Appendix: Proof of Spatial Derivative of Clear Raindrop

1. Geometric Derivative of Clear Raindrop Continuous mapping As shown in Fig. 1 (a), the appearance of each raindrop is a contracted image of the background, as if it is taken from a catadioptric camera. The numeric values indicated in Fig. 1 (c) are the contraction ratios between the original image and the image inside the raindrops, calculated from the black and white patterns. The contraction ratio is around 20 to 30, meaning that the motion observed inside the raindrops will be 1/30 to 1/20 smaller than the other areas in the image. This significant motion difference can be used as a clue for raindrop detection. Mathematically, for a given raindrop, we describe the smooth expand mapping start from raindrop area Ωr into the environment scene Ωe as φ: φ : Ωr → Ωe. (12)