Quadratic Transformation for Planar Mapping of Implicit Surfaces

Abstract

Level Set methods use a non-parametric, implicit representation of surfaces such as the signed distance function. These methods are applied to multiview 3D reconstruction and other machine vision problems that need correspondences between views of a surface. Correspondences can be found by matching surface texture patches of the size sufficient for their identification. Good matching requires precise mapping of a patch across the two image planes. Affine mapping is often used for this purpose assuming that the patches are small and nearly flat. However, this assumption is violated in locations of high surface curvature and/or when the surface is poorly textured and large windows are needed to provide enough information. Using second-order surface approximation, we derive a more precise, quadratic transformation for planar mapping of implicit surfaces. To validate this theoretical result, we apply it to correlation based, variational multiview 3D reconstruction using Level Sets. It is shown that a more detailed reconstruction can be achieved compared to the traditional affine mapping.

Appendix: Derivation of Equations (17) and (18)

Below, we briefly explain how equations (17) and (18) are derived. The explanations are not comprehensive as the main steps are only given, especially in the case of (18) whose derivation is quite involved. We consider the higher order terms of (11) that contain $\mathbf {J}_{i}^{-T} \mathbf {H}_{\alpha k} \mathbf {J}_{i}^{-1}$, α∈{x,y},k∈{i,j}. Calculating the partial derivatives in J _i and H _αk , we have

$$ \mathbf {J}_{i}^{-1} = \frac{1}{ [ \mbox{\boldmath$\nabla $} x_{i} , \mathbf {N}, \mbox{\boldmath$\nabla $} y_{i} ]} \begin{bmatrix} \mbox{\boldmath$\nabla $}y_i \cdot \mathbf {S}_v \quad& - \mbox{\boldmath$\nabla $}x_i \cdot \mathbf {S}_v \\ -\mbox{\boldmath$\nabla $}y_i \cdot \mathbf {S}_u \quad& \mbox{\boldmath$\nabla $}x_i \cdot \mathbf {S}_u \end{bmatrix} $$

(25)

and

$$ \mathbf {H}_{\alpha k} = \begin{bmatrix} \xi _{\alpha k}(u,u) \quad& \xi _{\alpha k}(u,v) \\ \xi _{\alpha k}(u,v) \quad& \xi _{\alpha k}(v,v) \end{bmatrix}, $$

(26)

where

$$\xi _{\alpha k}(u,v) = \mathbf {S}_u \cdot ( \mbox{\boldmath$\nabla $}\mbox{\boldmath$\nabla $}\alpha _k) \cdot \mathbf {S}_v + \mbox{\boldmath$\nabla $}\alpha _k\cdot \mathbf {S}_{uv}. $$

Substituting (25) and (26) into $\mathbf {J}_{i}^{-T} \mathbf {H}_{\alpha k} \mathbf {J}_{i}^{-1}$, after some straightforward re-arrangements we obtain equations (13)–(16) with

and

(27)

We seek a non-parametric form for the above quantities. After re-grouping, the first expression simplifies as

which leads to (17).

Equation (18) is more difficult to obtain. First, we will show that |N|² T _αk can be represented as

(28)

where II is the second fundamental form. Then we will prove that this expression is equivalent to

(29)

We start by introducing

$$ Z = \mbox{\boldmath$\nabla $}\alpha _k\cdot \bigl(\mathbf {S}_{uu} \mathbf {S}^u \mathbf {S}^u + \mathbf {S}_{uv} \mathbf {S}^u \mathbf {S}^v + \mathbf {S}_{vu} \mathbf {S}^v \mathbf {S}^u + \mathbf {S}_{vv} \mathbf {S}^v \mathbf {S}^v \bigr) , $$

(30)

where S _u ,S _v is the covariant, S ^u,S ^v the contravariant basis, and

$$\mbox{\boldmath$\nabla $}\alpha _k= (\mbox{\boldmath$\nabla $}\alpha _k\cdot \mathbf {n}) \mathbf {n}+ \bigl(\mbox{\boldmath$\nabla $}\alpha _k\cdot \mathbf {S}^u \bigr) \mathbf {S}_u + \bigl(\mbox{\boldmath$\nabla $}\alpha _k\cdot \mathbf {S}^v \bigr) \mathbf {S}_v . $$

Substituting this into (30), we obtain

(31)

where

and m,n∈{u,v}. Considering a coordinate transform u′=u′(u,v), v′=v′(u,v) and applying the chain rule to the derivatives, it is easy to show that n⋅S _mn are coordinates of a second-rank covariant tensor for every m,n∈{u,v}. Next, we observe that S _p ⋅S _mn can be expressed using the Christoffel symbols $\mathbf {\varGamma }^{q}_{mn}$:

$$\mathbf {S}_p \cdot \mathbf {S}_{mn} = \mathbf {\varGamma }^q_{mn} ( \mathbf {S}_q \cdot \mathbf {S}_p) , $$

where p,q∈{u,v}. For example,

$$\mathbf {S}_u \cdot \mathbf {S}_{vu} = \mathbf {\varGamma }^u_{vu} ( \mathbf {S}_u \cdot \mathbf {S}_u) + \mathbf {\varGamma }^v_{vu} ( \mathbf {S}_v \cdot \mathbf {S}_u). $$

It is known from tensor analysis [1] that a coordinate transformation exists in every point of a parametric surface that sets all Christoffel symbols to zero (geodesic coordinates). Applying such transformation, in (31) we can eliminate B _mn and C _mn , and this expression reduces to

(32)

where the first row is the original expression given in (30). (We do not introduce new notation for the transformed coordinates.) Applying to both sides the scalar product by −[N]_× and using (27), we obtain the representation (28). This representation is useful, but it contains the parametric second fundamental form. Now we will prove that the representation (29) is equivalent to (28). Consider the components of (29):

(33)

Multiplying (33) by $( \mbox{\boldmath$\nabla $}\alpha _{k}\cdot \mathbf {n})$, we obtain an expression identical to (28).