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Correlation



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Correlation

Disparity $ d$, in this case, is the relative displacement between two grayscale distributions. For every pixel $ p=(k,l)$ in the left image, a window with size $ m x n$ centered at the actual pixel is compared with a window which is centered at pixels
$ p' = (k+d,l+d)$ along the epipolar line. Figure 2.19 shows the shift of the window in the right image in case of a standard geometry. In this case
$ p'=(k+d,l)$.

The similarity with the highest correlation is chosen. The correlation is defined as


where

begin{equation*}begin{aligned}sigma_l^2 = sum limits_{i=k+d}^{k+m+d} sum ... ...um limits_{j=l}^{l+n} frac{(I_r(i,j)-mu)^2)}{mn} end{aligned}end{equation*}


are the variances of the intensity values of the left and right image and


is the covariance. The correlation is at its maximum if the variance is minimal and the covariance is maximal. A lower threshold $ Gamma$ is chosen, which decides whether the similarity is strong enough or not. Thus

$displaystyle d(x) = begin{cases}d & text{if $C(d) > Gamma$ and $d = argmaxlimits_{d}C(d)$} infty & text{else} end{cases}$ (2.22)


Because correlation takes a lot of time to compute, the sum of squared difference (SSD) is often used instead. Equation 2.23 shows the equation of SSD.


If the SSD is at its minimum, the best match has been found. In this case $ Gamma$ is an upper threshold and defines whether the result is small enough or not.

$displaystyle d(x) = begin{cases}d & text{if $SSD(d) < Gamma$ and $d = argminlimits_{d}SSD(d)$} infty & text{else} end{cases}$ (2.24)


Intensity differences in high contrast areas are more reliable than in low contrast areas. A possible solution is to normalize Equation 2.23 with the local variance.


Another problem is that cameras often have different sensitivities. A solution for this problem is to normalize the image with variance and intensity. Equation 2.25 would look like


So far we assumed that we have a fixed window size $ omega$ (m$ times$n). The choice of $ omega$ influences the resulting disparity map. If $ omega$ is very small, many false matches can occur, especially if the images are noisy. If $ omega$ is very big, then the optimum is flattened and the computation time increases. Some approaches use adaptive window sizes to gain their results [KO94]. After calculating disparity values for all pixels, the resulting disparity map should be convolved with a median filter so that single very unrepresentative pixel in a neighbour hood are deleted.

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Up: Intensity-based correspondence analysis