Synthetic Examples

Figure  shows an instructive example with a shifted rectangular block. The upper left panel is the original block position while the upper right panel shows the same block shifted down and to the right by one point. On the bottom, the disparity is shown as both a magnitude and vector plot of the shift.

 

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Figure 2 Top row: the input images. Bottom row: The disparity as a magnitude and a vector representation of disparity

 

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Figure 3 Top row: Images to be compared. Notice the appearance of the small block. Middle row: The magnitude and vector representations of the disparity using the pseudo-average. Bottom row: Zoomed image of the large block's and small block's, respectively, disparity in vector representation.

Figure  is similar to figure  except that, in addition to the larger rectangle, a smaller rectangle has appeared. If the disparity estimate were to run in the ``forward'' direction, then this block would be invisible. This is because it is based upon the derivative of the local phase of the left panel where the small block does not appear. In the ``reverse'' direction, however, the small block shows up. To remedy this, a pseudo-average is taken,

$$\textbf{d}(\textbf{x})=(\textbf{d}_f(\textbf{x})-\textbf{d}_r(\textbf{x}))/2,$$ (31)

where $\textbf{d}_f(\textbf{x})$ and $\textbf{d}_r(\textbf{x})$ are the disparities in the forward and reverse directions, respectively. The minus sign reverses the direction of the second disparity model to bring it into accordance with the first. Note that the block has no coherent direction as shown in the enlargement in figure . This should not be surprising since there is no sense of direction for something that materializes from nowhere. This technique of pseudo-averaging the forward and reverse disparities, however, has pitfalls. The image resulting from a single disparity estimate depends on the location of the original image. If the disparity between the images is too large, the forward and reverse will not coincide properly causing a ``shadow'' effect.

A map of the world is shown in figure . This image was subsequently shifted one percent to the west and two percent to the south. A clear disparity is shown on the bottom panel of figure . In figure , the same image is shown. The shift, however, is 15 percent to the east and 30 percent to the north. The shadow caused by the pseudo-average is clearly visible. There are two disparities that are equal but vary in position based upon where their respective image 1's were located. This pseudo-average has advantages, but for the purposes of this paper only the forward will be used from this point forward.

 

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Figure 4 Top row: A map of the world. This image has been shifted one percent to the west and two percent to the south.Bottom row: the magnitude of the shift and a vector plot of the shift of the Big Island of Hawaii.

 

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Figure 5 Top row: a map of the world. This will be shifted 15 percent to the east and 30 percent to the north. Bottom: The shadow effect caused by applying the pseudo-average for a large shift. Notice that there appears to be two disparity images varying only by a shift.