A New Convexity Measure With Distance Weighted Area Integral

Shape analysis is one of the most popular fields in computer vision,and its applications range from pattern recognition,calibration,shape decomposition,image registration,etc.It is often the case that there are interferences,such as distortion,occlusion,and noises in shape extraction.Accordingly,for theoretical studies and practical applications,it is significant to find a shape descriptor invariant to translation,scaling,and rotation,and robust to noise.So far,geometric characteristics,such as compactness,linearity,rectangularity,and convexity,have been widely explored domestically and abroad.Among them convexity,a global geometric characteristic indicating the convex degree of a shape,plays an important role in visual perception for its conspicuous visual feature.Existing convexity measures can be divided into three groups:the area-based,boundary-based,and probability-based convexity measures.Due to its computational simplicity and robustness against noise,the area-based measure has been used widely.Nevertheless,this method only considers the area of a shape and its convex hull,and some shapes will be measured with an unreasonable convexity value.This paper proposes a new area-based measure which is based on the following assumptions and easy to understand.First,we assume that dents of an arbitrary nonconvex shape are formed by region collapsing from its convex hull and different ways of collapsing have different influences on the original shape.If an area collapsed has a larger influence on the original shape,it is said that the shape has a smaller convexity,and vice versa.Second,this paper is dedicated to an implementation based on a notion,named the Distance Weighted Area Integral(DWAI),which distributes the influence of the dent in light of the distance between each patch of the dent and the geometric center of convex hull(GCCH).The further the patch from the GCCH,the lower influence it has.By adjusting the influence factor ? we can control the influence of different areas.For example,if we increase ? it will emphasize the influence of dent position.If we want to emphasize the influence of dent area,we can decrease the value of ?.When ? equals to 0,the new measure will degenerate into the traditional area-based convexity measure.Therefore,the new measure can completely replace the traditional area-based one.Besides,this paper also presents a new 3D convexity measure by generalizing the concept of DWAI to 3D space,which is perceptually more reasonable than Lian's method.By setting different parameters and extracting statistical information,this paper also proposes CS(Convexity Statistic),a three-dimensional model shape descriptor based on convexity.Compared with CD(Convexity Distribution),a shape descriptor based on convexity and propounded by Lian,CS has a better performance on retrieval.In summary,the theory deduction in this paper has verified the validity of the new method,and various experimental results have successfully indicated the method's excellences over other dominant convexity measures in either qualitative or quantitative aspects.