Shape recognition using metrics on the space of shapes
David Spotts Fry · 1993
This dissertation discusses the notion that two-dimensional shapes can be viewed as points in an infinite dimensional space of shapes. Furthermore, we can place a metric on this space, giving a mathematical meaning to the idea that two shapes are similar, or close. A variety of metrics are possible, each having its own strengths and weaknesses, each constructing a different topology on the shape space. The dimensions of the space represent features inherent in the shapes. We believe that these features are what enable humans to recognize shapes, and to partition them into classes by their common qualities. This thesis discusses results from psychological tests involving humans and pigeons that indicate our sense of similarity corresponds to a metric-like view of shape space. A neural network model is presented that tackles the same problems given to the humans and pigeons. Its results give some clues about how to construct a real metric to successfully discriminate planar objects. We briefly discuss some common metrics that can be used, and highlight some metric successes other researches have had. However, this thesis mainly deals with the development and implementation of the transport metric. The transport metric, having its roots in operation research, acts on shapes as if their boundaries were collections of tiny line segments. We can imagine the segments from one shape moving into place to form a second shape. The metric uses a user-definable cost function to place penalties on these motions; the cost of the most economical rearrangement is the distance between the two shapes in shape space. A cost function that places adjustable penalties on translational and rotational movement is featured. This is applied to a database of 365 leaves drawn from 16 common New England species. Five leaves from each species are used as prototypes and the remaining 285 leaves are classified by an algorithm that uses several copies of the transport metric. The thesis concludes with the results from this experiment, a discussion of their meaning, and directions for possible future work.