Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction

A polynomial transform is the multiplication of an input vector x\in\C^n by a matrix \PT_{b,\alpha}\in\C^{n\times n}, whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials p_\ell(x)\in\C[x] from a list $b=\{p_0(x),\dots,p_{n-1}(x)\}$ and sample points \alpha_k\in\C from a list $\alpha=\{\alpha_0,\dots,\alpha_{n-1}\}$. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. Important examples include the discrete Fourier and cosine transforms. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel $O(n\log{n})$ general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.Comment: 19 pages. Submitted to SIAM Journal on Matrix Analysis and Application

Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs

This paper presents a systematic methodology based on the algebraic theory of signal processing to classify and derive fast algorithms for linear transforms. Instead of manipulating the entries of transform matrices, our approach derives the algorithms by stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebraic principles that generalize the well-known Cooley-Tukey FFT and make the algorithms' derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast algorithms, many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2^n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of non-abelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2^n is O(n^2) in all cases.Comment: 16 pages, LaTeX2

Feasibility of Terrestrial laser scanning for collecting stem volume information from single trees

Interest in measuring forest biomass and carbon stock has increased as a result of the United Nations Framework Convention on Climate Change, and sustainable planning of forest resources is therefore essential. Biomass and carbon stock estimates are based on the large area estimates of growing stock volume provided by national forest inventories (NFIs). The estimates for growing stock volume based on the NFIs depend on stem volume estimates of individual trees. Data collection for formulating stem volume and biomass models is challenging, because the amount of data required is considerable, and the fact that the detailed destructive measurements required to provide these data are laborious. Due to natural diversity, sample size for developing allometric models should be rather large. Terrestrial laser scanning (TLS) has proved to be an efficient tool for collecting information on tree stems. Therefore, we investigated how TLS data for deriving stem volume information from single trees should be collected. The broader context of the study was to determine the feasibility of replacing destructive and laborious field measurements, which have been needed for development of empirical stem volume models, with TLS. The aim of the study was to investigate the effect of the TLS data captured at various distance (i.e. corresponding 25%, 50%, 75% and 100% of tree height) on the accuracy of the stem volume derived. In addition, we examined how multiple TLS point cloud data acquired at various distances improved the results. Analysis was carried out with two ways when multiple point clouds were used: individual tree attributes were derived from separate point clouds and the volume was estimated based on these separate values (multiple scan A), and point clouds were georeferenced as a combined point cloud from which the stem volume was estimated (multiple-scan B). This permitted us to deal with the practical aspects of TLS data collection and data processing for development of stem volume equations in boreal forests. The results indicated that a scanning distance of approximately 25% of tree height would be optimal for stem volume estimation with TLS if a single scan was utilized in boreal forest conditions studied here and scanning resolution employed. Larger distances increased the uncertainty, especially when the scanning distance was greater than approximately 50% of tree height, because the number of successfully measured diameters from the TLS point cloud was not sufficient for estimating the stem volume. When two TLS point clouds were utilized, the accuracy of stem volume estimates was improved: RMSE decreased from 12.4% to 6.8%. When two point clouds were processed separately (i.e. tree attributes were derived from separate point clouds and then combined) more accurate results were obtained; smaller RMSE and relative error were achieved compared to processing point clouds together (i.e. tree attributes were derived from a combined point cloud). TLS data collection and processing for the optimal setup in this study required only one sixth of time that was necessary to obtain the field reference. These results helped to further our knowledge on TLS in estimating stem volume in boreal forests studied here and brought us one step closer in providing best practices how a phase-shift TLS can be utilized in collecting data when developing stem volume models. (C) 2016 The Authors. Published by Elsevier B.V. on behalf of International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS).Peer reviewe

Terrestrial laser scanning in forest inventories

AbstractDecision making on forest resources relies on the precise information that is collected using inventory. There are many different kinds of forest inventory techniques that can be applied depending on the goal, scale, resources and the required accuracy. Most of the forest inventories are based on field sample. Therefore, the accuracy of the forest inventories depends on the quality and quantity of the field sample. Conventionally, field sample has been measured using simple tools. When map is required, remote sensing materials are needed. Terrestrial laser scanning (TLS) provides a measurement technique that can acquire millimeter-level of detail from the surrounding area, which allows rapid, automatic and periodical estimates of many important forest inventory attributes. It is expected that TLS will be operationally used in forest inventories as soon as the appropriate software becomes available, best practices become known and general knowledge of these findings becomes more wide spread. Meanwhile, mobile laser scanning, personal laser scanning, and image-based point clouds became capable of capturing similar terrestrial point cloud data as TLS. This paper reviews the advances of applying TLS in forest inventories, discusses its properties with reference to other related techniques and discusses the future prospects of this technique