Probabilistic Object Recognition:

In the context of probabilistic object recognition we are interested in the calculation of the probability of the object O_n given a certain local measurement M_k. This probability p(O_n|M_k) can be calculated by the Bayes rule:

$$\begin{eqnarray*}p( O_n \vert M_k ) &=& \frac{ p(M_k\vert O_n) p(O_n)}{ p(M_k)} \\ \end{eqnarray*}$$

with p(O_n) the a priori probability of the object O_n, p(M_k) the a priori probability of the filter output combination M_k, and p(M_k|O_n) is the probability density function of object O_n, which differs from the multidimensional histogram of an object O_n only by a normalization factor.

Having K independent local measurements M₁, M₂, $\dots,$ M_Kwe can calculate the probability of each object O_n by:

 

$$p( O_n \vert M_1, \dots, M_k ) \frac{\prod_k p(M_k\vert O_n) p(O_n) } { \prod_k p(M_k) }$$ = (1)

In our context the local measurement M_k corresponds to a single multidimensional receptive field vector. Therefore K local measurements M_k correspond to K receptive field vectors which are typically from the same region of the image. To guarantee the independence of the different local measurements we choose the minimal distance d(M_k,M_l) between two measurements M_k and M_l sufficiently large (in the experiments described below we choose the minimal distance $d(M_k,M_l) \geq 2 \sigma$).

For the experiments we can assume that all objects do have the same probability $p(O_n) = \frac{1}{N}$, where N is the number of objects. Therefore equation (1) simplifies to:

 

$$p(O_n \vert \bigwedge_k M_k ) \frac{\prod_k p(M_k\vert O_n)} {\sum_n \prod_k p(M_k\vert O_n)}$$ = (2)

In the following we assume the a priori probabilities p(O_n) to be known and use $p(M_k) = \sum_i p(M_k\vert O_i) p(O_i)$ for the calculation of the a priori probability p(M_k). Since the probabilities p(M_k|O_n) are directly given by the multidimensional receptive field histograms, equation (1) shows a calculation of the probability for each object O_n based on the multidimensional receptive field histograms of the N objects. Perhaps the most tempting property of equation (2) is that we do not need correspondence. That means that the probability can be calculated for arbitrary points in the image.

Equation (2) has been applied to a database of 103 objects. In an experiment 1327 test images of the 103 objects have been used which include scale changes up to $\pm 40$%, arbitrary image plane rotation and view point changes. Figure 6 shows results which were obtained for six-dimensional histograms, e.g. for the filter combination Dx-Dy-Lapat two different scales ( $\sigma = 2.0$ and = 4.0). The figure compares probabilistic object recognition and recognition by histogram matching: $\chi^2_{qv}$ (chstwo) and $\cap$ (inter). A visible object portion of approximately 62% is sufficient for the recognition of all 1327 test images (the same result is provided by histogram matching). With 33.6% visibility the recognition rate is still above 99% (10 errors in total). Using 13.5% of the object the recognition rate is still above 90%. More remarkably, the recognition rate is 76% with only 6.8% visibility of the object.

  

Figure: Recognition results of 103 objects.