Global optimization-based evolution of coupled contours

Here, we extend the optimization theory introduced by Yuan et al.[22] to evolve the two contours _CLIB and _CAB, which minimizes the energy function (4.8).

To motivate the evolution theory of coupled contours _CLIB and _CAB, we utilize the key observation of the single contour evolution: For the region _C and its com- plementary Ω_\C during each discrete time frame, the region shrinkage _C− actually corresponds to the expansion of the complementary region Ω_\C. Hence, the opti- mization principle (4.10) amounts to achieving the minimum total cost of the two region expansions w.r.t. _C and Ω_\C, which can be applied for the evolution of the coupled contours_CLIB and_CAB. Note that,_CLIB and_CAB partition the given image domain Ω into three regionsR_l,w,b(see Fig.4.2(a)). Given the current contours_CLIBt and _CABt at time t, the region changes w.r.t. _CLIBt and _CABt can be expressed using the region expansions R+_l,w,b w.r.t. the current regions _Rt_l,w,b (see Fig. 4.2(b)). Let e+_l,w,b(x) be the cost functions corresponding to the region expansions R+_l,w,b, such that e+_l,w,b(x) are the first-order derivatives of the intensity PDF matching function (4.7) [29] w.r.t. u_l,w,b(x), i.e. e+_i (x) = 1 2V_i X z_∈Z q h_i(z) ˆh_i(z)₋ s ˆ h_i(z) h_i(z)K(z−I(x)) (4.14)

whereV_i =R_Ωu_idx,i=l, w, b, is the volume of the current region_Rt_i. Similar to the method proposed by Yuan et al. [22], an additional distance term dist(x,_Rl,w,b)(x), which is the distance fromxto the boundary of the region _Rl,w,b, is added to the cost e+_l,w,b(x), respectively to constrain the contour movements during each time step. The cost functions (4.14) guide the evolution of the contours towards the minimization of the intensity PDF matching function (4.7).

We propose to propagate the two contours _CLIBt and _CABt to their new positions from timet tot+ 1 by achieving the minimum total cost of region expansions, i.e.

min CLIB,CAB Z R+_l e+_l (x)dx+ Z R+w e+_w(x)dx+ Z R+_b e+_b (x)dx (4.15) + Z ∂_CLIB g(s)ds+ Z ∂_CAB g(s)ds

subject to the geometrical inter-surface constraint (4.1). As shown in Fig. 4.2(b), the shrinkage region of the wall matches the union of the expansion region of the lumen and the expansion region of the background. However, we formulated (4.15) based on its current label after the evolution. For example, a voxel that moved to the lumen region pays the cost e+_l (x) irrespective of whether it is changed from the wall region or background region, which is counted once in the final energy function. We show the optimization problem (4.15) can be globally and exactly solved by means of convex relaxation. In order to achieve this, we first show that (4.15) can be equally reformulated as a spatially continuous min-cut problem with the linearly ordered labels proposed in [27].

We define the label assignment functions

D_i(x) :=

(

e+_i (x), wherex /_{∈ R}t_i

0, otherwise , i=l, w, b . (4.16)

In view of the labeling functions (4.3) and (4.4) of the two contours _CLIB and _CAB, we have

Proposition 1 The optimization problem (4.15) can be equivalently expressed by the following continuous min-cut problem with the linearly ordered labels:

min u1,2(x)∈{0,1} hu₁, D_li+_hu₂₋u₁, D_wi+_h1₋u₂, D_bi + X i=1,2 Z Ω g(x)_|∇u_i|dx (4.17)

subject to the label order constraint (4.5), i.e. u₁(x)_≤u₂(x).

proof 2 In view of (4.16), the label assignments functions D_l,w,b(x) = 0, for any x_{∈ R}t_l,w,b; and D_l,w,b(x) =e+_l,w,b(x), otherwise.

Hence, the integral _hu_i, D_ii, i = l, w, b, provides the exact value R

R+i

e+_i (x)dx. Moreover, the weighted total-variation functions in (4.17) correspond to the weighted perimeter/area of the contour _CLIB and _CAB. Then, the proposition is proved.

4.2.3

Convex relaxation and continuous max-flow approach

Bae et al. [27] proved that the combinatorial optimization problem (4.17) can be solved globally and exactly by its convex relaxation

min u1,2(x)∈[0,1] hu₁, D_li+_hu₂₋u₁, D_wi+_h1₋u₂, D_bi + X i=1,2 Z Ω g(x)_|∇u_i|dx (4.18)

subject to the ordered label constraint u₁(x)_≤u₂(x).

Note that, the binary constraint u_1,2(x) _{∈ {}0,1_} on the values of the labeling functions u₁(x) and u₂(x) in (4.17) is relaxed to be u_1,2(x) _∈ [0,1] in (4.18), which amounts to the convex optimization problem (4.18). Moreover, we have the following.

Proposition 3 Given the global optimum u∗_1,2(x) of the convex relaxation problem (4.18), the threshold of u∗_1,2(x)by any value γ _∈(0,1] provides the global binary opti- mum of the combinatorial optimization problem (4.17)using the thresholding theorem proposed by Chan et al. [40].

proof 4 Its proof directly follows [27].

The existence of the global and exact optimum to (4.17) indicates that the two contours _CLIBt and _CABt can be propagated to their globally optimal positions from the discrete time t tot+ 1, while preserving the inter-surface order constraint (4.1).