Rosenbrock Problem

The Rosenbrock problem [Rosenbrock 1960] is a classic optimisation problem, also known as the banana function. It is non-separable problem. The global optimum lays inside a long, narrow, relatively flat parabolic valley. To find the valley is trivial,

however convergence to the global optimum is difficult. TheNdimensional problem

can be defined as finding a vectorxthat minimises the equation:

f(x) = N−1

∑

Many researchers take the high-dimensional Rosenbrock function as a unimodal func- tion by instinct. However, the Rosenbrock function has been shown [Shang and Qiu 2006] to have exactly one minimum for N = 3 (at (1, 1, 1)) and exactly two minima

for 46 N630–the global minimum of all ones and a local minimum nearby. Figure

5.9 illustrates the Rosenbrock valley in two dimensions.

The four methods have the expected result of optimising the 30 dimensional Rosen- brock function as shown in Figure 5.10. Simulated Annealing converges relatively slowly, but it is more capable of escaping from the attraction of local minima. The other three methods have better convergence rates, however, they are trapped after 2000 evaluations. While CMA-Anealing has better convergence than CMA and PSO, its improvements in accuracy after 2000 evaluations are minor. It is difficult for the CMA based method to move along the Rosenbrock valley. Thus it is easy for CMA- Annealing to be trapped in local minima.

§5.4 Covariance Matrix Adaptation Annealing 117

Figure 5.9: The two dimensional Rosenbrock function. Although it is a unimodal problem, finding the global minimum is very difficult. For instance, if an optimisation method starts at the initial point located at (-1.2, 1), it has to find its way to the other side of a flat, curved valley to find the optimal point.

Figure 5.10: With the exception of Simulated Annealing, the methods become trapped in local minima. However, Simulated Annealing has the slowest convergence rate. Overall, none of the four methods are able to converge reasonably close to the global minimum after 4000 evaluations.

Chapter 6

Robust Evaluation Model

A general failure of tracking using the Annealing Particle Filter like optimisation method may indicate two possibilities: 1) The number of particles or layers is in- adequate to cover large enough search space to guarantee finding reasonably good solutions. This often requires increasing the computational time or effectively reduc- ing the computational complexity to expand coverage in search space. 2) The opti- misation process is converged to the global minimum, but it does not correspond to the true gesture. This is the consequence of the evaluation being modelled inaccu- rately. For instance, the converged estimate may correspond to the global minimum of corrupted data rather than the true posture. In this chapter, we propose several im- provements on the evaluation model. Incremental Relaxation by Fast March Method aims to avoid premature convergence; Colour and Texture Incorporation attempts to introduce extra information to resolve ambiguities and improve robustness to light variations. Maximisation of Mutual Information and Gradual Sampling together al- low the evaluations to concentrate on large errors (caused by misalignments of the tracking subject), and therefore be robust to other noise errors as well as being com- putationally efficient.

6.1

Incremental Relaxation by Fast March Method

The silhouette1 is often used in markerless human motion capture to describe the

shape of the human body. However, when lacking colour and texture information, the silhouette essentially describes an image as no more than a contour line which only contains partial information from the original image. Shape ambiguities can occur along the depth direction. This can result in multiple solutions when one attempts to fit the original human body to the visual hull computed by the shape-from-silhouette technique [Laurentini 1994]. The parameterised human pose usually resides in high dimensional space, and it turns out that the solution of human motion capture is sub- ject to non-convex and multi-modal optimisation in high dimensional space.

Many different approaches have attempted to breach this non-convex and multi- modal high dimensional problem. One of these ideas that is particularly intriguing, Graduated Optimisation, assumes that we can convert the original problem to a se- quence of designed problems which are ordered from simple to complex (equivalent to the original problem). Since those solutions can be progressively obtained and used as the initialisation for each successive problem, solving this sequence of problems is much simplified compared with solving the original problem. Take the example of a continuous multi-modal problem shown in Figure 6.1. Here the original optimisation problem is transformed into a sequence of optimisation problems, such that the first problem in the sequence is convex (or nearly convex), the solution to each problem gives a good starting point for the next problem in the sequence, and the last problem in the sequence is the difficult optimisation problem that it ultimately seeks to solve. If each problem in the sequence is locally convex around the optimal value and the so- lutions are good enough inside the the local convex region, then, it can be guaranteed that the optimal solution to the final problem in the sequence will be found.

