From a Non-Local Ambrosio-Tortorelli Phase Field to a Randomized Part Hierarchy Tree

Abstract

In its most widespread imaging and vision applications, Ambrosio and Tortorelli (AT) phase field is a technical device for applying gradient descent to Mumford and Shah simultaneous segmentation and restoration functional or its extensions. As such, it forms a diffuse alternative to sharp interfaces or level sets and parametric techniques. The functionality of the AT field, however, is not limited to segmentation and restoration applications. We demonstrate the possibility of coding parts—features that are higher level than edges and boundaries—after incorporating higher level influences via distances and averages. The iteratively extracted parts using the level curves with double point singularities are organized as a proper binary tree. Inconsistencies due to non-generic configurations for level curves as well as due to visual changes such as occlusion are successfully handled once the tree is endowed with a probabilistic structure. As a proof of concept, we present (1) the most probable configurations from our randomized trees; and (2) correspondence matching results between illustrative shape pairs.

The work is a significant step towards establishing exponentially decaying diffuse distance fields as bridges between low level visual processing and shape computations.

Appendix

Field Computation

To keep implementation simple we combine (9a) and (9b), and multiply the inhomogeneity function f by ρ ² as scaling affects neither the geometrical nor the topological features of the level curves. In discrete setting this gives

$$ \mathbb{L}_* \big( \omega_{i,j} \big) \,-\, \frac{1}{ \rho^2 } \omega _{i,j} \, - \, \left( \frac{1}{|\varOmega|}\sum_{(k,l) \in\varOmega} \omega _{k,l} \right) + f_{i,j} \,=\, 0 $$

(10)

where $\mathbb{L}_{*}$ denotes the discrete Laplace operator; ω _i,j ≈ω(x=i⋅h _x ,y=j⋅h _y ) with h _x and h _y are spatial discretization step sizes that are taken as the pixel width. Based on our discussions in Sect. 2, we set ρ ²=|Ω|. Next, we define a relaxed scheme:

$$\omega_{i,j}^{n+1} = \omega_{i,j}^{n} + \tau A $$

where A is the left hand side of (10) and τ is the relaxation parameter selected smaller than $\frac{|\varOmega|}{ 4|\varOmega|+2}$. When implemented in parallel the ω _i,j values required for A need to be taken from the n ^th step; if, however, the values are being updated sequentially, updated values are used for faster convergence.

Saddle Point Detection

For locating saddle points, we do not rely on indefiniteness of the Hessian. Instead, we find watershed regions by calling Matlab’s watershed routine, which uses Meyer’s method [30]. We eliminate all those watershed boundaries that do not neighbor Ω ⁺. On each of the remaining watershed boundaries, the saddle point is the minimum of the restriction of ω to the respective watershed boundary. Typically, the considered watershed boundaries extend from Ω ⁺ to the shape boundary. It may be possible, however, that a watershed boundary touching Ω ⁺ bifurcates before reaching the shape boundary. In this case there are indeed two watershed boundaries; the respective saddle points are given by the respective minima after the bifurcation.

Tree Matching

Let (V _i ,E _i ), i=1,2 be two rooted trees. Let k,l∈V ₁ and m,n∈V ₂ be distinct nodes of the respective trees. The tree association graph of the two trees is the graph (V,E) where V:=V ₁×V ₂, and the graph nodes (k,m)∈V and (l,n)∈V are adjacent when the connectivity between k and l is equivalent to that of m and n. Specifically, we say (k,m)∈V and (l,n)∈V are adjacent if level(k)−level(l)=level(m)−level(n) and the length of the path from k to l in the first tree is the same with the length of the path from m to n in the second tree.

Defining the equivalence between two sets of nodes in respective trees by comparing levels and path lengths, there exists a bijection between maximal subtree isomorphism and maximal clique of the association graph of the two trees; i.e., tree matching is equivalent to finding the maximal clique in the association graph. If the trees are attributed—e.g. in our case (V _i ,E _i ,α) where α is a function that assigns an attribute vector [α ⁽¹⁾(u) , α ⁽²⁾(u)]^T to each node u in either tree—then subtree isomorphism with the largest similarity is called maximum similarity subtree isomorphism. In this case, the weighted association graph is the weighted graph (V,E,c) such that c(z) for z≡(u,v), z∈V, u∈V ₁ and v∈V ₂ is defined via a similarity measure $\operatorname{sim}(\cdot,\cdot)$ in the attribute space: $c(z) = \operatorname{sim} (\alpha(u), \alpha(v))$. The attributes and similarity measure are calculated as described in Sect. 4.1.

Suitably defining a weight matrix M using node weights c(⋅), the global maximizer of x ^T Mx gives the maximum weight clique, which is solved iteratively:

$$ x_i^{n+1} = x_i^{n} \frac{\left(M \, x^n \right)_i}{(x^n)^T \, M x^n} $$

(11)

where n is the iteration variable. The matrix M=(m _ij ) is given via a matrix B=(b _ij ) as follows:

$$m_{ij} = \underset{i,j}{\operatorname{{max}}} (b_{ij}) - b_{ij} $$

where

$$b_{ij} = \begin{cases} 0 & \text{if } i \neq j, \text{and node } i \text{ is}\\[-3pt] & \text{adjacent to node } j \\[3pt] \frac{1}{2 c(u_i)} & \text{if } i = j \\[4pt] \frac{1}{2 c(u_i)}+\frac{1}{2 c(u_j)} & \text{otherwise} \end{cases} $$

Let maximum weighted clique be C⊂V. The solution x ^∗ to the maximization problem (via the iterative scheme (11)) is expected to be

$$ x_i^* = \begin{cases} \frac{c(u_i)}{\sum_{u_j \in C} c(u_j)} & \text{if } u_i \in C \\ 0 & \text{otherwise} \end{cases} $$

(12)

The iterative scheme (11) returns an approximation to the limit vector x ^∗ in (12). What remains is how to interpret this vector. The paper [35] provides no suggestion on this. We adopt the following strategy instead of simply thresholding.

We start with an empty clique. Then starting with the node with the highest x value, we gradually add nodes to the clique in the order of decreasing x value. After each inclusion we compute expected vector using (12) and check the difference between this estimate and the actual vector returned by the iterative scheme (11). We keep adding nodes till inclusion of nodes no longer decreases the difference.