A Recursive Taylor Method for Algebraic Curves and Surfaces

Shou, Huahao; Martin, Ralph R.; Wang, Guojin; Bowyer, Adrian; Voiculescu, Irina

doi:10.1007/3-540-27157-0_10

Cited by 4 publications

(5 citation statements)

References 10 publications

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“…The first two univariate case studies: polynomial approximation and B-splines come from [2]. Some of the multivariate polynomial functions are from [21][22][23] and some are global optimization benchmarks. System examples are from [26].…”

Section: Case Studies and Resultsmentioning

confidence: 99%

Tradeoff between Approximation Accuracy and Complexity for Range Analysis using Affine Arithmetic

Zhang

Zhou

2010

J Sign Process Syst

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Digital signal processing algorithms are usually developed in floating-point arithmetic. After that floating-point to fixed-point transformation is performed to implement them on fixed-point devices, for higher speed, smaller area and lower power. During this transformation, range analysis is to find the minimum integer bit-widths for signals to prevent overflow. Existing state-of-the-art analytical methods for range analysis are generally based on Affine Arithmetic, which presents two approximation methods for non-affine operations. The Chebyshev approximation provides the best approximation with prohibitive computation expense. The trivial range estimation, which is very efficient for computation, over-estimates the range four times at the worst case. This paper presents a novel approach to let user decide tradeoff between approximation accuracy and complexity of Affine Arithmetic. Case studies and experiments are carried out to demonstrate its efficiency.

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Section: Case Studies and Resultsmentioning

confidence: 99%

Tradeoff between Approximation Accuracy and Complexity for Range Analysis using Affine Arithmetic

Zhang

Zhou

2010

J Sign Process Syst

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“…According to Lemma 1,(29), and (30), the estimated maximum and estimated minimum of d f , d max and d min can be expressed as…”

Section: Expression Of the Equivalent Affine Form In Aaseementioning

confidence: 99%

“…The first two cases are univariate cases and come from [11]. The rest of cases are multivariate polynomial functions and come from [27][28][29]. …”

Section: Case Studiesmentioning

confidence: 99%

A refined affine approximation method of multiplication for range analysis in word-length optimization

Sun

Zhang

Cui

2014

EURASIP J. Adv. Signal Process.

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Affine arithmetic (AA) is widely used in range analysis in word-length optimization of hardware designs. To reduce the uncertainty in the AA and achieve efficient and accurate range analysis of multiplication, this paper presents a novel refined affine approximation method, Approximation Affine based on Space Extreme Estimation (AASEE). The affine form of multiplication is divided into two parts. The first part is the approximate affine form of the operation. In the second part, the equivalent affine form of the estimated range of the difference, which is introduced by the approximation, is represented by an extra noise symbol. In AASEE, it is proven that the proposed approximate affine form is the closest to the result of multiplication based on linear geometry. The proposed equivalent affine form of AASEE is more accurate since the extreme value theory of multivariable functions is used to minimize the difference between the result of multiplication and the approximate affine form. The computational complexity of AASEE is the same as that of trivial range estimation (AATRE) and lower than that of Chebyshev approximation (AACHA). The proposed affine form of multiplication is demonstrated with polynomial approximation, B-splines, and multivariate polynomial functions. In experiments, the average of the ranges derived by AASEE is 59% and 89% of that by AATRE and AACHA, respectively. The integer bits derived by AASEE are 2 and 1 b less than that by AATRE and AACHA at most, respectively.

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“…Thus, for example, a quantity x which is known to lie in the range [3,7] can be represented by the affine formx = 5 + 2 k , for some k. Conversely, the formx = 10 + 2 3 − 5 8 implies that the corresponding quantity x lies in the range [3,17]. The sharing of a symbol j among two affine formsx,ŷ implies that the corresponding quantities x, y are partially dependent, in the sense that their joint range is smaller than the Cartesian product of their separate ranges.…”

Section: Affine Arithmetic (Aa)mentioning

confidence: 99%

“…In a similar context, [17] brings a new recursive Taylor method for ray-casting algebraic surfaces and shows that this method works better than various versions of interval and affine arithmetic. In general, interval or affine arithmetic cannot be used to detect degenerate zeros, like that of the function f (x) = x 2 .…”

mentioning

confidence: 99%

Certified meshing of Radial Basis Function based isosurfaces

2011

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Certified meshing of Radial Basis Function based isosurfacesChattopadhyay, A.; Plantinga, S.; Vegter, G. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract Radial Basis Functions are widely used in scattered data interpolation. The surface-reconstruction method using radial basis functions consists of two steps: (i) computing an interpolating implicit function the zero set of which contains the points in the data set, followed by (ii) extraction of isocurves or isosurfaces. In this paper we focus on the second step, generalizing the work on certified meshing of implicit surfaces based on interval arithmetic (Plantinga and Vegter in Visual Comput. 23:45-58, 2007). It turns out that interval arithmetic, and even the usually faster affine arithmetic, are far too slow in the context of RBF-based implicit surface meshing. We present optimized strategies giving acceptable running times and better space complexity, exploiting special properties of RBF-interpolants. We present pictures and timing results confirming the improved quality of these optimized strategies.

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A Recursive Taylor Method for Algebraic Curves and Surfaces

Cited by 4 publications

References 10 publications

Tradeoff between Approximation Accuracy and Complexity for Range Analysis using Affine Arithmetic

Tradeoff between Approximation Accuracy and Complexity for Range Analysis using Affine Arithmetic

A refined affine approximation method of multiplication for range analysis in word-length optimization

Certified meshing of Radial Basis Function based isosurfaces

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