By Ales Jaklic;Ales Leonardis;F. Solina
ISBN-10: 9048155746
ISBN-13: 9789048155743
ISBN-10: 9401594562
ISBN-13: 9789401594561
A illustration of gadgets through their components is the dominant process for representing complicated 3D gadgets in lots of disciplines. In desktop imaginative and prescient and robotics, superquadrics are one of the such a lot common half types. Superquadrics are a kin of parametric types that hide a wide selection of easily altering 3D symmetric shapes, that are managed with a small variety of parameters and which are augmented with the addition of worldwide and native deformations. The booklet covers, intensive, the geometric houses of superquadrics. the most contribution of the booklet is an unique method of the restoration and segmentation of superquadrics from variety photographs. a number of purposes of superquadrics in computing device imaginative and prescient and robotics are completely mentioned and, particularly, using superquadrics for variety photograph registration is confirmed.
Audience: The publication is meant for readers of all degrees who're conversant in and drawn to desktop imaginative and prescient issues.
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A illustration of items by means of their elements is the dominant process for representing advanced 3D items in lots of disciplines. In machine imaginative and prescient and robotics, superquadrics are one of the such a lot common half types. Superquadrics are a kinfolk of parametric types that conceal a large choice of easily altering 3D symmetric shapes, that are managed with a small variety of parameters and which are augmented with the addition of worldwide and native deformations.
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Here we derive these properties analytically. 1 Normal vector at a point r(rJ, w) on the superellipsoid surface (Eq. 10) on page 19) is defined by a cross product of the tangent vectors along the co ordinate curves 1 [ -alEl sinrJcos€1-lrJcos€2w -a2El sinrJ cos€l-lrJ sin€2 w a3EI sin€! +lrJ sinw cos€2- l w -al a2EIE2 sinrJ cos 2€! 8. Every (a) superquadric has a (b) dual superquadric which is defined by the scaled normal vector of the original one. 38) The scalar function f ('17, w) term can be dropped out if we only need the surface normal direction.
45) a3 First, we will examine the ease when 'T] is not equal to o. 46) Note that the aretan function is restrieted to the main braneh, namely -~ :s; 'T] :s; ~. 46) ean be used for any -7T :s; W < 7T where the rim is r(w) = r(1](w), w) . 48) Orthographie projeetion of this rim to the (xy)-plane is the oecluding eontour of the superellipsoid in general position. 50) where the integration path C' is the eurve itself. To simplify the integral, we do not ehoose the eurve C', but rather the eurve C that eonsists of the three segments Cl, C 2 and C3 (Fig.
6 =0 EI =1 EI = 2 8ala2a3 ~ala2a3 ~ala2a3 27rala2a3 t7rala2a3 j 7ra la2 a 3 4ala2a3 ~ala2a3 ~ala2a3 MOMENTS OF INERTIA OF A SUPERELLIPSE As we have used the expression for area of a superellipse to derive the volume of a superellipsoid, we will use expressions for moments of inertia of a superellipse to derive expressions for moments of inertia of a superellipsoid. 64) Jfs I~z (x 2 + y2)dxdy = + I~y = ab(a2 + b2)cB (~c,;). 65) where B(x, y) is a beta function. 66) I~y = xydxdy = o. 2. 2. 69) Superquadries and Their Geometrie Properties 37 j +1r/2 I~z(1])i(1])d1] = -1r/2 1 2 2"ala2a3clc2(al 3 2 1 1 + a2)B(2"c2, 2"c2)B(2"cl , 2cl + 1).
Segmentation and Recovery of Superquadrics by Ales Jaklic;Ales Leonardis;F. Solina
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