# Formula for 2D Image Filter for Scratch Detection

In section 4.1 (Preprocessing), they gave us a formula of the kernel,

$$H(u, v) = \frac{1}{1+0.414 \times \left(\sqrt{\frac{u^*}{D_h}+\frac{v^*}{D_v}}\right)^{2n}}$$

Where, \begin{split} t_x & = \mathrm{center} \times \cos(\theta) \\ t_y & = \mathrm{center} \times \sin(\theta) \\ u^* & = \cos(\theta) \times (u+t_x) + \sin(\theta)\times(u+t_y)\\ v^* & = -\sin(\theta) \times (u+t_x) + \cos(\theta)\times(u+t_y)\\ \end{split}

The formula doesn’t look correct to me.

1. Why would someone apply $\mathrm{sqrt}()$ and subsequently apply a power of $2n$ as it makes the $\mathrm{sqrt}()$ operation redundant?
2. $H(u,v)$ transfer function looks like a high-pass filter rather than a band-pass filter.
3. Shouldn’t $\mathrm{center}$ be $\mathrm{center}_x$, and $\mathrm{center}_y?$
4. Shouldn’t $(u+t_y)$ be $(v+t_y)$?
• Without doing any checking, I'd guess that both $u^*$ and $v^*$ equations are missing a $v$ term. – Peter K. Jul 17 '16 at 13:28
• Yes it is "wrong". The 1/(1+(a fraction)^2*n) is very much reminiscent of the Butterworth response and the $cos,sin$ business is just a rotation. So, that should be $u$ to the $cos$ term and $v$ to the $sin$ term. – A_A Jul 18 '16 at 9:58
• @A_A, tx = centerX * cos; ty = centerY * sin; u* = cos . (u + tx) + sin . (v + ty); v* = -sin . (u + tx) + cos . (v + ty); ... ... ... ... ... Is this the correct one? – user18425 Jul 19 '16 at 5:31
• the question needs some help with $\LaTeX$. – robert bristow-johnson Jul 21 '17 at 5:59
• Reference 15 in that paper is this paper. I think they're trying to say that the equation you quote is the same as equation (1) in that paper... which I very much doubt: $F(r,\theta) = [e^{r^2\sigma_1^2} - e^{r^2\sigma_2^2}] 2 \pi i r \cos(\theta)$. – Peter K. Jul 21 '17 at 15:19

## 1 Answer

I agree with Peter K. It looks like the authors of the Multidirectional Scratch Detection and Restoration in Digitized Old Images (E. Ardizzone, H. Dindo, and G. Mazzola, EURASIP Journal on Image and Video Processing, 2010) paper tried to reformulate the equation (shown by Peter K.) from the Analyzing Oriented Patterns paper by Kass and Witkin (1987; Computer Vision, Graphics and Image Processing, Vol. 37. No 3, pages 362 to 385).

Comparing the equations posted by the OP with the equations of a Gabor Filter, it feels like the author's are trying to make a novel contribution with their paper by reformulating the work of Kass and Witkin. Although neither paper includes references to Gabor filters directly, Ardizzone et al. justify the use of their reformulated "steerable" filter by stating,

The filter order n can control the slope of the subband, so that we chose n=4 to concentrate the filtering in a precise zone of the spectrum. This explains the use of this bandpass filter instead of a Gabor one.

My experience with image filters is limited, but I believe a Gabor filter can realize the same apparent functionality as the equation given by the OP including bandwidth selection (see this entry, for example).

I think "subband" refers to one of the directionally dependent filters and the "center" is the origin of the Gabor kernel relative to the image on which the kernel is being applied.

I hope this helps.