A stochastic subspace system identification algorithm for state-space systems in the general 2-D Roesser model form

ABSTRACT

The stochastic realisation problem is associated with fitting a state-space model to a given data-set so that the second-order statistics of the output of the system match those of the data. This problem has been well studied and documented in the 1-D case, but unfortunately not so in the 2-D case, despite the similarities. Until now, the main reason behind the lack of 2-D stochastic realisation algorithms is the fact that there is a strong coupling between horizontal and vertical states, which are difficult to separate. The only known way to separate the states is to assume the model to be causal, recursive, and separable-in-denominator (CRSD). Nevertheless, there is currently no known algorithm that can solve the general 2-D stochastic realisation problem. Such problem arises naturally in image modelling, where, given an image, one needs to fit a 2-D Kalman filter model to it. In this paper, we introduce a 2-D stochastic realisation algorithm for state-space models in the general 2-D Roesser form without using the CRSD assumption. The algorithm constructs a positive real 2-D Kalman filter model. We test the algorithm with three case studies, one of which is an image example.

KEYWORDS:

• 2-D stochastic subspace system identification
• general 2-D stochastic Roesser model
• N4SID
• PO-MOESP
• CCA

Acknowledgments

The authors would like to thank the reviewers for the constructive criticism that led to a significant improvement of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. It is well known that images are nonstationary processes, however, under mild violation of these assumptions, the 2-D 4SID (4SID stands for State Space Subspace System I Dentification) algorithm performed very well with real images, as seen in Section 7.

2. Here, we make reference to the fact that since the vertical state estimates are known, they appear as deterministic inputs even though they are indeed stochastic terms. Thus, in the context of the problem, it can be seen as a deterministic–stochastic system identification problem (Van Overschee & De Moor, 1996).

3. Note that x^v _{k, m} = A ₃ x^h _{k, m − 1} + A ₄ x^v _{k, m − 1} + w _{k, m − 1} ^v has dynamics that imply correlation with x^h _{r, s} , unless we restrict {k, m} to k ≥ r and m = s, which will be guaranteed in horizontal data processing.

4. This is the case since the only data available to estimate ${\widehat{x}}_{r,s}^v$ is y _k,t , for 0 ≤ k ≤ r and 0 ≤ t ≤ s, thus ${\mathcal{F}}_r\left(s\right)$ -measurable.

5. In order to avoid an abuse of notation when there is no time index involved, and for lack of better words, we still use the terms past and future as is commonly used in time domain subspace system identification.

6. These equations are needed to prove the main results. However, for the 2-D system identification algorithm, we only need (41d(41d) $$\begin{array}{ccc}\hfill {\mathbf{Y}}_p^h& =& {\Gamma }_i^h{\widehat{\mathbf{X}}}_p^h+{\Gamma }_i^{vh}{\widehat{\mathbf{X}}}_p^{vh}+K_i^h{\mathbf{E}}_p^h\hfill \end{array}$$(41d) )–(41f(41f) $$\begin{array}{ccc}\hfill {\widehat{\mathbf{X}}}_f^h& =& A_1^i{\widehat{\mathbf{X}}}_p^h+{\Phi }_i^{vh}{\widehat{\mathbf{X}}}_p^{vh}+{\mathcal{L}}_i^h{\mathbf{E}}_p^h,\hfill \end{array}$$(41f) ).

7. For zero mean processes, A/B = A D B ^⊤(B D B ^⊤)^{− 1} B denotes the orthogonal projection of $\mathbf{A}$ onto $\mathbf{B}$, where $\mathbf{D}$ is a diagonal weight matrix.

8. Note that ${\tilde{R}}_{i,i}^h=R_{i,i}^{hs}$, ${\tilde{\Omega }}_i^h={\Omega }_i^{hs}$ and ${\tilde{\Delta }}_i^h={\Delta }_i^{hs}$ when $S^{hd,vh}=0_{n_h\times n_{v}i}$.

9. There is no need to compute the Q part, just the triangular factor L will suffice, thus achieving a significant reduction in the number of computations.

10. Here, we use reshape to change the shape of a matrix, i.e. reshape {M,n,m} takes vector M ∈ R ^mn and converts it into an n × m matrix.

11. We should point out that the development of these metrics led Alan Bovik and three of his colleagues to receiving a Primetime Emmy Award for Outstanding Achievement in Engineering Development from the Television Academy.