D.W. Jacobs. Space efficient 3D model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439-444, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....into use although the topic makes a very appealing case for applications. In many instances, one would like to achieve a direct representation of 3D shape from images without the need to recover the camera geometry as an intermediate step. This includes indexing into a library of ob jects (cf. [13]) multi body segmentation (collection of points belong to the same structure when the shape invariants hold) and even for tracking applications (which traditionally use multi view constraints) where features may get lost due to occlusions and later reappear. A direct shape constraint is ....

D.W. Jacobs. Space efficient 3D model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439-444, 1992.

....is not required, more freedom in picture taking is allowed such as taking pictures of pictures of objects and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) [17, 2, 30, 12,46, 47, 13, 26,7,32, 34, 36, 25, 45, 29,8,10, 23, 31,16,15,48] and relevant to this paper are the work described in [17,7,13,34,36] The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective framework [7, 13, 36] or the affine framework. The latter requires either assuming parallel projection ....

D.W. Jacobs. Space efficient 3D model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....4.3. Affine Objects In this section we extend the LC scheme to objects that undergo general affine transformations in space. In addition to the rigid transformations affine transformations include stretching and shearing. They are important since tilted pictures of objects appear to be stretched [46]. This effect is known as the La Gournerie Paradox (see [47] In order to extend the LC method to include affine transformations the same scheme can be used, but with the quadratic constraints ignored. Namely, the fourdimensional linear space contains all and only the affine views of the object. ....

D. W. Jacobs. Space efficient 3d model indexing. In Proceedings of CVPR Conference, Urbana, IL, 1992.

....in a single index space [3] requiring an inordinate number of table entries. Jacobs developed an indexing algorithm for the more general class of affine transformations, which separates the two dimensional surface in indexing space into two one dimensional surfaces (lines) in two smaller spaces [7]. This separation reduces the size of the tables and also simplifies table construction. The drawback of this approach is that for affine transformations, having eight degrees of freedom, overconstraint requires five point groups. This work also stems from work in the field of coding theory. A ....

D. W. Jacobs. Space efficient 3d model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....is not required, more freedom in picture taking is allowed such as taking pictures of pictures of objects and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) [17, 2, 30, 12, 46, 47, 13, 26, 7, 32, 34, 36, 25, 45, 29, 8, 10, 23, 31, 16, 15, 48] and relevant to this paper are the work described in [17, 7, 13, 34, 36] The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective framework [7, 13, 36] or the affine framework. The latter requires either assuming parallel ....

D.W. Jacobs. Space efficient 3D model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

.... for the problem of recognizing three dimensional objects from single two dimensional images require groups of four points to generate a key into the table of model groups and each model group must be represented over an infinite subspace of a multi dimensional table [ Clemens and Jacobs, 1991, Jacobs, 1992 ] We present a system that is capable of indexing using groups of three points by taking advantage of the probabilistic peaking effect [ BenArie, 1990 ] Each model group need only be represented at one point in the index table. To be able to index using groups of three points, we must allow ....

....(i.e. viewing direction in the weak perspective model, thus they must represent each group from each viewing direction in their index table. Since the viewing direction has two degrees of freedom, this means that each group must be represented on a two dimensional surface in the table. Jacobs [1992] has shown that groups of four three dimensional points can be indexed from two dimensional data by representing each group as one dimensional surfaces in two orthogonal two dimensional tables. To determine which model groups may have projected to an image group, model groups are indexed in both ....

D. W. Jacobs. Space efficient 3d model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....of the orthographic projection model only a relatively small number of studies have been devoted to it. Recently, however, starting from the work on affine shape by Koenderink and van Doorn [1991] orthographic projection models have again been used in various applications [Ullman and Basri, 1991; Jacobs, 1992] It turns out that two views are enough to reconstruct affine shape, that is, to reconstruct the affine coordinates of every object point with respect to three linearly independent vectors attached to the object. Actually, affine coordinates remain constant not only under rigid motion but under ....

D.W. Jacobs. Space efficient 3D model indexing. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....of models. Using hash tables or trees, indexing can occur in time that is sublinear with respect to database size [10] However, some of the enthusiasm for invariant based recognition was dampened by the observation that there are no nontrivial invariants for the image of an arbitrary 3D point set [1, 5, 11]. Consequently, a 3D point set cannot be represented by a single vector which is invariant from all viewpoints. For smooth curved 3D objects, the line drawing is the projection of visible points on the occluding contour (surface points where the line of sight lies in the tangent plane) Because ....

D. W. Jacobs. Space efficient 3D model indexing. In Proc. IEEE Conf. on Comp. Vision and Patt. Recog., pages 439--444, 1992.

.... literature of structure from motion, that show that under parallel projection any view of an object can be constructed as a linear combination of a small number of views of the same object, a series of studies on recognition using orthographic and paraperspective projections have been conducted [31, 64]. The body of projective geometry has been investigated to prove results about the computation of structure and motion from a set of views under perspective projection [14] The learning of object recognition capabilities has been studied for neuronal networks using nodes that store ....

