Smooth and Inextensible Surface Reconstruction

Although the strategem of maximizing the sum of depths described in the previous section gives reasonable results, it is merely a heuristic, not based on any valid principle related to surface properties. We therefore consider next a new formulation based on the principle of surface inextensibility.

Let the surface be modelled as a function , mapping the planar template to -dimensional space. The inextensibility constraint is equivalent to saying that the map must be everywhere a local isometry. This condition may be expressed in terms of its Jacobian. Let be the Jacobian matrix evaluated at the point . The map is an isometry at if the columns of are orthonormal. This local isometry can be enforced for the whole surface with the following least-squares constraint:

In practice, we consider a discretization of the quantity in equation (7.7), namely

where is a set of 2D points in the template image space taken on a fine and regular grid (for instance, a grid of size ). This term measures the departure from inextensibility of the surface .

Our minimization problem is then to minimize this quantity, over all possible surfaces, subject to the projection constraints, namely that point projects to (or near to) the image point , for all .

The problem just described involves a minimization over all possible surfaces. Instead of considering this as a variational problem over all possible surfaces, we consider a parametrized family of surfaces. For this purpose, we chose Free-Form Deformations (FFD) (162) based on uniform cubic B-splines (61). All the details on this parametric model have been given in section 2.3.2. Here, we just precise the notation used in this chapter. Let be the parametric FFD, parametrized by a family of 3D points , which act as `attractors' for the surface.

For a point in the template, the surface point is explicitly given as

(7.9) |

The functions are the B-spline basis functions (61) which are polynomials of degree . If point is fixed and known then the surface point is expressed as a linear combination of the points

Surface Reconstruction as a Least-Squares Problem

By replacing
by
** ** in equation (7.6)
we may arrive at a constraint:

We may then formulate the optimization problem as minimizing the inextensibility cost given in equation (7.8) over all choices of parameters

To simplify the formulation of the reprojection error, we introduce the depths as subsidiary variables, for reasons that become evident below. This is not strictly necessary, but reduces the degree of the reprojection-error term. The minimization problem now takes the form:

where , , are the

The inextensibility term has been described previously. We now describe the two other terms in equation (7.11).

(7.12) |

which measures the distance between the point on the surface and the point at depth along the ray defined by .

where is the -th coordinate of the point, and is the Frobenius norm of the Hessian matrix. With FFD, there exists a simple and linear closed-form expression for the bending energy:

(7.14) |

where is a symmetric, positive, and semi-definite matrix which can be easily computed from the second derivatives of the B-spline basis functions.

An alternative is to modify the problem (7.6),
expressing
in terms of
the required parameters
**, according to
**** **.
Then one may solve for
** directly using SOCP.
If necessary, the linear smoothing term of equation (7.13) can
be included in equation (7.15).
**

Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010) - Florent Brunet

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