Showing papers on "Hadamard transform published in 1971"
TL;DR: The feasibility of coding two-dimensional data arrays by first performing a two- dimensional linear transformation on the data and then block quantizing the transformed data is investigated.
Abstract: The feasibility of coding two-dimensional data arrays by first performing a two-dimensional linear transformation on the data and then block quantizing the transformed data is investigated. The Fourier, Hadamard, and Karhunen-Loeve transformations are considered. Theoretical results for Markov data and experimental results for four pictures comparing these transform methods to the standard method of raster scanning, sampling, and pulse-count modulation code are presented.
184 citations
TL;DR: The performance of an n th order DPCM system is studied and it is compared to the performance of the unitary-transform techniques (Hadamard, Fourier, and KarhunenLoeve) in coding two monochrome still pictures.
Abstract: Two important classes of coding schemes that use the spatial correlation of picture elements in reducing data redundancy are the differential pulse-code modulation (DPCM) and the unitary transform coding techniques. We will study the performance of an n th order DPCM system for n ranging from 1 to 22 and compare it to the performance of the unitary-transform techniques (Hadamard, Fourier, and KarhunenLoeve) in coding two monochrome still pictures. We will also consider the sensitivities of the coding systems to picture-to-picture variations.
116 citations
TL;DR: A 255-slot Hadamard-transform spectrometer is described which has experimentally verified the multiplex advantage in signal-to-noise ratio previously predicted for this class of spectrometers.
Abstract: A 255-slot Hadamard-transform spectrometer is described which has experimentally verified the multiplex advantage in signal-to-noise ratio previously predicted for this class of spectrometer.
109 citations
TL;DR: A character recognition experiment is selected for exemplary purposes and the use of features in the rotated spaces results in effective minimum distance classification.
Abstract: An important aspect in mathematical pattern recognition is the usually noninvertible transformation from the pattern space to a reduced dimensionality feature space that allows a classification process to be implemented on a reasonable number of features. Such feature-selecting transformations range from simple coordinate stretching and shrinking to highly complex nonlinear extraction algorithms. A class of feature-selection transformations to which this note addresses itself is that given by multidimensional rotations. Unitary transformations of particular interest are the Karhunen-Loeve, Fourier, Hadamard or Walsh, and the Haar transforms. A character recognition experiment is selected for exemplary purposes and the use of features in the rotated spaces results in effective minimum distance classification.
104 citations
TL;DR: Multidimensional BIFore transform is defined and a physical interpretation of its power spectrum is presented andvantages and as well the limitations of the BIFORE transform in its application in information processing are cited.
Abstract: BIFORE or Hadamard transform is defined and several of its properties are developed. BIFORE power and phase spectra are developed and their frequency-sequency composition is explored. Using matrix partitioning, fast algorithms for efficient computation of BIFORE coefficients and power and phase spectra are developed. Multidimensional BIFORE transform is defined and a physical interpretation of its power spectrum is presented. Advantages and as well the limitations of the BIFORE transform in its application in information processing are cited.
89 citations
TL;DR: It is indicated that the slant orthogonal transformation derived from Hadamard matrix is most advantageous for compression of TV signal bandwidth.
Abstract: We discuss orthogonal transformation as a method to compress frequency bandwidth for TV signals. We indicate that the slant orthogonal transformation derived from Hadamard matrix is most advantageous for compression of TV signal bandwidth. Results of simulation show that unweighted Sp-p/Nq rms of approximately 32 dB was obtained with 2.5 bits per sample on picture showing human face.
75 citations
26 Apr 1971
69 citations
TL;DR: Discrete forms of the Fourier, Hadamard, and Karhunen-Loeve transforms are examined for their capacity to reduce the bit rate necessary to transmit speech signals and these bit-rate reductions are shown to be somewhat independent of the transmission bit rate.
Abstract: Discrete forms of the Fourier, Hadamard, and Karhunen-Loeve transforms are examined for their capacity to reduce the bit rate necessary to transmit speech signals. To rate their effectiveness in accomplishing this goal the quantizing error (or noise) resulting for each transformation method at various bit rates is computed and compared with that for conventional companded PCM processing. Based on this comparison, it is found that Karhunen-Loeve provides a reduction in bit rate of 13.5 kbits/s, Fourier 10 kbits/s, and Hadamard 7.5 kbits/s as compared with the bit rate required for companded PCM. These bit-rate reductions are shown to be somewhat independent of the transmission bit rate.
49 citations
01 Apr 1971
TL;DR: It is shown that moments can be generated taking the Gibb's derivative of the Walsh spectrum and that products of Walsh spectra yield the distribution of dyadic sums.
