Errors in Image Intensity

Errors in Line Estimation

Errors in Movement

Erroneous shape estimation

Errors in Image Intensity
How images change when smoothed

We can think of the images on our eyes as a noisy version of the ideal image signal from the scene. Consider a ray from a point on the scene which ideally should impinge the image at location x. But because of imperfections of the visual apparatus, we can only say that the point in the image is with large probability at location x, with smaller probability it could be at the neighboring location, x+dx, and with even smaller probability at the even farther location, x+2dx. By considering the uncertainties at every image point, it is mathematically like smoothing the image. Therefore, we can create the most likely image the vision system works with by smoothing the image.

Many illusions can be understood from the behavior of straight lines and edges. There are three cases to consider. First is an edge at the border between regions of different intensity, such as black and white.

Figure 1: Type 1 Edge

This edge can be represented by a function. The picture below shows the gray value changes from black to white in the horizontal direction.

Figure 2

The edge is at the location of the inflection point on the curve, indicated by the black dot. If we were to smooth this image, it would look like this.

Figure 3

As you can see, smoothing the image does not change the location of the edge.

The second case is a line on a background of different intensity.

Figure 4: Type 2 Edge

This again can be represented as a function with two inflection points representing the edges at the boundaries of the line and the background regions.

Figure 5

When the image is smoothed the edges drift apart as shown.


Figure 6

The third case is a gray line between a bright and a dark region.

Figure 7: Type 3 Edge

Figure 8

When this image is smoothed the edges at the boundary of the line move toward each other.

Figure 9