# Methods of Theoretical Physics. By P.M. Morse and H. Feschbach. 2vols., Pp.xxii, 1978. 120s. each vol. 1953.(McGraw-Hill)

About: This article is published in The Mathematical Gazette.The article was published on 1955-02-01. It has received 9185 citations till now.

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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.

7,412 citations

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TL;DR: This paper presents a new external force for active contours, which is computed as a diffusion of the gradient vectors of a gray-level or binary edge map derived from the image, and has a large capture range and is able to move snakes into boundary concavities.

Abstract: Snakes, or active contours, are used extensively in computer vision and image processing applications, particularly to locate object boundaries. Problems associated with initialization and poor convergence to boundary concavities, however, have limited their utility. This paper presents a new external force for active contours, largely solving both problems. This external force, which we call gradient vector flow (GVF), is computed as a diffusion of the gradient vectors of a gray-level or binary edge map derived from the image. It differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function, and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation. Using several two-dimensional (2-D) examples and one three-dimensional (3-D) example, we show that GVF has a large capture range and is able to move snakes into boundary concavities.

4,071 citations

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TL;DR: Quantitative-diffusion-tensor MRI consists of deriving and displaying parameters that resemble histological or physiological stains, i.e., that characterize intrinsic features of tissue microstructure and microdynamics that are objective, and insensitive to the choice of laboratory coordinate system.

4,064 citations

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TL;DR: The authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively steer a filter to any orientation, and to determine analytically the filter output as a function of orientation.

Abstract: The authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively steer a filter to any orientation, and to determine analytically the filter output as a function of orientation. Steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation. The authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase, angularly adaptive filtering, edge detection, and shape from shading. One can also build a self-similar steerable pyramid representation. The same concepts can be generalized to the design of 3-D steerable filters. >

3,365 citations

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TL;DR: In this paper, the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation, which is based on the generalized least squares criterion, and it can handle errors in the data set and a priori information on the model.

Abstract: The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commerc...

3,198 citations

### Cites background from "Methods of Theoretical Physics. By ..."

...…by equation (5), the solution for 6p will be given by 6p(r, t; rs) = dr’ g(r, t; r’, 0) * As(r’, t; r.), or using the reciprocity theorem for the Green’s function (see Morse and Feshbach, 1953; Aki and Richards, 1980; BenMenahem and Singh, 1981) 6p(r, t; r,) = s dr’ g(r’, t; r, 0) * As(r’, t; r,)....

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...In Tarantola (1984) herein referred as “paper I,” I discussed the philosophy of inverse theory, compared to the philosophy of “migration” or “direct inversion....

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