Showing papers on "Hadamard transform published in 1976"
TL;DR: A unified matrix treatment for the various orderings of the Walsh-Hadamard (WH) functions using a general framework is presented in this article, which clarifies the different definitions of the WH matrix, the various fast algorithms and the reorderings of WH functions.
Abstract: A unified matrix treatment is presented for the various orderings of the Walsh-Hadamard (WH) functions using a general framework. This approach clarifies the different definitions of the WH matrix, the various fast algorithms and the reorderings of the WH functions.
186 citations
TL;DR: This paper gives a brief survey of the design of masks for Hadamard spectrometers and image scanners, and three different criteria are described for judging a mask.
Abstract: This paper gives a brief survey of the design of masks for Hadamard spectrometers and image scanners. Three different criteria are described for judging a mask, as well as techniques for choosing masks that are not too far from the optimum.
125 citations
TL;DR: The effect of multiplexing in Fourier transform spectroscopy (FTS) and other coding techniques such as Hadamard transform spectral analysis (HTA) on the signal/noise ratio arises from two factors as mentioned in this paper.
Abstract: The effect of multiplexing in Fourier transform spectroscopy (FTS) (and other coding techniques such as Hadamard transform spectroscopy) on the signal/noise ratio arises from two factors. The first of these is a N½ improvement for a spectrum containing N resolution elements since each individual resolution element is now viewed for an N-fold longer time. The second factor is due to the simultaneous observation of all resolution elements, raising the signal level by a factor N for an uniform spectrum.
77 citations
TL;DR: In this paper, it was shown that for any natural number q > 3, there exists an integer t ⩽ [2log2(q − 3)] such that an Hadamard matrix exists for every order 2sq where s > t.
69 citations
TL;DR: An analysis of the situations for which the multiplex advantage may be gained and of the limitations of the technique is presented, including potential applications of the spectrometric imager.
Abstract: An imager and a spectrometric imager, which achieve multiplexing by the use of binary optical encoding masks, have been built and tested. The masks are based on orthogonal, pseudorandom digital codes derived from Hadamard matrices. The spatial (and/or spectral) data are therefore obtained in the form of a Hadamard transform of the spatial (and/or spectral) scene; computer algorithms are used to decode the data and reconstruct images of the original scene. The hardware, algorithms processing and display facility are described. A number of spatial and spatial/spectral images, obtained in the laboratory, are presented. We present an analysis of the situations for which the multiplex advantage may be gained and of the limitations of the technique. Potential applications of the spectrometric imager are discussed. The spectrometric imager is covered by U.S. Patent 3,720,469 assigned to Spectral Imaging Inc., Concord, Mass.
66 citations
TL;DR: This paper is intended as a tutorial review of certain digital image processing transform techniques utilizing the notion of outer product expansions and implementation of the singular value decomposition (SVD) of large sized images.
Abstract: This paper is intended as a tutorial review of certain digital image processing transform techniques utilizing the notion of outer product expansions. Examples from Fourier, Walsh, Haar, and other well known transforms are reviewed in the notation of matrix-vector outer products; and implementation of the singular value decomposition (SVD) of large sized images is presented. The use of the SVD as an aid in image restoration utilizing the pseudoinverses is presented. Conditions on the point spread matrix are investigated in the light of singular value decomposition, Kronecker products, and general imaging conditions.
51 citations
TL;DR: In this paper, the Hadamard-Fischer inequality was shown to hold for the case of BE W(n) and A E 7(1I) if A E W( n) and 0, A.
Abstract: 1) Spec A[Jl.l n IR =1= t/>, for t/> c Jl. S (n), 2) I(A[J-L]) « I(A[v]), if t/> c v S Jl. S (n), where I(A[Jl.]) = min(Spec A[Jl.l n IR). For A, BE W(n), define A «, B by I(A[J-L]) « I(B[J-L]), for t/> c Jl. S ( n ). By definition, A E7(1I) if A E W(n) and 0 «, A. For 0 « , A «t B (where A, BE W(n» it is shown that 3) 0 « det A « det B-det(B-I(A)I) « det B. For A E 7(11 ) , A «, A[Jl.l El1 A{J1.), and hence we obtain the Hadamard-Fischer inequality 4) 0 « det A « det A[Jl.l det A{J1.)
50 citations
TL;DR: It is shown that energy contained in the lowest 1/2^{j} ( j : positive integer) of a signal's sequency spectrum can be explicitly evaluated in terms of covariances of the signal.
