Shape matching plays important roles in many fields such as object recognition and image retrieval. A recently proposed novel shape matching algorithm called the "shape context" utilizes the relationship of the points on the boundary of shape to all the other points on the boundary as a shape descriptor. The magnitude of the alignment between two shapes so described is the distance between the shape contexts of the comparison shapes. The shape context was shown to be an information rich descriptor that is invariant to translation, scale, and rotation. To determine the distance between two shapes that have been abstracted into shape contexts the problem was modeled as a bi-partite matching problem know as "the assignment problem". While performing well, the nature of the assignment problem limits the effectiveness of the matching. Using graph theory, a proof is provided that shows that certain geometrically different shapes are considered identical by the shape context algorithm. By limiting the domain of the shapes to those of continuous boundaries and using the fact that an order does exit for the points on the boundary, a more effective matching algorithm, entitled "the least cost diagonal" is presented and explored. Finally, the least cost method is applied in a real world application of automobile identification and compared to the same application but using the assignment problem model for the matching.
