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In image processing, if we call $I(x):\mathbb{R}^2\mapsto \mathbb{R}$ to the function that gives the brightness value at an image location $x=(u,v)^\top\in\mathbb{R}^2$, then the structure tensor in a neighbourhood of pixels $\Omega$ (centered around a point $x_0$), is given by the following sum of matrices of 1st order derivatives: $$ \text{structure tensor} = S(x_0) = \sum_{x\in\Omega} \begin{bmatrix} I_u(x)^2 & I_u(x)I_v(x)\\ I_u(x)I_v(x) & I_v(x)^2 \end{bmatrix}, $$ where $I_u(x)=\partial I(x)/\partial u, \,\,I_v(x)=\partial I(x)/\partial v$ (both evaluated at the point $x$).

Given this, as noted in this work$^1$, (page 9, eq.31-35), to derive statistical properties of this structure tensor, we can assume that each image value $I(x)$ is perturbed with some Gaussian noise. Thereby, considering the image values $I(x)$ as observations, we can use the following model: $$ I(x) = I(x+\Delta x)+\varepsilon_x, \qquad \varepsilon_x\sim\mathcal{N}(0,\sigma_x) $$ to set up a statistical estimation problem for the variable $\Delta x\in\mathbb{R}^2$ (where $\Delta x=0$ is the maximum-likelihood estimate -MLE).

Given this, considering the neighbourhood $\Omega$ and asumming independence between observations, the log-likelihood $\ell(\Delta x)$ is given by: $$ \ell(\Delta x) = \sum_{x\in\Omega} \log(\frac{1}{\sigma_x\sqrt{2\pi}}) - \frac{1}{2\sigma_x^2}(I(x) - I(x+\Delta x))^2. $$ Thereby, the score $l=\partial\ell(\Delta x)/\partial\Delta x$ is: $$ \begin{align} l &= \sum_{x\in\Omega} \frac{1}{\sigma_x^2}(I(x)-I(x+\Delta x))\frac{\partial I(x+\Delta x)}{\partial \Delta x},\\ &= \sum_{x\in\Omega} \frac{1}{\sigma_x^2} \frac{\partial I(x+\Delta x)}{\partial\Delta x} \varepsilon_x. \end{align} $$ Therefore, the Fisher information matrix, $\mathcal{I}(\Delta x)=\mathbb{E}[{ll}^\top]$, evaluated at the MLE ($\Delta x=0$), is given by (since independence between observations): $$ \mathcal{I}(\Delta x=0) = \frac{1}{\sigma_x^2} \sum_{x\in\Omega} \begin{bmatrix} I_u(x)^2 & I_u(x)I_v(x)\\ I_u(x)I_v(x) & I_v(x)^2 \end{bmatrix} = \frac{1}{\sigma_x^2} S(x_0) $$ where the fact that $\mathbb{E}[\varepsilon_x \varepsilon_x]=\sigma_x^2$ has been used. So there exists a direct relation between the Fisher information matrix and the structure tensor, $S$, defined above.

Question

However, as it is commonly considered in several works (e.g. here$^2$), a Gaussian weighting $w(x)\sim\mathcal{N}(x_0,\sigma)$ (with known $\sigma$) is used to weigh each of the matrices of 1st-order derivatives, such that: $$ S_w(x_0) = \sum_{x\in\Omega} w(x) \begin{bmatrix} I_u(x)^2 & I_u(x)I_v(x)\\ I_u(x)I_v(x) & I_v(x)^2 \end{bmatrix}, $$ Given this, is still correct to consider $(1/\sigma_x^2) S(x_0)$ as the Fisher information matrix?

My intuition is that it still holds since weighting each of the log-likelihood terms, according to $w(x)$, does not affect the score $l$ (it dissappears for being in the constant term). But I'm not sure since I've not seen this reasoning before.

Any help is highly appreciated.


References:

  1. Do We Really Have to Consider Covariance Matrices for Image Feature Points?, Kanazawa and Kanatani, page 9 eqs. 31-35,

  2. The Harris Corner Detector, K.G. Derpanis.

Javier TG
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