This project is concerned with the Apparent Contour Reconstruction Problem, specifically in two dimensions. We base this on the three-dimensional version from Bellettini, Beorchia, Paolini, and Pasquarelli. In their research, they take a picture of a 3D object, take information from the picture, specifically the edges of the object as seen in the picture, and from this information, they see if they can reconstruct the original 3D object. This is a pivotal problem in computer vision and optics. Our project looks at a similar question, only in 2D instead of 3D.
We start with a curve in the xy-plane and have the curve project onto the x-axis. We can consider this as either casting a shadow or squishing the curve onto the x-axis. We note that there are turning points on the curve, points where the curve switches from right to left or left to right, and we call the set of these points the contour generator. These points map to points on the x-axis and we call this set of points the apparent contour. We can label the points of the apparent contour and the intervals that they define; these labels contain information about the projection, and we call this labeling the Apparent Contour Graph.
In this project, we asked the following three questions:
1) What conditions are there on an Apparent Contour Graph?
2) Given an admissible Apparent Contour Graph, is there a curve with this graph as its apparent contour?
3) How do we reconstruct that curve?
In my poster, I present the answers to these three questions. For the first question, we provide a list of conditions that the Apparent Contour Graph must satisfy. For the second question, all admissible apparent contour graphs can be realized by a curve. For the third question, we describe an algorithm that reconstructs such a curve.