Abstract: This concise paper presents a theoretical evaluation of energy packing of the Hadamard transform which is often used in signal processing. It is shown that energy contained in the lowest 1/2^{j} ( j : positive integer) of a signal's sequency spectrum can be explicitly evaluated in terms of covariances of the signal.
38 citations
TL;DR: A real-time digital video processor using Hadamard transform techniques to reduce video bandwidth is described and algorithms related to spatial compression, temporal compression, and the adaptive selection of parameter sets are described.
Abstract: A real-time digital video processor using Hadamard transform techniques to reduce video bandwidth is described. The processor can be programmed with different parameters to investigate various algorithms for bandwidth compression. The processor is also adaptive in that it can select different parameter sets to trade-off spatial resolution for temporal resolution in the regions of the picture that are moving. Algorithms used in programming the system are described along with results achieved at various levels of compression. The algorithms relate to spatial compression, temporal compression, and the adaptive selection of parameter sets.
30 citations
TL;DR: An encoding figure of merit is established for a detector-noise limited Fourier transform spectrometer (FTS) and it is compared to the comparable figure for a Hadamard transform spectrumeter (HTS) to establish the mean square errors.
Abstract: We establish an encoding figure of merit for a detector-noise limited Fourier transform spectrometer (FTS) and compare it to the comparable figure for a Hadamard transform spectrometer (HTS). If N measurements are made to establish N spectral densities, the mean square errors obtained with the Fourier system are a factor of 2 greater than for the analogous Hadamard system. The limitation of the Fourier system is partly that it does not truly Fourier analyze the radiation. Instead a cosine squared modulation is imposed on the different spectral frequencies. An additional difficulty is that neither the cosine nor the cosine squared functions form an orthonormal set. This makes the Fellgett's advantage (root-mean-squared figure of merit) for a single detector Michelson interferometer a factor of (N/8)(1/2) greater than for a conventional grating instrument-rather than (N/2)(1/2) as maintained in standard texts. The theoretical limit, which may not be realizable with practical instruments, would be (N)(1/2).
23 citations
TL;DR: In this article, it was shown that there exist four symmetric circulant matrices A, B, C, D of order υ such that where Iv is the identity matrix of order ǫ.
Abstract: Let p be a prime ≡ 1 (mod 4) and put v = p(p + 1)/2. It is proved in this paper that there exist four symmetric circulant matrices A, B, C, D of order υ such that where Iv is the identity matrix of order υ. This result is used to construct Hadamard matrices of order 4υ that are of the type originally prescribed by Williamson.
TL;DR: In this article, a slightly different approach was used to extend a (4 t − k, 4 t ) row-orthogonal matrix to a Hadamard matrix of order 4 t.
Abstract: Shrinkande and Bhagwan Das (1970) showed how to extend a (4 t – 1, 4 t ) row-orthogonal matrix with entries ± 1 to a Hadamard matrix of order 4 t . Using a slightly different approach we consider extensions of (4 t – k , 4 t ) row-orthogonal matrix to a Hadamard matrix of order 4 t .
TL;DR: A fast sequency ordered Walsh transform was proposed in this paper, which accepts data in normal order, returning the coefficients in bit-reversed sequency order, and is the complement to one developed by Manz.
Abstract: A fast sequency ordered Walsh Transform algorithm is presented, which is the complement to one developed by Manz. It is in place, is its own inverse, and accepts data in normal order, returning the coefficients in bit-reversed sequency order.
TL;DR: Equivalence of Hadamard matrices is described in terms of designs and it is shown that nonisomorphic designs may arise from the same matrix.
TL;DR: In this article, the authors used broad bandwidth Walsh-Hadamard sequences as test signals to identify the dynamics of a gas phase catalytic reactor using broad-band Walsh functions as the Hadamard matrices.
13 Dec 1976
TL;DR: An operational video compressor was built using field-to-field differencing, on fields processed using Landau and Slepian's Hadamard transform method, with a substantial performance improvement.
Abstract: An adaptive video compressor was designed as part of the CTS Digital Video Curriculum Sharing Experiment. The compressor was constructed using field-to-field differencing on fields processed using a Hadamard transform method. The spatial resolution was improved using a fixed-rate, three-mode adaptive system to compress each field.
TL;DR: In this paper, the authors developed general methods for generating complete orthonormal systems of Walsh-like functions from certain permuted Kronecker products of Hadamard matrices. Some examples are included.
Abstract: We develop general methods for generating complete orthonormal systems of Walsh-like functions from certain permuted Kronecker products of Hadamard matrices. Some examples are included.
