Pattern Recoglnition, Vol. 26, No. 11, pp. 1603 1618, 1993 Printed in Great Britain 0031 3203/93 $6.00+.00 Pergamon Press Lid Pattern Recognition Society THE PLANAR THREE-LINE JUNCTION PERSPECTIVE PROBLEM WITH APPLICATION TO THE RECOGNITION OF POLYGONAL PATTERNS VINCENZOCAGLIOTI Artificial Intelligence and Robotics Project, Dipartimento di Elettronica, Politecnico di Milano, P. za Leonardo da Vinci, 32, 20133 Milano, Italy (Received 10 September 1991; in revised form 29 May 1992; received for publication 4 May 1993) A~tract--A planar pattern called a three-line junction is considered and a straightforward method is pres- ented for its localization. This method can be utilized to recognize polygonal patterns of minimum com- plexity from single perspective views. Three-line junction Polygonal patterns Inverseperspective Compatibility surface Recognition uncertainty Approximations of perspective Orthoperspective Aftine approximation error 1. INTRODUCTION Many approaches to the recognition of a modeled object from an unconstrained viewpoint involve the matching between a pattern extracted from the image and a homologous pattern contained in the object model. This matching process becomes simpler as the complexity of the image pattern (i.e. the number of independent parameters needed to univocally specify it) increases: polygonal patterns of complexity at least nine (notice that the complexity of an n-vertices polygon is 2n) can be simply recognized") by exploiting the cross ratio invariance. ~2) Since the extraction of an image pattern is sensitive to noise and occlusions, we are here concerned with patterns of minimum complexity. Generally, the minimum complexity required for pattern recognition is 7. In fact, while a given 6-complex image pattern can constitute an image of any homologous pattern from a suitable viewpoint, a given 7-complex pattern can be matched with almost none of its homologous patterns. A connected planar pattern of minimum complexity is shown in Fig. l(a). Image patterns of subminimum complexity, such as triangles ~a)or line triplets, c*)cannot be matched directly: in this case the recognition is based on laborious cumulative approaches. The matching of a minimum complexity pattern often requires the localization of (a part of) it. 13- s) A typical approach to the matching of a polygonal pattern (A', B', C', q') of minimum complexity with a homologous model pattern (A, B, C, q) involves the localization of the triangle (A, B, C) using (A', B', C') as its image. Thereafter the semi-line q' is matched against the projection of q from the determined position. This paper addresses the recognition of polygonal patterns of minimum complexity from single perspective views: a straightforward method for the localization of a planar pattern called "three-line junction" (3LJ) (see Fig. l(b)) is presented and applied to the recognition of polygonal patterns of minimum complexity. This method, which reduces to the solution of a two-degree equation while working in exact perspective, is then compared with other recognition methods with respect to two aspects: analytical complexity and recognition precision. Triangle localization methods working in exact per- spective, while exploiting the recognition precision of the exact perspective, require the solution of a four- degree equation. 15'6)On the other hand, the analytical complexity of triangle localization methods working in affine approximations of perspective, ~7) such as ortho- perspective, 16~ is comparable with that of the presented method. However, their recognition precision is worst due to the adopted approximation of the perspective transformation. In fact the recognition error relative to our method, which works in exact perspective, is due to the errors on the estimation of the parameters of the image features, while for methods working with approximate perspective an additional contribution to the recognition error derives from the approximation introduced by the adopted relation between image and scene. The recognition of an image pattern, which is il- lustrated in Section 3, is based on the determination of the orientation parameters of the model pattern as a function of some of its geometric parameters (Sec- tion 2). In Section 4 the recognition precision is in- troduced, and this feature is evaluated for recognition methods based on the orthoperspective approximation of perspective. 1603