The present study furthcr explores the fundamental singular solutions for Stokes flow that can be useful £or constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associatedwith a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, strcsslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of cxterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary ilow. The different primary flows considered here include the uniform strcam, shear flows, parabolic profiles and extensional flows (hyperbolic profiles), while the body shapcs cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for tlhe construction of a solution in each specific case, their distribution densities and the rangc of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, arc discussed.

This article addresses the problem of natural shadow matting , the removal or extraction of natural shadows from a single image. Because textures are maintained in the shadowless image after the extraction process, our approach produces some of the best results to date among shadow removal techniques. Using the image formation equation typical of computer vision, we advocate a new model for shadow formation where shadow effect is understood as light attenuation instead of a mixture of two colors governed by the conventional matting equation. This leads to a new shadow equation with fewer unknowns to solve, where a three-channel shadow matte and a shadowless image are considered in our optimization. Our problem is formulated as one of energy minimization guided by user-supplied hints in the form of a quadmap which can be specified easily by the user. This formulation allows for robust shadow matte extraction while maintaining texture in the shadowed region by considering color transfer, texture gradient, and shadow smoothness. We demonstrate the usefulness of our approach in shadow removal, image matting, and compositing.

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The optimum shape problems considered in this part are for those profiles of a two-dimensional flexible plate in time-harmonic motion that will minimize the energy loss under the condition of fixed thrust and possibly also under other isoperimetric constraints. First, the optimum movement of a rigid plate is completely determined; it is necessary first to reduce the original singular form representing the energy loss to a regular one of a lower order, which is then tractable by usual variational methods. A favourable range of the frequency is found in which the thrust contribution coming from the leading-edge suction is as small as possible under the prescribed conditions, outside of which this contribution becomes so large as to be hard to realize in practice without stalling. This optimum solution is compared with the recent theory of Lighthill (1970); these independently arrived-a6 conclusions are found to be virtually in agreement.The present theory is further applied 60 predict the movement of a porpoise tail of large aspect-ratio and is found in satisfactory agreement with the experimental measurements. A qualitative discussion of the wing movement in flapping flight of birds is also given on the basis of optimum efficiency.The optimum shape of a flexible plate is analysed for the most general case of infinite degrees of freedom. It is shown that the solution can be determined to a certain extent, but the exact shape is not always uniquely determinate.