Wavelet Processing Implementation in Digital Hardware

Szecowka, Przemyslaw M.; Kowalski, Maciej; Krysztoforski, K.; Wołczowski, Andrzej

doi:10.1109/mixdes.2007.4286243

Cited by 2 publications

(7 citation statements)

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“…Apart from the accuracy and stability of Jacobi algorithm, it also has high degree potential for parallelism, and hence can be implemented on FPGA [5], [6]. In [3], [5], [4], [6], [7], [8], [9], [10] this algorithm is implemented on FPGA with fixed-point arithmetic to reduce power consumption and silicon area. However, in all the works, fixed-point implementation of Jacobi algorithm uses the simulation-based approach for estimating the ranges of variables.…”

Section: Motivationmentioning

confidence: 99%

“…Eigenvalue decomposition (EVD) is a key building block in signal processing and control applications. The fixed-point development of eigenvalue decomposition (EVD) algorithm have been extensively studied in the past few years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] because fixed-point circuitry is significantly simpler and faster. Owing to its simplicity, fixed-point arithmetic is ubiquitous in low cost embedded platforms.…”

Section: Introductionmentioning

confidence: 99%

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An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

Kabi

Sahadevan

Pradhan

2017

2017 Conference on Design and Architectures for Signal and Image Processing (DASIP)

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Abstract-We consider the problem of enabling robust range estimation of eigenvalue decomposition (EVD) algorithm for a reliable fixed-point design. The simplicity of fixed-point circuitry has always been so tempting to implement EVD algorithms in fixed-point arithmetic. Working towards an effective fixed-point design, integer bit-width allocation is a significant step which has a crucial impact on accuracy and hardware efficiency. This paper investigates the shortcomings of the existing range estimation methods while deriving bounds for the variables of the EVD algorithm. In light of the circumstances, we introduce a range estimation approach based on vector and matrix norm properties together with a scaling procedure that maintains all the assets of an analytical method. The method could derive robust and tight bounds for the variables of EVD algorithm. The bounds derived using the proposed approach remain same for any input matrix and are also independent of the number of iterations or size of the problem. Some benchmark hyperspectral data sets have been used to evaluate the efficiency of the proposed technique. It was found that by the proposed range estimation approach, all the variables generated during the computation of Jacobi EVD is bounded within ±1.

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Section: Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

Kabi

Sahadevan

Pradhan

2017

2017 Conference on Design and Architectures for Signal and Image Processing (DASIP)

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“…f j and f k are round-off errors for j = 1, …, n and k = 1, …, j À 1. Multiplying the aforementioned two equations with q T k and q T j , respectively, and subtracting them, we obtain the following recursion given by β j w k;jþ1 ¼ β k w j;kþ1 þ α k w j;k À α j w k;j þ β kÀ1 w j;kÀ1 À β jÀ1 w k;jÀ1 À q T k f j þ q T j f k ; (11) Algorithm 2 Lanczos tridiagonalization with partial orthogonalization 1: q 0 = 0; 2: β 0 = 0; 3: w 1,1 = 1; 4:…”

Section: Lanczos Tridiagonalization With Partial Orthogonalizationmentioning

confidence: 99%

“…is defined for the round-off error term q T j f k À q T k f j of Equation (11). Using Equation (14), w k,j + 1 can be calculated as w k;jþ1 ¼ β À1 j β k w j;kþ1 þ α k w j;k À α j w k;j þ β kÀ1 w j;kÀ1 À β jÀ1 w k;jÀ1 þ θ k;j ;…”

Section: Lanczos Tridiagonalization With Partial Orthogonalizationmentioning

confidence: 99%

“…In the last decade, quick and accurate computation of symmetric SVD using fixed-point algorithms has been one of the prime concerns of the researchers. Considerable effort has been devoted to the implementation issues and architectural development of SVD and their deployment on embedded platforms [10,11]. The class of SVD algorithms using Lanczos tridiagonalization has been found to be faster and more accurate as compared with all other forms of algorithms proposed till date [12][13][14].…”

mentioning

confidence: 99%

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Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Pradhan

Kabi

Mohanty

et al. 2015

Circuit Theory & Apps

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SUMMARYProper range and precision analysis play an important role in the development of fixed-point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image-processing applications. These algorithms have not been attempted much in fixed-point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round-off errors. For any real-time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed-point formats to decide on integer wordlengths during float-to-fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed-point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed-point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed-point Lanczos algorithm in three different formats has been compared on the basis of signal-toquantization-noise-ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed-point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean-square-error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed-point formats produce vectors with lesser round-off error than constant format. All the three fixed-point formats of Lanczos algorithm have been synthesized on Virtex 7 field-programmable gate array using Vivado high-level synthesis design tool. A comparative study of resource usage and power consumption is carried out.

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Wavelet Processing Implementation in Digital Hardware

Cited by 2 publications

References 1 publication

An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

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