Analytical Proof of LoG Filter Separability

Question

I've been studying the Laplacian of Gaussian (LoG) filter, and I understand its importance in edge and blob detection. However, I’ve come across some references that mention the LoG filter being separable, which would allow for more efficient computation by splitting the filtering process into separate 1D convolutions.

While I get the intuition behind separability for the Gaussian filter, I’m struggling to understand how this concept applies to the LoG filter. Specifically, I’m looking for a detailed analytical proof that shows how the LoG can be decomposed into separable components.

Here’s the starting point I’ve been working with:

$$ \text{LoG}(x, y) = \nabla^2 G(x, y) = \frac{\partial^2 G(x, y)}{\partial x^2} + \frac{\partial^2 G(x, y)}{\partial y^2} $$

Where $G(x, y)$ is the 2D Gaussian. I’ve seen claims that this can be separated into components along $x$ and $y$, but I’m not sure how to derive that step-by-step.

Can anyone provide or guide me through the analytical derivation that proves the separability of the LoG filter? Any help would be greatly appreciated!

Thanks in advance!

Answer 1

The centered Gaussian Kernel can be written, in its general form, as (Up to a scale):

$$ G \left( x, y \right) = \exp \left( -{ \begin{bmatrix} x \\ y \end{bmatrix} }^{T} \boldsymbol{C}^{-1} \begin{bmatrix} x \\ y \end{bmatrix} \right) $$

In the case the matrix $\boldsymbol{C}$ is a diagonal matrix it can be separated into a product of 2 functions:

$$ G \left( x, y \right) = \exp \left( - \frac{ {x}^{2} }{ {C}_{1, 1} } \right) \exp \left( - \frac{ {y}^{2} }{ {C}_{2, 2} } \right)$$

The linearity of the partial derivative keeps the separable form even in the Laplacian of Gaussian (LoG).

You may look at:

• How Is Laplacian of Gaussian (LoG) Implemented as Four 1D Convolutions.
• How to Decompose a Separable Filter.
• How to Prove a 2D Filter Is Separable.
• Mathematical Approach to Detect If a 2D Signal Is Separable.
• How to Check Separability of 2D Filter / Signal / Matrix.

Separability of the Function

Specifically one you have:

$$ \nabla^2 G(x, y) = \left( \frac{-1}{\sigma^2} + \frac{x^2}{\sigma^4} \right) G(x, y) + \left( \frac{-1}{\sigma^2} + \frac{y^2}{\sigma^4} \right) G(x, y) $$

You may write it as 4 functions, each separable in $x$ and $y$.
Let $G(x, y) = g(x) g(y)$ the above becomes:

$$ -\frac{1}{\sigma^2} g(x) g(y) + \frac{x^2}{\sigma^4} g(x) g(y) -\frac{1}{\sigma^2} g(x) g(y) + \frac{y^2}{\sigma^4} g(x) g(y) $$

Let $f(x) = \frac{x^2}{\sigma^4} g(x)$ and $h(y) = \frac{y^2}{\sigma^4} g(y)$ then the above becomes:

$$ -\frac{2}{\sigma^2} g(x) g(y) + f(x) g(y) + g(x) h(y) $$

You could narrow them down to a sum of 2 separable functions. Like:

$$ (-\frac{2}{\sigma^2} g(x) + f(x) ) g(y) + g(x) h(y) $$

So now you have a sum of 2 separable functions where each is 2 1D convolutions. So it can be done by 4 1D convolutions.

Yet in How Is Laplacian of Gaussian (LoG) Implemented as Four 1D Convolutions there are better approaches how to apply this in 4 convolutions.