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It can also be expressed as a dual functional: TVε (u) = sup Ω u div v − χ∗ε (v) dx v → Cc1 (Γ, Rd ) , with χ∗ε (t) = ψ 1− ◦ 1 − |t| 2 for |t| < 1, else. This reduces the tendency towards piecewise constant solutions. However, as χε still grows as fast as | · |, discontinuities and consequently, the staircasing eﬀect still appears. Such an observation can generally be made for ﬁrst-order functionals penalizing the measure-valued gradient with linear growth at ◦. One approach to overcome these defects is to incorporate higher-order derivatives into the image model.
46] with ϕ = 100, T = 425. 0449. These results demonstrate the eﬀectiveness of our method for direction diﬀusion, even in cases where the staircasing eﬀect may cause unwanted artifacts. Fast Regularization of Matrix-Valued Images 33 Fig. 1. TV regularization of SO(n) data. Left-to-right, top-to-bottom: the initial estimated ﬁeld for a 4-piece piecewise constant motion ﬁeld, a concentric motion ﬁeld, the denoised images for the piecewise constant ﬁeld and the concentric motion ﬁeld. Diﬀerent colors mark diﬀerent orientations of the initial/estimated dense ﬁeld, black arrows signify the measured motion vectors, and blue arrows demonstrate the estimated ﬁeld after sampling.
5. It can be seen that for a careful choice of the regularization parameter, total variation in the group elements is seen to significantly reduce rigid motion estimation errors. Furthermore, it allows us to discern the main rigidly moving parts in the sequence by producing a scale-space of rigid motions. Visualization is accomplished by projecting the embedded matrix 34 G. Rosman et al. Fig. 2. TV regularization of SO(n) data, based on the same data from Fig. 1, with a higher-order regularity term.