On a Decomposition Model for Optical Flow

Decomposition of optical flow on the sphere

“…In some of them the problematic g-norm was approximated by an H −1 norm [22,29]. Recently, u + v models have been extended to the R 2 optical flow setting [1]. Hierarchical models, originally introduced in [27] for image analysis, have not yet been tried in combination with optical flow.…”

“…Directly solving the optical flow equation for u is infeasible. We therefore use Tikhonov regularisation to compute an approximate solution to (1). Tikhonov regularisation consists in minimising a functional which is a weighted sum of two terms.…”

On a spatial-temporal decomposition of optical flow

“…In the context of total variation regularization, reconstructions of patterns was studied first in [25]. Here, we are implementing similar ideas as have been used before for variational image denoising [3,4,5,6,7,13,26] and optical flow decomposition [1,18,31,32,33]. However, a conceptual difference is that we aim for extracting temporal features, while, in all the cited papers the decomposition was for finding spatial components of the flow.…”

“…Registration and optical flow. The problem of aligning dynamic sequences f (•, t), t ∈ (0, 1) can be formulated as the operator equation for finding a flow Ψ of diffeomorphisms, Ψ(•, t) : Ω → Ω, ∀ t ∈ (0, 1), such that (1) f (Ψ( x, t), t) = f ( x, 0), ∀ x ∈ Ω and t ∈ (0, 1).…”

“…j , i, j = 1, 2, and thus is highly under-determined. To overcome the underdeterminacy, the problem is formulated as a constrained optimization problem, to determine, for some fixed α > 0, (9) argmin R (1)…”