Experimental results - Random Fields

In this section, we report the experimental evaluation of our algorithm. We first discuss the parameter settings used for the results presented in the this chapter.

Parameter settings :

The disparity range and therefore the value of L is fixed to different values depending on each image pair. The parameters used in the experiments are found heuristically. In the data term (4.6), the parameter λ₁ determines the number of intensity level differences permitted with a low penalty. Depending on the noise level in the image,λ1 and alsoT1 are set within the range ofλ1= 0.4 to 1.4 andT1= 5 to 10, respectively. The parameterλ2 in the interaction function (4.11) determines the smoothness of truncated-linear (figure2.3(b), 21). As small value for this parameters means that large differences in the neighbouring disparities will have smaller penalties and therefore will favour smoother disparity maps.

The value T₂ gives a constant penalty for disparity difference greater than itself. A small value for this parameter favours discontinuities and therefore may result in noisy disparity map, whereas a very large value would result in an effect similar to that of linear function, i.e., very smooth disparity maps. The T₂ parameter is also used for the discontinuity chain formation, where it influences the number of chains formed. This is further discussed in section 4.6. In the following results, λ2 is set to 1 and T1 ranges between 2.0 to 2.4. The search range forU_c, given by an integerw in (4.20) determines how far the function should search for a gradient maximum in the image. This mainly depends on how textured or noisy the image is. If the image fairly smooth and this parameter is chosen to be small it may not find any global maxima in its vicinity and therefore, may favour no correction.

On the other hand, if the image is textured and a w is set to a large value then it may favour the movement of the discontinuity chain to a gradient maximum which does not belong to a object boundary, but to a texture. In our case for textured images w value is set to 2 for textured images and to 5 otherwise. The alternation carried out for 4 levels.

The parameters β_d and β_c, which determine the influence of the data and the interaction term, are set to1. With these parameters we will now present the result obtained using the proposed approach.

We first show the result on a simple texture stereo-image pair, where the left reference image is shown in figure4.8(a). This simple stereo pair consists of only two disparity values:

0and30. Even though for this simple example it is possible to find the exact disparity map without the boundary extraction, we use it to illustrate the evolution of the algorithm. figure 4.8(b) shows the corresponding gradient image used as evidence/data for the displacement estimation. In figures 4.8(c) and 4.8(d) we show the disparity and corresponding object boundary iteration q = 0 at the coarsest scale (l = 3). The final results of disparity and boundary for the original image (finest scale l= 0) are shown in figures. 4.8(e)and 4.8(f).

We see how the boundary and disparity are simultaneous corrected to give a much accurate result.

We will now present the result of the proposed approach on more realistic images. Two of the pairs correspond to themap(figure4.9(a)) and thevenus(figure4.10(a)) images from the Middlebury database. Two others are images acquired in our laboratory; book (figure 4.11(a)) and hat (figure 4.12(a)) . It is to be noted that for all the stereo pairs only the original reference images (left) are presented in the figures.

The results for themapstereo pair clearly show the advantage of including the boundary estimation. In the case of highly textured images, such as the map image, the standard boundary detection approaches usually fail to segregate the objects. The disparity map in figure 4.9(c)is the one obtained without boundary estimation. As can be seen in figure 4.9(d)the disparity discontinuities are at improper locations but in the vicinity of the actual object boundaries. The proposed cooperative approach, using the gradient information (figure 4.9(b)), is able to obtain a corrected disparity map, and the object boundaries as shown in figures4.9(e)and 4.9(f)respectively.

Similar results are observed for the venus and book images. With the standard Mean Field algorithm the disparity is correct at almost all regions except at the boundaries. This is illustrated in figures 4.10(c) and 4.11(c) and the boundaries corresponding to them are shown in figures 4.10(d) and 4.11(d) respectively. The corrected disparity maps obtained using our approach can be seen in figures4.10(e)and4.11(e). The actual object boundaries are shown in figures4.10(f)and 4.11(f).

Finally, we present results on the hat image, which is particularly interesting as the object of interest has smooth boundaries. As can be seen from figure 4.12(c), the standard Mean Field algorithm that only estimates disparity information fails to obtain good results.

Our method provides more satisfactory results for disparity (figure 4.12(e)) and boundary estimation (figure4.12(f)).

All the previous examples show results of our approach considering only Mean Field for the optimization of the disparity-MRF. The figures 4.13 and 4.14 show results of the proposed approach on map and venus using BP instead. As can be seen from the results for disparity (figures 4.13(b) and 4.14(b)) and boundary (figures 4.13(c) and 4.14(c)) the