Motion-Separable backbone structure

Based on the theoretical analyses in RAN paper, a novel multi-scale backbone structure is designed in the paper. This structure enables the network to efficiently predict motion patterns with larger separable upper bounds by using optimized dilation convolution on high-resolution feature maps, while maintaining a capturable range of motion with low computational complexity.

To quantify the network's capacity for large deformation capturing, the accessible motion capture range is defined as:

Definition 1: Accessible Motion Range

The radius of capture range of the kᵗʰ-level registration by the registration moduleℛₖ is defined as the smallest upper bound of its accessible Deformation Displacement Field:

aₖ := min_𝐱({sup(φₖ[𝐱]_∞)})

where ·_∞ denotes the L-∞ norm of a vector, sup(·) denotes the supremum or the maximum value of a given function with varying inputs and trainable weights of networks, and 𝐱 denotes one coordinate entry of the images or Deformation Displacement Fields.

To quantify the Degree-of-Freedom limitation in the discontinuity of the estimated Deformation Displacement Field, we define the separability of the predicted motion:

Definition 2: Separability Bottleneck of Predicted Motion

The motion separability bottleneck is defined as the minimum value of the upper bound of the Chebyshev difference of a network's predicted DDF ϕ between two locations 𝐱, 𝐲 ∈ℤᵈ with the specific Chebyshev distance p ∈ℤᵈ:

Δ_∞(p) := min_(𝐱, 𝐲){sup(ϕ[𝐱] - ϕ[𝐲]_∞) : 𝐱 - 𝐲_∞ = p}

where p denotes the L-∞ distance between the two pixels.

Theorem: Regional Dependency

The upper boundary of motion difference is related to aₖ and pₖ:

∀𝐱, 𝐲∈ℤᵈ, 𝐱 - 𝐲_∞ ≥ p_(k″) + 2∑_(k′=k″+1)ᵏ a_(k′), sup(ϕₖ[𝐱] - ϕₖ[𝐲]_∞) ≥ 2∑_(k′=k″)ᵏ a_(k′); ∃𝐱, 𝐲∈ℤᵈ, 𝐱 - 𝐲_∞ < p_(k″-1) + 2∑_(k′=k″)ᵏ a_(k′), sup(ϕₖ[𝐱] - ϕₖ[𝐲]_∞) = 2∑_(k′=k″)ᵏ a_(k′);

where k″, k, denote two recursive numbers satisfying 0 ≤k″ < k, and 𝐱, 𝐲 denote two coordinate entries of images or DDFs.

Thus a Motion-Separable structure is designed with the upsampled feature maps processed by the corresponding atrous convolution layers.