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A sequential application of two global models defined on a variational framework is proposed for the detection of polygonal shapes in3D models of complex architectures.As a first step, the procedure involves the use of the Mumford and Shah (1989) 1st-order variational model in dimension two (griddedheight data are processed). In the Mumford-Shah model an auxiliary function detects the sharp changes, i.e., the discontinuities, of apiecewise smooth approximation of the data. The Mumford-Shah model requires the global minimization of a specific functional tosimultaneously produce both the smooth approximation and its discontinuities.  In the proposed procedure, the edges of the smooth approximation derived by a specific processing of the auxiliary function are thenprocessed using the Blake and Zisserman (1987) 2nd-order variational model in dimension one (edges are processed in the plane). Thissecond step permits to describe the edges of an object by means of piecewise almost-linear approximation of the input edges themselvesand to detects sharp changes of the first-derivative of the edges so to detect corners. The Mumford-Shah variational model is used intwo dimensions accepting the original data as primary input. The Blake-Zisserman variational model is used in one dimension for therefinement of the description of the edges.  The selection among all the boundaries detected by the Mumford-Shah model of those that present a shape close to a polygon isperformed by considering only those boundaries for which the Blake-Zisserman model identified discontinuities in their first derivative.The output of the procedure are hence shapes, coming from 3D geometric data, that can be considered as polygons. The applicationof the procedure is suitable for, but not limited to, the detection of objects such as foot-print of polygonal buildings, building facadeboundaries or windows contours.vThe procedure is applied to a height model of the building of the Engineering Faculty of the University of Trento and to other data.The obtained results show the effectiveness of the variational methods in handling geometric data describing complex objects.The most original and innovative contributions of this work are:  - the consecutive and coordinated use of the proper variational models (of first and second order respectively) at first in two dimensionand subsequently on the one dimensional sets arising in the first segmentation,  - the application to geometrical data that represent a quite complex architecture

the international archives of the photogrammetry, remote sensing and spatial information sciences/international archives of the photogrammetry, remote sensing and spatial information sciences

POLYGONAL SHAPES DETECTION IN 3D MODELS OF COMPLEX ARCHITECTURES