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I am reading the SURF paper (Speeded-Up Robust Features (SURF)) but can't understand two things. In 3.2 it says:

Furthermore, the filter responses are normalised with respect to their size. This guarantees a constant Frobenius norm for any filter size, an important aspect for the scale space analysis as discussed in the next section.

How is the normalisation done? Dividing each element of the matrix by width * height? This doesn't work because the Frobenius norm isn't constant.

The second thing I dont understand is in 4.1:

In keeping with the rest, also the size of the wavelets are scale dependent and set to a side length of 4s. Therefore, we can again use integral images for fast filtering.

Why we can use integral image and what size is the filter of Haar?

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For the first question, I understand that the filters are normalized in energy. Suppose that we first consider uniform or box filters of size $(2L+1)\times(2L+1)$ and unit amplitude. Their Frobenius norm are $2L+1$. If you divide the amplitude by $(2L+1)^2$, then the Frobenius not of all filters will be exactly one.

For other filters, such a scale normalization will result in either exact or approximate unit energy (I may add simulation on Gaussians later).

I am note sure to understand the second question. You may find useful explanations on the SURF (Speeded-Up Robust Features) in the paper: An Analysis of the SURF Method, IPOL Journal,Image Processing On Line (Edouard Oyallon, Julien Rabin)

The SURF method (Speeded Up Robust Features) is a fast and robust algorithm for local, similarity invariant representation and comparison of images. Similarly to many other local descriptor-based approaches, interest points of a given image are defined as salient features from a scale-invariant representation. Such a multiple-scale analysis is provided by the convolution of the initial image with discrete kernels at several scales (box filters). The second step consists in building orientation invariant descriptors, by using local gradient statistics (intensity and orientation). The main interest of the SURF approach lies in its fast computation of operators using box filters, thus enabling real-time applications such as tracking and object recognition. The SURF framework described in this paper is based on the PhD thesis of H. Bay [ETH Zurich, 2009], and more specifically on the paper co-written by H. Bay, A. Ess, T. Tuytelaars and L. Van Gool [Computer Vision and Image Understanding, 110 (2008), pp. 346–359]. An implementation is proposed and used to illustrate the approach for image matching. A short comparison with a state-of-the-art approach is also presented, the SIFT algorithm of D. Lowe [International Journal of Computer Vision, 60 (2004), pp. 91–110], with which SURF shares a lot in common.

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    $\begingroup$ thanks for the link $\endgroup$
    – sucicf1
    Commented Nov 7, 2021 at 21:36
  • $\begingroup$ Would be a good source. They have code as well $\endgroup$ Commented Nov 7, 2021 at 22:14

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