Recovering shape and motion by a dynamic system for low-rank matrix approximation in L ₁ norm

Abstract

To recover motion and shape matrices from a matrix of tracking feature points on a rigid object under orthography, we can do low-rank matrix approximation of the tracking matrix with its each column minus the row mean vector of the matrix. To obtain the row mean vector, usually 4-rank matrix approximation is used to recover the missing entries. Then, 3-rank matrix approximation is used to recover the shape and motion. Obviously, the procedure is not convenient. In this paper, we build a cost function which calculates the shape matrix, motion matrix as well as the row mean vector at the same time. The function is in L ₁ norm, and is not smooth everywhere. To optimize the function, a continuous-time dynamic system is newly proposed. With time going on, the product of the shape and rotation matrices becomes closer and closer, in L ₁-norm sense, to the tracking matrix with each its column minus the mean vector. A parameter is implanted into the system for improving the calculating efficiency, and the influence of the parameter on approximation accuracy and computational efficiency are theoretically studied and experimentally confirmed. The experimental results on a large number of synthetic data and a real application of structure from motion demonstrate the effectiveness and efficiency of the proposed method. The proposed system is also applicable to general low-rank matrix approximation in L ₁ norm, and this is also experimentally demonstrated.

Appendices

Appendix A: Proof of Proposition 1

Under (4), the vector x(t) changes with time t, and V(x(t)) changes accordingly. To study the variation of V(x(t)) with time t, the upper right Dini derivative ‘D ⁺’ is used, which is the generalization of ordinary derivative and is applicable to continuous functions [36]. Based on (4), we have

(6)

where we have applied Dini derivative in (1) to differentiate an absolute function, |g(x)|, that is, $D^{+}|g(x)|= \operatorname {sgn}(g(x))\dot{g}(x)$. In 2) we have inserted (4).

Equation (6) means that if $\|\dot{\mathbf{x}}^{T}(t)\|^{2}\neq0$, D ⁺ V(x(t))<0 will decease V(x(t)). When $\|\dot{\mathbf{x}}^{T}(t)\|^{2}=0$, x(t) will keep the same value at time t, and from (6) we can see D ⁺ V(x(t))=0, which implies that V(x(t)) attains a local minimal value (not a local maximal value due to the restriction of (6)). Thus, under the control of (4), if x(t) does not change, V(x(t)) is locally optimized. If $\|\dot{\mathbf{x}}^{T}(t)\|^{2}$ does not converge to zero all the time, we can see V(x(t)) will decrease ceaselessly. As we know, V(x(t))≥0 is lowly bounded, and V(x(t)) cannot decrease forever. Thus finally $\|\dot{\mathbf{x}}^{T}(t)\|^{2}$ will converge to zero, and accordingly D ⁺ V(x(t)) will get to 0 in view of (6). So, with the evolution of (6), V(x(t)) will go to a local minimal value.

Appendix B: Proof of Proposition 2

To address the evolution of V(x(t)) under (5), we need to study the derivative of V(x(t)) with respect to time t when x(t) evolves under (5). Like in the proof of Theorem 1, the Dini derivative is used.

(7)

where we have inserted (5) in 1). To discuss the influence of σ, we first study the following relation for σ>0 and x∈R:

(8)

Equation (8) indicates that $\operatorname {sgn}(\sigma x )=\tanh (\sigma x )$ when σ=0 or x=0, otherwise tanh(σx) become closer and closer to $\operatorname {sgn}(\sigma x )$ when σ|x| becomes larger. From (7) we have

(9)

where in (1) and (2) we have used $\operatorname {sgn}(x )= \operatorname {sgn}(\sigma x )$ for σ≠0 and (8), respectively. In terms of whether f _kl (x(t))=0 holds, there are two cases: (i) f _kl (x(t))=0, which leaves

$$\operatorname {sgn}\bigl(f_{kl}\bigl(\mathbf{x}(t)\bigr) \bigr)\frac{2}{ \exp (2\sigma|f_{kl}(\mathbf{x}(t))| )+1} \textbf{g}_{kl}(\mathbf{x}(t))=0; $$

and (ii) f _kl (x(t))≠0, which decreases $\frac{2}{ \exp (2\sigma|f_{kl}(\mathbf{x}(t))| )+1}$ with the increase of σ. Combining the two cases, we see that with the increase of σ, the probability, in which D ⁺ V(x(t))≤0 holds, will increase. Especially, when σ goes to infinity, D ⁺ V(x(t))≤0 holds definitely as

goes to zero in this case.

Appendix C: Proof of Proposition 3

We use reduction ad absurdum to prove Proposition 3. Assume it is right that V(x(t)) will keep on increasing, which means

(10)

From (3), we know $V(\mathbf{x})\triangleq\sum_{i=1}^{m}\sum_{j=1}^{n} |f_{ij}(\mathbf{x})|$. The condition that V(x) keeps on increasing means that there is |f _ij (x)| (the corresponding ω _ij ≠0) which will keep on increasing, and so are σ|f _ij (x)|. The condition that σ|f _ij (x)| keeps on increasing will make a time t _h available, and at t _h we have

(11)

In view of (9), (11) means

(12)

Both equations, (10) and (11), are all derived from the assumption that V(x(t)) keeps on increasing, and the contradiction between the both equations implies that the assumption is not right. Thus, V(x(t)) does not keep on increasing. When σ is large enough, in view of Proposition 2, a decrease of V(x(t)) occurs more often than increase, and in addition to what we have proved, V(x(t)) cannot keep on increasing; we can say that V(x(t)) falls, on the whole.