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17
Measurement and modeling of depth cue combination: in defense of weak fusion
 Vision Research
, 1995
"... Various visual cues provide information about depth and shape in a scene. When several of these cues are simultaneously available in a single location in the scene, the visual system attempts to combine them. In this paper, we discuss three key issues relevant to the experimental analysis of depth c ..."
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Cited by 137 (21 self)
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Various visual cues provide information about depth and shape in a scene. When several of these cues are simultaneously available in a single location in the scene, the visual system attempts to combine them. In this paper, we discuss three key issues relevant to the experimental analysis of depth cue combination in human vision: cue promotion, dynamic weighting of cues, and robustness of cue combination. We review recent psychophysical studies of human depth cue combination in light of these issues. We organize the discussion and review as the development of a model of the depth cue combination process termed modified weak fusion (MWF). We relate the MWF framework to Bayesian theories of cue combination. We argue that the MWF model is consistent with previous experimental results and is a parsimonious summary of these results. While the MWF model is motivated by normative considerations, it is primarily intended to guide experimental analysis of depth cue combination in human vision. We describe experimental methods, analogous to perturbation analysis, that permit us to analyze depth cue combination in novel ways. In particular these methods allow us to investigate the key issues we have raised. We summarize recent experimental tests of the MWF framework that use these methods. Depth Multiple cues Sensor fusion
On stable numerical differentiation
 Mathem. of Computation
, 1968
"... Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
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Cited by 39 (22 self)
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Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.
On the Design of Optimal Filters for GradientBased Motion Estimation
 In: Proc. Intern. Conf. on Computer Vision
, 2002
"... Gradient based approaches for motion estimation (OpticalFlow) refer to those techniques that estimate the motion of an image sequence based on local changes in the image intensities. ..."
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Cited by 26 (0 self)
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Gradient based approaches for motion estimation (OpticalFlow) refer to those techniques that estimate the motion of an image sequence based on local changes in the image intensities.
Optimal Differentiation Based On Stochastic Signal Models
 IEEE Transactions on Signal Processing
, 1991
"... The problem of estimating the time derivative of a signal from sampled measurements is addressed. The measurements may be corrupted by coloured noise. A key idea is to use stochastic models of the signal to be differentiated and of the measurement noise. Two approaches are suggested. The first is ba ..."
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Cited by 13 (3 self)
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The problem of estimating the time derivative of a signal from sampled measurements is addressed. The measurements may be corrupted by coloured noise. A key idea is to use stochastic models of the signal to be differentiated and of the measurement noise. Two approaches are suggested. The first is based on a continuoustime stochastic process as model of the signal. The second approach uses a discretetime ARMA model of the signal and a discretetime approximation of the derivative operator. The introduction of this approximation normally causes a small performance degradation, compared to the first approach. There exists an optimal (signal dependent) derivative approximation, for which the performance degradation vanishes. Digital differentiators are presented in a shift operator polynomial form. They minimize the mean square estimation error. In both approaches, they are calculated from a linear polynomial equation and a polynomial spectral factorization. (The first approach also requ...
Stable numerical differentiation: when is it possible
 Jour. Korean SIAM
"... Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrate ..."
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Cited by 11 (8 self)
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Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrated by examples. These examples show the practical value of the new understanding of the problem of stable differentiation. 1.
A Variational Method for Numerical Differentiation
, 1995
"... this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed interval [a; b], we construct an associated functional, ..."
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Cited by 4 (0 self)
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this paper, we suggest a method for numerical differentiation that we believe avoids some of the limitations mentioned above. Given a real valued, smooth function u defined on the closed interval [a; b], we construct an associated functional,
A Stable Finite Difference Ansatz for Higher Order Differentiation of NonExact Data
 Bull. Austral. Math. Soc
, 1996
"... If standard central difference formulas are used to compute second or third order derivatives from measured data even quite precise data can lead to totally unusable results due to the basic instability of the differentiation process. Here an averaging procedure is presented and analysed which allow ..."
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Cited by 4 (1 self)
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If standard central difference formulas are used to compute second or third order derivatives from measured data even quite precise data can lead to totally unusable results due to the basic instability of the differentiation process. Here an averaging procedure is presented and analysed which allows the stable computation of low order derivatives from measured data. The new method first averages the data, then samples the averages and finally applies standard difference formulas. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. 1991 Mathematics Subject Classification. 65D25. 1. Introduction Let the given (observational or nonexact) data be defined by d := fd j = f(t j ) + ffl j ; t j = jh; h = 1=n; j = 0; 1; 2; \Delta \Delta \Delta ; ng; (1) where f(t) denotes the underlying, but unknown, signal process and the ffl j denote the (observational or nonexact) errors which are assumed to be identical and independe...
A scheme for stable numerical differentiation
 J. COMP. APPL. MATH.
, 2006
"... A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are ..."
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Cited by 2 (1 self)
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A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are presented. These examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods, in particular, than variational regularization.
ON COMPLEXVALUED 2D EIKONALS. PART FOUR: CONTINUATION PAST A CAUSTIC
, 905
"... Abstract. Theories of monochromatic highfrequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude ..."
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Abstract. Theories of monochromatic highfrequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude — in other words, are complexvalued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a nonzero imaginary part, comes on stage. In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complexvalued twodimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, illposedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasireversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. 1.
Extremum Estimation and Numerical Derivatives
, 2010
"... Many empirical researchers rely on the use of finitedifference approximation to evaluate derivatives of estimated functions. For instance many optimization routines implicitly use finitedifference formulas for the gradients. Such routines frequently require the choice of step size parameters for fi ..."
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Many empirical researchers rely on the use of finitedifference approximation to evaluate derivatives of estimated functions. For instance many optimization routines implicitly use finitedifference formulas for the gradients. Such routines frequently require the choice of step size parameters for finitedifference numerical gradients or similar parameters such as the computing tolerance. This paper investigates the statistical properties numerically evaluated gradients and the properties of extremum estimators computed using numerical gradients. We find that first, for unbiased inference one needs to adjust the step size or the tolerance as a function of the sample size. Second, higherorder finite difference formulas reduce the asymptotic bias analogous to higher order kernels in kernel smoothing. Third, we provide weak sufficient conditions for uniform consistency of the finitedifference approximations for gradients and directional derivatives. Fourth, we analyze the numerical gradientbased extremum estimators and find that the asymptotic distribution of the resulting estimators can be a hybrid between the asymptotics of the original extremum estimators and the asymptotics of the kernel smoothers. Fifth, we state conditions under which the numerical derivative estimator is consistent and asymptotically normal. Sixth, we generalize our results to semiparametric estimation problems. Finally, we show that the theory is also useful in a range of nonstandard estimation procedures. JEL Classification: C14; C52