gradient-analysis.md

@@ -4,15 +4,15 @@ Photoelatic image: gradient analysis
 ## 1. Overview
 The gradient square method, or $`G^2`$ method, is an empirical measure used to estimate the stress from the photoelastic fringes. For a given photoelastic image, the value of gradient square for a particular pixel $`(i,j)`$ is defined as (taken from [*the PhD Thesis of T. S. Majmudar*](http://webhome.phy.duke.edu/~jt41/MYDKTH.pdf)): 
 ```math
-\nabla I^2_{ij}=\frac{1}{4}[(\frac{I_{i-1,j}-I_{i+1,j}}{2})^2+(\frac{I_{i-1,j-1}-I_{i+1,j+1}}{2\sqrt{2}})^2+(\frac{I_{i,j-1}-I_{i,j+1}}{2})^2+(\frac{I_{i+1,j-1}-I_{i-1,j+1}}{2\sqrt{2}})^2]
+|\nabla I_{ij}|^2=\frac{1}{4}[(\frac{I_{i-1,j}-I_{i+1,j}}{2})^2+(\frac{I_{i-1,j-1}-I_{i+1,j+1}}{2\sqrt{2}})^2+(\frac{I_{i,j-1}-I_{i,j+1}}{2})^2+(\frac{I_{i+1,j-1}-I_{i-1,j+1}}{2\sqrt{2}})^2]
 ```
 And the value of $`G^2`$ for a region of interest (ROI) is 
 ```math
-G^2 = \frac{1}{N}\sum_{ij\in ROI} \nabla I^2_{ij}
+G^2 = \frac{1}{N}\sum_{ij\in ROI} |\nabla I_{ij}|^2
 ```
 where $`N`$ is the number of pixels inside the ROI. As an example, the top-left figure below shows an experimental image for a disc under diametric loading and the lower-left figure below shows the $`G^2`$ image calculated based on the experimental image. Both figures are colormap images: red means large value and blue means small value. For the $`G^2`$ image, the value of each pixel is the gradient square $`\nabla I^2`$ of the experimental image. The $`G^2`$ value for this disc is calculated by averaging the $`\nabla I`$ values inside the disc, which is the ROI of the problem.
 
-![demo1](uploads/922ae0b296375c84ed95e7045279231a/demo1.png)
+![demo1](uploads/6a5c9c83f30e1cd99837e8d869de385b/demo1.png)
 
 Although vector forces on contacts of discs can be solved by the [nonlinear fitting algorithm](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/inverse-analysis), $`G^2`$ method remains a popular choice to estimate stress based on photoelastic fringes because the following reasons: (1) $`G^2`$ calculation runs much faster (2) $`G^2`$ works for relatively low resolutions and (3) $`G^2`$ does not rely on the initial guess of contact forces, which can be a problem for the vector force fitting algorithm. For example, the right figure above shows the fit error for diametrically loaded disc under different loading force (real force) and using different initial guess force for the nonlinear force fitting algorithm. It is clear that only when the gauss force is close to the real force that the fitting algorithm gives accurate results.