01 Jan 1976
TL;DR: The recent development of an approximation to a least-difference method has led to an appreciation of the effects of data bounding and to the development of a more powerful process.
Abstract: Processing of nuclear medicine images is generally performed by essentially linear methods with the non-negativity condition being applied as the only non-linear process. The various methods used: matrix methods in signal space and Fourier or Hadamard transforms in frequency or sequency space are essentially equivalent. Further improvement in images can be obtained by the use of inherently non-linear methods. The recent development of an approximation to a least-difference method (as opposed to a least-square method) has led to an appreciation of the effects of data bounding and to the development of a more powerful process. Data bounding (modification of statistically improbable data values) is an inherently non-linear method with considerable promise. Strong bounding depending on two-dimensional least-squares fitting yields a reduction of mottling (buttermilk effect) not attainable with linear processes. A pre- bounding process removing very bad points is used to protect the strong bounding process from incorrectly modifying data points due to the weight of an extreme but yet unbounded point as the fitting area approaches it. (auth)
TL;DR: It is shown that the Williamson series of H -matrices can also be constructed by using modular Hadamard matrices and resolvable semi-regular group divisible designs.
TL;DR: In this paper, it was shown that the orthogonal designs derived from the above are the only possible designs for At = -A, Bt = B, and ABt = ABt.
Abstract: A powerful tool in the construction of orthogonal designs has been amicable orthogonal designs. Recent results in the construction of Hadamard matrices has led to the need to find amicable orthogonal designs A, B in order n and of types (u1, U2, …, u6) and (ν1, ν2, …, νr) respectively satisfying At = -A, Bt = B, and ABt = BAt withFor simplicity, we say A, B are amicable orthogonal designs of type (u1, u2, …, us; v1, v2, …, vr).We completely answer the question in order 8 by showing (1, 2, 2, 2; 8), (1, 2, 4; 2, 2, 4), (2, 2, 3; 2, 6), (7, 1, 7) and those designs derived from the above are the only possible.We use our results to obtain new orthogonal designs in order 32.
TL;DR: A special case of FWT is examined where a scheme is devised to replace a single high-order FWT by consecutive FWT's of much lower order, which results in a reduction of transform order which may have useful applications in processing signals in localized sequency regions.
Abstract: A special case of FWT is examined where a scheme is devised to replace a single high-order FWT by consecutive FWT's of much lower order. This special case of the usual decimation-in-frequency approach to fast transforms [1] results in a reduction of transform order, which may have useful applications in processing signals in localized sequency regions.
13 Dec 1976
TL;DR: Three modifications of the Constant Area Quantization (CAQ) image bandwidth compression technique have been developed and tested in order to broaden the range of compression ratios obtainable with acceptable image quality.
Abstract: Three modifications of the Constant Area Quantization (CAQ) image bandwidth compression technique have been developed and tested in order to broaden the range of compression ratios obtainable with acceptable image quality. The first modification involved the introduction of an adaptive area threshold, the second was a two-threshold algorithm and the third was a hybrid of the CAQ with a Hadamard transform technique. Using these three algorithms together with the basic CAQ, images spanning the range from 0. 2 to 2 bits per picture element were obtained from an 8 bit original.© (1976) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: In this paper, the author shows how to simplify the complex response of the Hadamard variance by analog filtering to design a spectrum analyzer.
Abstract: In this paper, the author shows how to simplify the complex response of the Hadamard variance by analog filtering to design a spectrum analyzer. The realized spectrum analyzer is described.
TL;DR: In this article, the authors studied n × n Hermitian semidefinite positive matrices which are infinitely divisible in a sense that they define in Sec. 1.
TL;DR: In this paper, the Walsh transform matrices are used to obtain the summing or the differencing of a discrete-time function f from the WT of f, for which recursive relationships have been found.
Abstract: To obtain the Walsh transform (WT) of the summing or the differencing of a discrete-time function f from the WT of f, transform matrices may be used, for which recursive relationships have been found The present letter is intended to state them rigorously
01 Jul 1976
TL;DR: Some peculiarities of the Walsh functions and the rather unpromising results, achieved in many attempts to use the Walsh-Fourier transform in solving engineering problems, are explained.
Abstract: This communication intends to set the Walsh-Fourier transform and the concept of sequency in proper perspective with regard to the other Fourier transforms and the concept of generalized frequency. In doing this, it explains some peculiarities of the Walsh functions and the rather unpromising results, in spite of its computational advantages, achieved in many attempts to use the Walsh-Fourier transform in solving engineering problems.