Extracting a polyhedron from a single-view sketch: Topological construction of a wireframe sketch with minimal hidden elements

Abstract

An essential prerequisite to construct a manifold trihedral polyhedron from a given natural (or partial-view) sketch is solution of the “wireframe sketch from a single natural sketch (WSS)” problem, which is the subject of this paper. Published solutions view WSS as an “image-processing”/“computer vision” problem where emphasis is placed on analyzing the given input (natural sketch) using various heuristics. This paper proposes a new WSS method based on robust tools from graph theory, solid modeling and Euclidean geometry. Focus is placed on producing a minimal wireframe sketch that corresponds to a topologically correct polyhedron.

Introduction

The subject of this paper is related to 3D geometric modeling and to sketch-based CAD, with an emphasis on natural (or partial-view) sketch (Fig. 1(a)), i.e., a sketch without any hidden lines. More specifically, this work deals with the problem of automatically constructing a “polyhedron from a single natural sketch (PSS)”. The principal sub-problem of PSS is topological construction of a “wireframe sketch from a single natural sketch (WSS)”, for which a solution is presented here.

Regarding published research on the PSS/WSS problem, one observes the following: numerous papers have appeared from 1980 until today on this problem, offering various solutions; see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9] and references therein. Despite that, it is fair to say that existing solutions are far from satisfactory, as, even in 2008, new methods are appearing (see, e.g., [10] and references therein) for the plainest case of the PSS/WSS problem, where the polyhedron (to be constructed) has only planar faces and is trihedral. Current methods view PSS/WSS as “image-processing” or “computer vision” problems, where emphasis is placed at analyzing the given input (sketch) using tools and techniques lacking a robust mathematical foundation. More specifically, many published methods are based on the “line-labeling (LL) methodology” (initiated in [11], [12], [13]), which tries to associate each sketch line to a “label” (with possible values=“convex”, “concave”, “occluding”) so that the whole set of labels is compatible with a predetermined set of “labeling rules”. Numerous authors have explored this idea (see [5], [14], [15], [16], [17], [18] and references therein). After presenting, in a series of papers (see references in [17]) improved versions of the methods in [5], Varley et al. conclude [17] that LL has very little to offer towards solution of the PSS/WSS problem.

It must be emphasized that the predominant “image-processing”/“computer vision” methodologies not only place focus on poor heuristics like LL, but also they clearly place PSS/WSS out of the correct context, which obviously is “graph theory” (for analyzing the sketches) and “3D solid modeling” (for constructing the corresponding polyhedron). Recently, Cao et al. [10] published a paper on the PSS problem that indeed moves away from the classical “image-processing” methodology and adopts “graph theory” to solve the WSS problem. The proposed method includes two main steps: (a) construction of an initial hidden structure, and (b) reduction of this structure to the “most plausible one” according to “human visual perception” (these two main steps correspond to Steps 3 and 4 in the description given in Section 5.1. of [10]). Unfortunately, this method is far from being complete as both steps (a) and (b) are based on heuristic criteria/processes lacking a solid justification, e.g., realization of step (b) is solely based on the heuristic criterion “the human visual system … tends to interpret a figure in such a way as to produce an object that is as symmetrical as possible” (see Section 5.4 in [10]). This approach is problematic as it cannot handle objects that are far from symmetric; indeed the test sketches used in [10] are either symmetric or “almost symmetric”.

The present research aims at producing a PSS algorithm that is free from the shortcomings of the “image-processing” methodology as well as of a technique like [10] based solely on “graph theory”. Indeed, an improved WSS algorithm is presented here employing robust tools from graph theory, 3D solid modeling and Euclidean geometry. A detailed topological analysis of sketches is given, followed by an efficient technique to complement a given “visible sketch” with appropriate hidden parts. The scope of the present research covers manifold polyhedra without holes, which are also trihedral. The employed sketch is a natural sketch, where:

• No element of the natural sketch causes two or more visible regions of it to correspond to one visible region in the wireframe sketch to be constructed.

• At most one T-junction (: this is defined in Section 2) exists in each region of the sketch.

The proposed methodology aims at producing a polyhedron, which is CAD-usable, i.e., a valid 3D solid model. This objective, combined with the fact that no information is available regarding the hidden part of the polyhedron, leads to the conclusion that this hidden part should be minimal (e.g., a single planar face) and at the same time sufficient to define a valid solid model. This “minimal-completion strategy” implies that from the given natural sketch a wireframe sketch should be derived where the number of hidden lines/junctions/regions is as small as possible.