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I think this part in article is starting point of the confusion : See how the $I(x+u, y+v)$ changed into a totally different form $I(x,y)+ uI_x + vI_y)$? The author should have write a bit more detailed notation as: $I(x,y)+ uI_x(x,y) + vI_y(x,y)$ i.e. pointing out that these are element-wise spatial derivatives (like Sobel) not matrix derivatives ( ...

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