D. Jacobs. Space efficient 3d model indexing. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....of the table, which is not one dimensional anymore but multi dimensional. We also pay in a higher frequency of inevitable false positive matches (or accidental matches) As an example, we will derive the lookup table for the affine model based invariant function (a similar scheme is described in [10]) Let b denote the affine representation of five non coplanar image points, as defined in Eq (3) There are three unknown parameters in this representation, b 1 ; b 2 ; b 3 , while a single image gives only two constraints on these unknowns. We project the two constraints to a lower dimension by ....

D. W. Jacobs. Space efficient 3D model indexing. In Proceedings Image Understanding Workshop, January 1992.

....that might have produced them. A system based on this idea has been implemented in [12] assuming orthographic projection. This system has been improved in the case of 3D linear transformations so that the hash table is built using analytical formulas, without having to sample the viewing sphere [13,14]. In particular, it was shown in [14] that the images of groups of 3D points can be represented as a pair of 1D lines in two highdimensional spaces. During preprocessing, each group of model points is represented by a line in each of the two spaces. During recognition, groups of scene points are ....

....to at least one of the reference views. In our case, new views can be constructed by combining a small number of reference views. Furthermore, our approach for generating the images that a model group can produce during preprocessing is more practical since it does not require 3D models. In [13,14] for example, the lines which represent the images of a model group can be found easily only if the 3D structure of the object is known. Since this information is not always available, a set of different 2D images, containing the group, is used instead [13,14] Each image defines a point in each ....

[Article contains additional citation context not shown here]

D. Jacobs, Space efficient 3D model indexing, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 439--444, 1992.

....is not required, more freedom in picture taking is allowed such as taking pictures of pictures of objects, and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) [14, 26, 33, 34, 9, 20, 3, 4, 28, 29, 19, 31, 23, 5, 6, 18, 27, 13, 12] and relevant to this paper are the work described in [14, 4, 26, 28, 29] This paper has two parts. In Part I we investigate the intrinsic differences conceptually and algorithmically between an affine framework for recognition reconstruction and a projective framework. Although the ....

D.W. Jacobs. Space efficient 3D model indexing. In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....is not required, more freedom in picture taking is allowed such as taking pictures of pictures of objects and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) [21, 2, 13, 45, 46, 16, 32, 7, 39, 31, 44, 36, 9, 11, 29, 20, 19, 47] and relevant to this paper are the work described in [21, 8, 14, 39] The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective framework [8, 14, 39] or the affine framework. The latter requires either assuming parallel projection ....

D.W. Jacobs. Space efficient 3D model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992. 6 SUMMARY 19

....linear skewing in the x and y directions, yielding a transformation with eight degrees of freedom. An advantage to this class of transformations is that they model the transformations 16 (neglecting perspective effects) that would take place when capturing an image of a picture of an object [Jacobs, 1992]. The transformation is similar to weak perspective except that 2 4 R 11 R 12 R 13 R 21 R 22 R 23 3 5 is no longer constrained to be the first two rows of a rotation matrix, thus allowing skewing and implicitly scaling the object. 2 4 i x i y 3 5 = 2 4 R 11 R 12 R 13 R 21 R 22 R 23 3 5 ....

....over the remaining viewing parameters (i.e. viewing direction, thus they must represent each group from each viewing direction in their index table. Since the viewing direction has two degrees of freedom, this means that each group must be represented on a two dimensional surface in the table. Jacobs [1992] has shown that groups of five three dimensional points undergoing affine transformations can be indexed from two dimensional data by representing each group as lines in two orthogonal two dimensional tables. To determine which model 28 groups may have projected to an image group, model groups ....

[Article contains additional citation context not shown here]

D. W. Jacobs. Space efficient 3d model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

....specificity in the indices, a voting scheme must be used to collect the noisy pieces of information into meaningful hypotheses. Clemens and Jacobs [CJ91] show that the view variation of indices derived from point sets generate 2 D sheets embedded in 4 (or higher) dimensional index spaces. Jacobs [Jac92] provides a more space efficient construction which reduces the 2D sheets to two 1 D lines, each embedded in a 2 D space. These methods use hash tables to store and access their indices. Attempting to cover all possible appearances of a feature set by filling in a discrete look up table is a ....

D.W. Jacobs. Space efficient 3d model indexing. In Proceedings CVPR '92, pages 439--444, 1992.

....are sought. Solutions to these problems have been only partially successful. Two methods that have been useful are perceptual grouping and indexing. Grouping methods (e.g. 1, 2, 3, 4, 5, 6, 7] attempt to determine which features in an image belong to the same object, while indexing methods (e.g. [8, 9, 10, 11, 12]) determine which sets of model features may have projected to various sets of image features. These methods can be powerful when they are used together [9] Several studies have analyzed the power of indexing, by itself, as a means to reduce the search space for object recognition [8, 9, 12, 13, ....

D. W. Jacobs. Space efficient 3d model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439--444, 1992.

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