Experimental issues and practical concerns

Occlusion difficulty

It is clear that most of the object specific performance differences that we observe in these experiments must be attributed to what we call the “occlusion difficulty”, i.e. how much does a particular object suffer from the artificial degradation it has been subjected to. For example, consider object 12 or object 20 with respect to object 1 in Tab. 5.2. Whereas it is extremely difficult, even for a human observer, to estimate the correct poses of the degraded images 12 and 20, this is a fairly simple task for object 1. Even so, it is interesting to note that there are large differences in performance on these objects (RMMS, Tab. 5.1 and Fig. 5.4): whereas the pose of object 12 was correctly estimated for all 72 poses – indeed very impressive, this number is only 20 (27.78 %) for object 20. For 30 % occlusions, the number of correct pose classifications for object 20 becomes 67 (93.06 %). It is therefore quite reasonable that the degradations we have applied to the images – even though identical – lead to quite heterogeneous results across the objects.

Computational time

The drawback of the RMMS model is that the reconstruction becomes fairly cost intensive. We have not made any theoretical analysis of the computational complexity of these algorithms,

86 Experiments
        
      
   
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Figure 5.7: Top: percentage of variance explained by 15 and 30 eigenvectors (Eq. 3.5, p. 35) for each of the COIL-20 objects. Bottom: absolute variance that is not explained by 15 and 30 eigenvectors.

5.5 Conclusion 87

Occlusion size Object 15 eigenvectors 30 eigenvectors

RML RMMS RML RMMS 10 % 1 1.6 8.9 2.9 24 8 1.2 9.6 3.3 29 30 % 1 1.3 11 3.6 36 8 6.8 88 11 344 40 % 1 1.6 12 3.5 37 8 6.0 109 11 312

Table 5.3: Example of computation times in seconds as experimentally observed for object 1 and 8 at pose 0. We see that the computation time depends strongly on the object and the occlusion. Generally, the computation time increases with increasing occlusion size and with an increasing number of eigenvectors used in the model. ML reconstruction is always fast (∼ ms).

nor any convergence analysis. However, it can be seen from Eq. 4.34, p. 69, that the subspace mean shifting (prior term) depends on the number of samples and the subspace dimension. With the modified noise variance matrix due to the robust weights, there is also a dependence on the observation dimension (data term in Eq. 4.34). Furthermore, the computation depends on the type and severity of degradation of the image. In practice we observed that the reconstruction times varied greatly with the RMMS algorithm. Some examples that were obtained with our implementation are shown in Tab. 5.3 (code in C++, running on a Linux operating system over a 2.4 GHz Pentium 4 with 2GB of memory).

Other models

In these experiments we chose to concentrate on the three models ML, RML and RMMS instead of evaluating every model that was derived in Ch. 4 (and listed in Tab. 4.1, p. 72). This was done because of practical concerns (too many results can occlude principal trends), but also because we considered that these models would be most interesting to contrast in this particular experiment. The GMAP and RGMAP models were not considered because we already knew that the subspace distribution was far from Gaussian. The “Weighted” (non- isotropic) models were not considered because we have not derived the maximum likelihood estimates of the model parameters (notably the image generating matrix W ) for all these models yet. For the models we have used, we considered that the eigenvectors were sufficiently good estimates of W since these models are closer to the PPCA model and for this model the eigenvectors asymptotically becomes the maximum likelihood estimate of W .

5.5

Conclusion

We have performed a series of experiments on the COIL-20 database. We conclude from these that a general learning approach to object modeling is indeed difficult. This is evident from the high variability of results across the different objects of the database. The results do however show that on average, the added prior modeling does significantly improve performance. In general, added modeling does in many cases lead to a more constrained model. However, in our case, because of the general form of this prior distribution, we have quite a versatile model

88 Experiments

that does not seem to be constrained in any particular way with respect to the other models we have considered. Several paths for further investigation are open. In particular, we would like to continue investigating the results for the objects 15, 16 and 17 of the database. It would also be interesting to compare these results to the non-isotropic models developed in Ch. 4.

These experiments show that the improved modeling of images as well as the accurate reconstruction of new images under the model is important in order to obtain high recognition performances when the images are highly degraded. In the next part of this document, we shall be concerned with the comparison of images of brain perfusion of potential patients with an atlas of normal subjects. These images might potentially contain large zones of abnormal brain perfusion. As we shall use the residual between the observed image and the reconstructed image as a measure of abnormality, it is important that the reconstruction task be performed as accurately as possible. The recognition task in this medical setting can be interpreted as the problem of determining the nearest equivalent normal image to the image under evaluation and corresponds to solving the reconstruction problem.

Part III

Brain Perfusion Atlas: Construction

and Evaluation

Chapter 6

Models and preprocessing: overview

and state of the art

The goal of this work has been the creation of an atlas of brain perfusion as seen in SPECT images (Ch. 2). We chose to do this using statistical models from the domain of computer vi- sion/pattern recognition (Part II). In this chapter, we start by defining an atlas and explaining the connection to the problem of pattern recognition. We then review statistical models used in the analysis of functional brain images. In order to situate these models, it is necessary to understand the different clinical or research questions for which answers are sought. Attempts to answer these questions lead to different experimental designs. An atlas can be considered to be a special case in experimental design. The notion of experimental design comes from statistical hypothesis testing and has an analogue in the formulation of statistical pattern recognition systems. Finally, we review image preprocessing techniques that are necessary in order to make statistical modeling possible: registration, segmentation and normalization.

6.1

Atlas, definition

What is a probabilistic atlas of brain perfusion? An atlas can best be described by answering the following three questions:

1. What does it describe? 2. How does it describe it? 3. For what can it be used?

For example a geographical map is a description of a part of the earth. It describes the spatial distribution of hills, forests, roads, houses etc. This is done by means of an image using specific symbols to signify different objects or elements being described. Finally the map can have a multitude of uses, like finding the shortest path between two houses.

Neurologists are interested in atlases of the brain. There exist anatomical and functional atlases. Anatomical atlases are maps describing the shape, size, spatial extent and relative locations of different brain structures. Functional atlases uses the same features to describe different physiological processes in the brain and how they are related. A functional atlas of the brain contributes to explain how the brain works, a question that has thrilled human

92 Models and preprocessing: overview and state of the art

curiosity for at least a few thousand years1_!

In our application, we can answer the questions posed in the introduction of this section as follows: (1) The atlas describes patterns of brain perfusion. By perfusion pattern, we understand the level of blood flow (intensity) and the spatial distribution of it in the brain. Measures of these brain perfusion patterns are obtained from SPECT images. (2) The atlas describes patterns of brain perfusion by statistically modeling the intensity values observed at each voxel. (3) The purpose of the atlas is to detect abnormal perfusion patterns in an image by assigning to each voxel a measure of normality.

We thus arrive at the following definition: A probabilistic atlas of brain perfusion is a statistical model that describes patterns of brain perfusion in a population and allows for inferences about new, unseen patterns relative to this population.