与系统科学研究院交叉科学中心）；School of Information and Communication Engineering, Beijing University of Posts and Telecommunications（北京邮电大学信息与通信工程学院）；University of Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Science（中国科学院大学数学与系统科学研究院）
Erasure codes are widely used in today's storage systems to cope with failures. Most of them use the finite field arithmetic. In this paper, we propose an implementation and a coding speed evaluation of an original method called PYRIT (PolYnomial RIng Transform) to perform operations between elements of a finite field into a bigger ring by using fast transforms between these two structures. Working in such a ring is much easier than working in a finite field. Firstly, it reduces the coding complexity by design. Secondly, it allows simple but efficient \texttt{xor}-based implementations by unro…
The complexity of software implementations of MDS erasure codes mainly depends on the efficiency of the finite field operations implementation. In this paper, we propose a method to reduce the complexity of the finite field multiplication by using fast transforms between a field and a ring to perform the multiplication in a ring. We show that moving to a ring reduces the complexity of the operations. Then, we show that this construction allows the use of simple scheduling to reduce the number of operations.
It is well-known for transform coding of multivariate Gaussian sources, that the Karhunen-Loève transform (KLT) minimizes the mean square error distortion. However, finding the optimal transform for general non-Gaussian sources has been an open problem for decades, despite several important advances that provide some partial answers regarding KLT optimality. In this paper, we present a necessary and sufficient condition for optimality of a transform when high resolution, variable rate quantizers are employed. We hence present not only a complete characterization of when KLT is optimal, but als…
The problem of characterizing the optimal rate achievable with analog network coding (ANC) for a unicast communication over general wireless relay networks is computationally hard. A relay node performing ANC scales and forwards its input signals. The source-destination channel in such communication scenarios is, in general, an intersymbol interference (ISI) channel which leads to the single-letter characterization of the optimal rate in terms of an optimization problem with nonconvex, non closed-form objective function and non-convex constraints. For a special class of such networks, called l…
Fourier Transform Interferometry (FTI) is an interferometric procedure for acquiring HyperSpectral (HS) data. Recently, it has been observed that the light source highlighting a (biologic) sample can be coded before the FTI acquisition in a procedure called Coded Illumination-FTI (CI-FTI). This turns HS data reconstruction into a Compressive Sensing (CS) problem regularized by the sparsity of the HS data. CI-FTI combines the high spectral resolution of FTI with the advantages of reduced-light-exposure imaging in biology. In this paper, we leverage multilevel sampling scheme recently developed …
A new coding scheme for image transmission over noisy channel is proposed. Similar to standard image compression, the scheme includes a linear transform followed by successive refinement scalar quantization. Unlike conventional schemes, in the proposed system the quantized transform coefficients are linearly mapped into channel symbols using systematic linear encoders. This fixed-to-fixed length "linear index coding" approach avoids the use of an explicit entropy coding stage (e.g., arithmetic or Huffman coding), which is typically fragile to channel post-decoding residual errors. We use linea…
A model, called the linear transform network (LTN), is proposed to analyze the compression and estimation of correlated signals transmitted over directed acyclic graphs (DAGs). An LTN is a DAG network with multiple source and receiver nodes. Source nodes transmit subspace projections of random correlated signals by applying reduced-dimension linear transforms. The subspace projections are linearly processed by multiple relays and routed to intended receivers. Each receiver applies a linear estimator to approximate a subset of the sources with minimum mean squared error (MSE) distortion. The mo…
Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. There are NO constraints on the distribution of the transformation matrix and the field size. Specifically, the optimality of subspace …
Indian Institute of Technology, Kharagpur（印度理工学院卡拉格普尔分校）；Azure Software Systems Ltd.（Azure软件系统有限公司）
In this paper, we propose a new feedback-channel-free Distributed Video Coding (DVC) algorithm using Local Rank Transform (LRT). The encoder computes LRT by considering selected neighborhood pixels of Wyner-Ziv frame. The ranks from the modified LRT are merged, and their positions are entropy coded and sent to the decoder. In addition, means of each block of Wyner-Ziv frame are also transmitted to assist motion estimation. Using these measurements, the decoder generates side information (SI) by implementing motion estimation and compensation in LRT domain. An iterative algorithm is executed on…
The algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes, at any given time instant, is a \mathbb{F}_{q}$-linear combination of the input symbols across different generations, where $\mathbb{F}_{q}$ denotes the field over which the network operates. We use finite-field discrete fourier transform (DFT) to convert the output symbols at the sink nodes, at any give…
A novel channel coding scheme for progressive transmission of large images is proposed. The transmission time, low distortion reconstructed image and low complexity are most concerned in this paper. In the case of medical data transmission, it is vital to keep the distortion level under control as in most of the cases certain clinically important regions have to be transmitted without any visible error. The proposed system significantly reduces the transmission time and error. The progressive transmission is based on the process that the input image is decomposed into many subblocks each to be…
This paper is concerned with scalar quantization of transform coefficients in an audio codec. The generalized Gaussian distribution (GGD) is used as an approximation of one-dimensional probability density function for transform coefficients obtained by modulated lapped transform (MLT) or modified cosine transform (MDCT) filterbank. The rationale of the model is provided in comparison with theoretically achievable rate-distortion function. The rate-distortion function computed for the random sequence obtained from a real sequence of samples from a large database is compared with that computed f…
Remarkable mathematical properties of the integrable nonlinear Schrödinger equation (NLSE) can offer advanced solutions for the mitigation of nonlinear signal distortions in optical fibre links. Fundamental optical soliton, continuous and discrete eigenvalues of the nonlinear spectrum have already been considered for transmission of information in fibre-optic channels. Here we propose to apply signal modulation to the kernel of the Gelfand-Levitan-Marchenko equations (GLME) that offers the advantage of a relatively simple decoder design. First, we describe an approach based on exploiting the g…
Inverse probability problems whose generative models are given by strictly nonlinear Gaussian random fields show the all-or-nothing behavior: There exists a critical rate at which Bayesian inference exhibits a phase transition. Below this rate, the optimal Bayesian estimator recovers the data perfectly, and above it the recovered data becomes uncorrelated. This study uses the replica method from the theory of spin glasses to show that this critical rate is the channel capacity. This interesting finding has a particular application to the problem of secure transmission: A strictly nonlinear Gau…
A subspace-based approach for rate-adaptive distributed source coding (DSC) based on discrete Fourier transform (DFT) codes is developed. Punctured DFT codes can be used to implement rate-adaptive source coding, however they perform poorly after even moderate puncturing since the performance of the subspace error localization degrades severely. The proposed subspace-based error localization extends and improves the existing one, based on additional syndrome, and is naturally suitable for rate-adaptive distributed source coding architecture.
