On the convergence of planar curves under smoothing

Curve smoothing has two important applications in computer vision and image processing: 1) the curvature scale-space (CSS) technique for shape analysis, and 2) the Gaussian filter for noise suppression. In this paper, we study how planar curves converge as they are smoothed with increasing scales. First, two types of convergence behavior are clarified. The coined term shrinkage refers to the reduction of arc-length of a smoothed planar curve, which describes the convergence of the curve latitudinally; and another coined term collapse refers to the movement of each point to its limiting position, which describes the convergence of the curve longitudinally. A systematic study on the shrinkage and collapse of three categories of curve models is then presented. The corner models helps to reveal how the local structures of planar curves collapse and what the smoothed curves may converge to. The sawtooth models allows us to gain insights regarding how noise is suppressed from noisy planar curves by the Gaussian filter. Our investigation on the closed curves shows that each curve collapses to a point at its center of mass. However, different curves may yield different limiting shapes at the infinity scale. Finally, based upon the derived results the performance of the CSS technique in corner detection and shape representation is analyzed, and a fast implementation method of the Gaussian filter for noise suppression is proposed.