Simulated Annealing [Kirkpatrick et al. 1983] has similar behaviour when the

1_{The silhouette is more robust to illumination variation and easier to match than colour and texture}

§6.1 Incremental Relaxation by Fast March Method 121

annealling variable is gradually evaluated, causing the energy function to be trans- formed from a single peak to the multiple peaks in the landscape of the global opti- mum. In Chapter 7 of [Blake and Zisserman 1987], the authors introduce the Grad- uated Non-Convexity Algorithm. The algorithm first constructs a convex approxi- mation to the non-convex energy function, and then proceeds to find its minimum. In order to achieve the objective of a convex approximation, several constraints on line process interactions in the form of penalties levied on broken contours, etc., are added. The energy function becomes a function of the penalty intensities and a control parameter. In successive steps a sequence of energy functions generated by varying this control parameter are minimized, starting from the initial convex stage. Such a procedure is certainly intuitively appealing. It is unlikely that very general statements can be made about its effectiveness for an arbitrary non-convex cost function. But in the case of the energy functions that describe the weak string and membrane mod- els2, the algorithm can be shown to be correct for significant class of signals [Blake and Zisserman 1987].

An incremental method introduced in this section shares a similar spirit, but it works on the data domain. Although this approach is designed for our particular problem, its insight is generally applicable to any contexts and applications. The coarse-to-fine operation is performed incrementally on the data. The coarse and fine data correspond to a simple energy function (ideally with roughly convex shape) and a complex energy function (the original energy function). The proposed method seamlessly incorporates a control parameter for an isotropic distance map to APF, al- lowing incremental data relaxation to work consistently with the annealling schedule. The basic idea is that the fitness test criteria are relaxed to allow a large number of particles to survive at an early stage and encourage broad exploration. The complete fitness test is delayed until adequate information is gathered. This method enables

2_{The weak string, however, preserves discontinuities without any prior information about their exis-}

Figure 6.1: Graduated Optimisation for a continuous multi-modal problem: the multi- modal problem is reshaped and simplified as a sequence of easier subproblems. The solution of the current problem provides a good initial position to the successive prob- lem, which simplifies optimisation dramatically. (courtesy of [Wikipedia 2011])

a better chance than APF to escape the attraction of local minima and converge to the global minimum. As shown in Figure 6.2, the contour of the distance map is relaxed (thickened), allowing a larger number of particles to survive and explore a larger search space. As it proceeds, the contour is gradually contracted to approach the original silhouette shape. Probabilistically, it selects better fit particles, and grad- ually concentrates them on the region which contains the global minimum.

(a) (b) (c) (d) (e)

Figure 6.2: Level Set and Silhouette Images: (a) the original silhouette, (b) outwards relaxed silhouette, (c) outwards and inwards fixed silhouette, (d) outwards/inwards relaxed contour, and (e) the original contour

On the basis of this idea, an extra termDc(rel(yt,m),xt)is introduced for the pur- pose of relaxing the fitness test criteria. The energy functionE(yt,xt)is then given by:

§6.1 Incremental Relaxation by Fast March Method 123 1 Nv Nv

∑

where Nv is the number of views. Ds(yt,xt)measures differences between the ob- served silhouetteyt and the silhouette generated by the particlext. Dc(rel(yt,m),xt) measures how well the contour (generated by the particlext) fits the relaxed contour of the isotropic distance map.αis a factor3_{to adjust the influence of the relaxing term}

on the energy function.mcontrols the degree of relaxation. As the annealing schedule proceeds, the contour of the isotropic distance map is changed from the thick curve (corresponding to the loosest criteria) at the beginning to the original thin curve (cor- responding to the original criteria) at the end shown in Figures 6.2d and 6.2e, respec- tively. The relaxing operationrel(yt,m)is computed by Fast March Method, whose details are described in the next section. Furthermore, noisy silhouette images can be smoothed by marching Fast March Method outwards and then inwards as shown in Figures 6.2a, 6.2b and 6.2c, respectively. Overall, the relaxed APF algorithm in a typical annealing phase is outlined in Algorithm 10.