# LoG filter creating additional maxima in scale space

To create a scale space, I applied a Laplacian of Gaussian filter on the following image:

After the scale space was created, I plotted circles around local maxima in scale space. However, instead of $9$ circles, I got many more as shown below(image below not to scale).

The following video shows the LoG filter responses as scale is varied from $\sigma=1$ to $16$ in steps of $0.01$ -- http://youtu.be/m4LyXBI_6w0

It shows additional maxima being created at higher scales. But this should not be the case since LoG was especially chosen so as not to create additional maxima. What could be the explanation here ?

NOTE: Thresholding the magnitude of scale space responses eliminates the additional maxima as shown below. Could it be that the additional maxima being absent holds only up to a certain degree ?

Without exactly seeing your code it is hard to know for certain, but I suspect this is because you are not tracking your maxima well through scale space.

The position of the maxima will move in scale space. At the smallest scales you get the 4 small blobs around each circle. These are due to digitization and possibly noise/irregularities in the boundary edge.

These maxima will expand as the scale increases and merge together to form the "correct" blobs around the circles.

However, the other side of the peaks will also merge together resulting in the 12 medium size blobs and finally these will merge to form the four largest blobs.

Note at each stage here the number of maxima decreases 36->21 (9+12) -> 13 (9+4)

The problem is as you only look for local maxima in scale-space you lose a lot of the relationship between the maxima.

One option may be to look for all local maxima at each scale and track these through scales. However, I'm not sure how straightforward this would be or how you determine which are the significant peaks using this approach.

A possibly better option may be similar to what you suggested by thresholding the responses. This article does basically the same thing as you. It applies a threshold on the mean blob volume over all scales. Which it claims it better than thresholding the peak response.

My interpretation would be that:

• the 4 large circles really make sense. Think of it like a Kanisza's Triangle or a Negative Space optical illusion: You can either look at 9 white points, or at a grid of 4 black rectangles. If you look at the image on a coarse scale (e.g. through an out of focus lens) you really would see 4 black blobs in a thin white grid.
• the medium-sized 12 circles between each pair of white blobs probably have a similar explanation: there's a black blob between two white blobs. Or, alternatively: If you think of two "mexican hat" functions centered at two of the white circles, then that's where the mexican hats would intersect, and thus form a small peak.
• the 4 small circles around each of the white circles are probably artifacts: Again, if you think about a "mexican hat" centered on a circle, there would be a circular "moat" around each white circle. But since neither the circles nor the LoG filter are perfectly isotropic, the moat isn't perfectly symmetric either, and contains points that are slightly higher (or lower?) than the points around them.