GRAFICO N° 7 RELACION ENTRE DIETA Y PROGRAMACION QUIRURGICA

The first step in trying to assess whether an optimization approach can be implemented in real-time is to compute the size of the search space to be explored and the computational complexity of various approaches to exploring that space.

Intuitively, the optimal solution at each time instant would be the camera configuration

(set of all camera setting values) that results in the optimal objective function value. However, in order to accommodate the constraints stemming from the dynamic nature of active cameras and the events they capture, as well as the special requirements of computer vision algorithms, a simple search for the best-performing camera configuration at each time instant is no longer sufficient. Instead, the optimization method and its associated performance metric must also include time as a dimension of the search space. The performance metric must not only ag- gregate camera contributions over all cameras and all ROIs, but also over time. The following examples illustrate a few constraints that can only be incorporated into the performance metric and the planning method when time is a dimension of the search space:

• Cameras cannot change their settings instantaneously [CDBF04, BNB04], so the setting values of an optimal camera configuration have to be computed not for the cur- rent time instant, but for a future time instant that takes into account the time it takes the cameras involved to apply the desired camera settings. This constraint can be taken

into account by having configuration changes incur temporal costs: during the time when cameras change their settings, they do not contribute to the capture of any ROI, resulting in worse performance when aggregated over time.

• Computer vision algorithms have special requirements: many need accurate geo- metric camera calibration such as the one described in [Zha99], and some rely on in- ternal models [BC08] that need to be re-initialized when camera parameters change. These operations require image sequences captured with constant camera settings. This constraint can be taken into account by having configuration changes also incuralgo- rithmic costs: the control method would have cameras fixed during these calibration and re-initialization procedures. Moreover, many computer vision algorithms, such as the optical flow-based algorithm in [BC08], work better with continuous image sequences captured by fixed cameras than with separate images captured with different camera settings. This constraint can be taken into account by having the performance metric accumulate camera contributions over time.

These and other requirements of computer vision algorithms affect the optimization process: instead of finding the optimal configuration at each time instant, it needs to look ahead in time to find the configuration that would become optimal at a future time instant and remain optimal for a specific duration. Consequently, for my approach, the search space is the space of possible camera configurations over time. Constraints are derived from knowledge about the cameras, the predicted ROI trajectories and the computer vision algorithm the captured images are intended for.

Let nCon f igs be the total number of possible camera configurations at each evaluation. LetnSteps be the number of steps in the plan, lStep be the average length of a plan step in cycles, andlCyclebe the length of a planning cycle in time units. The number of plan cycles

nCyclesup to the planning horizon is related to the number of stepsnStepsas follows:

nCycles=nSteps·lStep (3.2.1)

The number of time units nU nits up to the planning horizon is related to the number of stepsnStepsas follows:

nU nits=nSteps·lStep·lCycle (3.2.2)

The planning process can be represented as a perfect n-ary tree of height nSteps, with

n=nCon f igs possible alternatives to be evaluated for each plan step. The size of the search space in terms of the number of plans to evaluate,nPlans, can be computed as the number of leaf nodes in the tree:

nPlans=

nSteps

∏

k=1

nCon f igs=nCon f igsnSteps (3.2.3)

A single performance metric evaluation aggregates the contributions of each camera to the capture of each ROI over thenU nitstime units until the planning horizon. In the compu- tational complexity analysis in this thesis, I consider the contribution computationto be the atomic operation to perform when evaluating the metric. All computations are expressed in terms of this atomic operation.

LetnROIsbe the total number of ROIs. The number of camera contributions evaluated for a single metric computation is:

nContribs=nCams·nROIs·nU nits (3.2.4)

Let SC be the computational complexity of an exhaustive search. Using nPlans from Equation 3.2.3 andnContribsfrom Equation 3.2.4, the general formula forSCin terms of the number of contribution computations is:

SC=nPlans·nContribs=nCon f igsnSteps·nCams·nROIs·nU nits (3.2.5)

in the Plans Equation 3.2.3 that grows the fastest. In the following sections, I look at the complexity of exhaustively exploring two different search spaces. In Chapter 4, I show the effect rules and heuristics can have on reducing the search complexitySC.