Abstract: Harmonic analysis of probability distribution functions has long served an important function in the treatment of stochastic systems. The tasks of generating moments and distributions of sums have been effectively executed in the Fourier spectrum. The properties of the Walsh-Hadamard transform of probability functions of discrete random variables is explored. Many analogies can be drawn between Fourier and Walsh analysis. In particular, it is shown that moments can be generated taking the Gibb's derivative of the Walsh spectrum and that products of Walsh spectra yield the distribution of dyadic sums. Stochastic systems with dyadic symmetry would benefit most from the properties of Walsh analysis and the computational advantages it offers. Some applications in the areas of information theory and pattern recognition are demonstrated.
43 citations
TL;DR: This paper presents a straightforward procedure using Walsh functions to determine the pattern in a binary sequence and when extended to infinite sequences it yields results agreeing with those by the classical probability theory.
Abstract: The concept of a finite binary random sequence does not seem to be covered in the classical foundations of the theory of probability. Solomonoff, Kolmogorov and Chaitin have tried to include this case by considering the lengths of programs required to generate these sequences: a longer program implying more randomness. However this definition is difficult to apply. This paper presents a straightforward procedure using Walsh functions to determine the pattern in a binary sequence. A quantitative measure of randomness has also been proposed. This has been defined as the number of independent data (via the Walsh transform) required to generate the sequence divided by the length of the sequence. However at present this classification procedure is restricted to sequences of length 2k only. When extended to infinite sequences it yields results agreeing with those by the classical probability theory.
41 citations
TL;DR: In this paper, the authors define ann-type (1, −1) matrix N = I + R of ordern ≡ 2 (mod 4) to haveR symmetric andR2 = (n − 1)In.
TL;DR: Complex BIFORE transform (CBT) as mentioned in this paper belongs to the family of discrete orthogonal transformations and is analogous to discrete Fourier transform (DFT) when dealing with complex inputs.
Abstract: Complex BIFORE (Binary FOurior REpresentation) transform belongs to the family of discrete orthogonal transformations and is analogous to discrete Fourier transform (DFT) when dealing with complex inputs. For real inputs, complex BIFORE transform (CBT) reduces to BIFORE or Hadamard transform (BT or HT) whose bases are Walsh functions. BT has been applied in several phases of information processing and sequency filters and sequency multiplexing equipment have also been built. When dealing with complex signals, CBT has some inherent computational advantages, and can be used to analyse and synthesize complex input functions. In the present paper, CBT is defined and its relationship to BT is shown. Several properties of CBT are developed. Invariance of power spectrum to sequential shift of the sampled data is shown. Using matrix factoring, fast algorithms suitable for digital computation of CBT and its inverse are developed. CBT is extended to multiple dimensions. Fast algorithms and corresponding fl...
TL;DR: The present Letter reports the initial operation of an advanced version of this spectrometer which uses a 2047-slot multiplexing mask and quantitatively verified the multiplex signal-to-noise ratio advantage for this class of spec trometer.
Abstract: Hadamard-transform spectrometers (HTS)—that is, dispersive multiplex spectrometers using binary orthogonal codes—have re ceived considerable at tent ion 8 since they were first proposed by Fellgett. We have recently described the successful operation of a 255-slot HTS which quantitatively verified the multiplex (or Fellgett) signal-to-noise ratio advantage for this class of spec trometer. The present Letter reports the initial operation of an advanced version of this spectrometer which uses a 2047-slot multiplexing mask.
TL;DR: Applications of Walsh functions, a subset of the Reed Muller Codes and Hadamard matrices, are described with emphasis on a correlation analysis for justification purposes.
Abstract: Walsh functions have become quite useful in the applications of image processing and feature selection. Due to their inherent efficiency of implementation, (they are a subset of the Reed Muller Codes and Hadamard matrices), they have become popular for coding, enhancement and other signal processing tasks. This paper will briefly describe applications in some of these areas with emphasis on a correlation analysis for justification purposes.
TL;DR: In this paper, a new method is presented for the generation of three families of discrete Walsh functions, each function is generated without reference to any other sequency representation, and sequencies up to 10 × 106 zero crossings per second are possible with currently available t.t.l. devices.
Abstract: A new method is presented for the generation of three families of discrete Walsh functions. Each function is generated without reference to any other sequency representation. Sequencies up to 10 × 106 zero crossings per second are possible with currently available t.t.l. devices.
TL;DR: In this paper, the concept of position spectrum for discrete orthogonal transformations of N-periodic sequences is introduced, and it is shown that a position spectrum is analogous to the conventional Fourier phase spectrum.
Abstract: The concept of "position spectrum" for discrete orthogonal transformations of N-periodic sequences is introduced. It is shown that a position spectrum is analogous to the conventional Fourier phase spectrum. As an illustration, the position spectrum for a modified Hadamard or BIFORE (binary Fourier representation) transform is developed.
TL;DR: A construction for amicable Hadamard matrices is given, and are used to generalize a construction for skew-Hadamards matrices.
