# normalized Laplacian of Gaussian

Laplacian of Gaussian formula for 2d case is $$\operatorname{LoG}(x,y) = \frac{1}{\pi\sigma^4}\left(\frac{x^2+y^2}{2\sigma^2} - 1\right)e^{-\frac{x^2+y^2}{2\sigma^2}},$$ in scale-space related processing of digital images, to make the Laplacian of Gaussian operator invariant to scales, it is always said to normalize $LoG$ by multiplying $\sigma^2$, that is $$\operatorname{LoG}_\text{normalized}(x,y) = \sigma^2\cdot \operatorname{LoG}(x,y) = \frac{1}{\pi\sigma^2}\left(\frac{x^2+y^2}{2\sigma^2} - 1\right)e^{-\frac{x^2+y^2}{2\sigma^2}}.$$ I wonder why multiply by $\sigma^2$ not $\sigma^4$ or anything else?

Laplacian response decays as $\sigma$ increases, and it is the second Gaussian derivative so it is multiplied by $\sigma^2$.

LOG is defined as $\bigtriangledown^2G$ so the scale normalized LOG would be $\sigma^2\bigtriangledown^2G$. You need to get rid of the scaling factor of the Gaussian which is $\sigma^2$.

• Could you please give more details? – avocado Sep 11 '13 at 13:17

What @bruvel says is right, but there's a much simpler way to understand this. Imagine you have a single white dot convolved with a gaussian with size $$\sigma$$:

$$f(x,y)=\frac{e^{-\frac{x^2+y^2}{2 \sigma }}}{2 \pi \sqrt{\sigma ^2}}$$

Now, if you convolve this with a LoG with sigma $$scale$$, you get at x/y=0:

$$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y) \log (x,y)dydx$$

$$= -\frac{1}{\pi (\sigma +\text{scale})^2}$$

Obviously, for a dot convolved with a gaussian with size $$\sigma$$, we would like to have a scale space maximum at $$scale = \sigma$$. If you multiply the LoG output with $$scale$$, you get just that:

$$\frac{\partial \frac{-\text{scale}}{\pi (\sigma +\text{scale})^2}}{\partial \text{scale}}=0\Longrightarrow \text{scale}=\sigma$$

So if you want to detect "dot-like" features at the right location in scale space (e.g. for keypoint detection), multiplying the LoG result with $$\sigma$$ gives the right results. (But this is not necessarily true for line or step-edge features! For those, multiplying with $$\sigma ^ k$$ for some $$k>0$$ might give more accurate results.)

• Thank you, by now I just can say that I understand your answer to some extent, not fully. Why the convolution of $f(x,y)$ with $LoG$ gives $-\frac{1}{\pi(\sigma + scale)^2}$, how is the computation of double integral done? – avocado Sep 11 '13 at 13:12
• I read a paper by Lindeberg, in which he introduced a concept $\gamma$-normalized derivative, and based on this concept, he discussed scale-selection. Is this normalized derivative related to my problem? – avocado Sep 11 '13 at 14:12
• Is this all about dimensionless differential operator? Like this en.wikipedia.org/wiki/Nondimensionalization#Substitutions – avocado Sep 11 '13 at 14:22
• Convolve with white "dot" has only sense in discrete case. About what convolution we are talking about discrete or continious? Where is a derivation of second formula? – bruziuz May 14 '16 at 0:26
• @bruziuz: It's been some time since I wrote this, I guess I meant discrete convolution. But if we called the "dot" a "Dirac delta peak" in a continuous function, wouldn't the math exactly the same? – Niki Estner May 14 '16 at 6:00