3D Reconstruction Using Multiple View Stereo and a Brief Introduction to Kinect

Why Projective Geometry?

Image formation is a perspective transformation from 3D world scene to 2D plane and most of the Euclidean geometric property is lost due to this transformation. One of the most familiar properties is that parallel lines which never meet, or some fancy way of saying they meets at infinity in Euclidean space, doesn’t hold true after perspective transformation in image as shown in Figure 1. This figure shows two parallel rail lines meet at a point called vanishing point. This phenomenon distinguishes projective geometry from Euclidean geometry. The notion of distance and parallelism is destroyed in projective transformation. When the notion of distance is removed from Euclidean geometry keeping only parallelism, the structure becomes affine. Further, removing the preservation of parallelism property gives rise to projective structure. A more comprehensive detail can be found in (Hartley & Zisserman, 2000).

Figure 1.

(a) Parallel lines meet at vanishing point, (b) construction of image point in perspective drawing and vanishing point (image courtesy: geometric invariance in computer vision by Zisserman and Mundy), (c) sequence of two or more perspective projections (image courtesy: geometric invariance in computer vision by Zisserman and Mundy), (d) projective model (image courtesy: geometric invariance in computer vision by Zisserman and Mundy)