TL;DR: In this article, it was shown that the character table of the abstract Abelian group Ck generated by k elements of order two can also be derived from the Slepians modular representation table.
Abstract: The set of Walsh functions, wal(j,?), is the character group of the dyadic group. For O?j?2k it is shown that they may also be derived from the character table of the abstract Abelian group Ck generated by k elements of order two. The method uses Slepians modular representation table[3] to compute the 2k irreducible representations (each of degree one) of Ck. The character table, K, is a 2kx2k square array of +1's and -l's and, considered as a matrix, the orthogonality relationships for characters show that K has the Hadamard property, [K][K]T = 2K [I]. In fact, for the proper ordering of the group elements in the construction of the modular representation table it is the Hadamard matrix, the entries of whose ith row take on the values of the Walsh function wal (i,?) in each of ?/2k subintervals. In a similar way other permutations of the modular representation table define different functions taking on the values +l, -l, also orthogonal and in a one to one relationship to the Walsh functions. Since an n place binary group code with k information places is isomorphic to Ck,[3] each code can thus be used to generate real functions orthogonal over a given interval or period ?. In the special case of cyclic codes where the elements of the code interpreted as polynomials form an ideal in a polynomial ring of characteristic two, the group operation used in deriving the character table is of course, addition.
01 Jan 1971
TL;DR: In this paper, the authors describe a method of defining Walsh functions by using orthogonal code blocks or the so-called Hadamard matrices, which is based on the multiplication law of Walsh functions which is a binary addition modulo 2 (no carry).
Abstract: In recent years several papers dealing with the mathematical theory as well as the technical applications of Walsh functions have been published. One method of defining Walsh functions is by using the Rademacher functions and the multiplication law of Walsh functions which is in fact a binary addition modulo 2 (no carry). This letter, however, describes in detail a method of defining Walsh functions by using orthogonal code blocks or the so-called Hadamard matrices. Some advantages of this method are its simplicity and its straightforward hardware implementation.
TL;DR: A technique is presented for the generation of any finite set of Walsh functions in both serial and parallel form and uses a straightforward constructive definition of these functions.
Abstract: A technique is presented for the generation of any finite set of Walsh functions in both serial and parallel form. It uses a straightforward constructive definition of these functions. In order to simultaneously generate the first 2nfunctions, [(22n-1)/3] storage devices and (2n+2-5) logic gates are required.
01 Sep 1971
TL;DR: In this paper, the Laplace transforms of generalized Walsh functions are presented, and from these the Fourier transforms and z transforms are easily deduced, and the bandwidth of finite-termed approximations is determined.
Abstract: The Laplace transforms of the generalized Walsh functions are presented, and from these the Fourier transforms and z transforms are easily deduced. The Fourier series representations of those generalized functions that are periodic are also given, and the bandwidth of finite-termed approximations is determined.
TL;DR: In this paper, the discrete two-dimensional Walsh transform is presented as a specific example of a generalized harmonic analysis approach to (visual) image classification, and some computer experiments are referenced and offered as validation of the approach.
Abstract: The discrete two-dimensional Walsh transform is presented as a specific example of a generalized harmonic analysis approach to (visual) image classification. Certain known aspects of biological systems are briefly presented as motivation for this approach, and some computer experiments are referenced and offered as validation of the approach. It is suggested that the mathematics typified by the Walsh transform provides an appropriate description of neural interactions in the primate visual system.
TL;DR: A recently proposed matrix factorization for a Hadamard matrix of order twelve is shown to be invalid in that the factored matrix is not hadamard.
Abstract: A recently proposed matrix factorization for a Hadamard matrix of order twelve is shown to be invalid in that the factored matrix is not Hadamard.
TL;DR: In this article, a variation of Hadamard's finite part integrals is described which can be used to solve a class of problems in a direct manner, eliminating the "ascent-descent" technique which had been used, and the Green's formula derived from the standard singular solution for the scalar wave equation yields the Kirchhoff formulas, if, when the space dimension is odd, one retains the logarithmically infinite part of the principal value integral.
Abstract: A variation of Hadamard’s finite part integrals is described which can be used to solve a class of problems in a direct manner, eliminating the “ascent-descent” technique which Hadamard used. It is shown that the Green’s formula derived from the standard singular solution for the scalar wave equation yields the Kirchhoff formulas, if, when the space dimension is odd, one retains the logarithmically infinite part of the principal value integral.
TL;DR: This note proves it is impossible to construct a Hadamard matrix of order twelve with this increasing sequency property and gives a corrected algorithm of order 12.
Abstract: An algorithm for a fast matrix transform of order twelve was given by Pratt [1]. The resulting transform matrix consists of plus and minus ones, is in order of sequency, but is not orthogonal, and hence is not a Hadamard matrix. This note proves it is impossible to construct a Hadamard matrix of order twelve with this increasing sequency property and gives a corrected algorithm of order twelve.