#### Keywords

3-D Shape, 3D shape, Symmetry, 3-D Reconstruction, Inverse Problem, Constrained Optimization

#### Abstract

3-D shape recovery is an ill-posed inverse problem which must be solved by using a priori constraints. We use symmetry and planarity constraints to recover 3-D shapes from a single image. Once we assume that the object to be reconstructed is symmetric, all that is left to do is to estimate the plane of symmetry and establish the symmetry correspondence between the various parts of the object. The edge map of the image of an object serves as a good representation of its 2-D shape and establishing symmetry correspondence means identifying pairs of symmetric curves in the edge map. The vanishing points define the symmetry planes up to a scale factor. In this work, we have assumed that we know the vanishing points. In order to be able to match curves, we should first extract some meaningful curves, where the word meaningful implies that the curve should make sense to a human observer. Connected components obtained after canny edge detection are broken down, based on gradient orientation, to get small curve pieces which can be then combined to form meaningful curves. In order to obtain longer pieces of curves, we find the shortest paths between all pairs of short pieces of curves with a cost function that penalizes spatial separation and large turning angles. In the next step, we find the optimal curve matches that minimize the number of planes required to fit the final 3-D reconstruction while simultaneously ensuring that a substantial portion of the object is reconstructed. This optimization problem is converted to a binary integer program which is then solved using the Gurobi optimization framework. Symmetry and planarity in many ways represent the simplicity of an object and by applying these constraints we are attempting to reconstruct a simple 3-D shape that can explain the image.

#### Start Date

12-5-2016 3:15 PM

#### End Date

12-5-2016 3:40 PM

3-D Shape Recovery from a Single Camera Image

3-D shape recovery is an ill-posed inverse problem which must be solved by using a priori constraints. We use symmetry and planarity constraints to recover 3-D shapes from a single image. Once we assume that the object to be reconstructed is symmetric, all that is left to do is to estimate the plane of symmetry and establish the symmetry correspondence between the various parts of the object. The edge map of the image of an object serves as a good representation of its 2-D shape and establishing symmetry correspondence means identifying pairs of symmetric curves in the edge map. The vanishing points define the symmetry planes up to a scale factor. In this work, we have assumed that we know the vanishing points. In order to be able to match curves, we should first extract some meaningful curves, where the word meaningful implies that the curve should make sense to a human observer. Connected components obtained after canny edge detection are broken down, based on gradient orientation, to get small curve pieces which can be then combined to form meaningful curves. In order to obtain longer pieces of curves, we find the shortest paths between all pairs of short pieces of curves with a cost function that penalizes spatial separation and large turning angles. In the next step, we find the optimal curve matches that minimize the number of planes required to fit the final 3-D reconstruction while simultaneously ensuring that a substantial portion of the object is reconstructed. This optimization problem is converted to a binary integer program which is then solved using the Gurobi optimization framework. Symmetry and planarity in many ways represent the simplicity of an object and by applying these constraints we are attempting to reconstruct a simple 3-D shape that can explain the image.