I'm detecting features in an image by computing the least-squares error between the filter and a region of the image. This approach is identical to convoluting the filter with the image, but instead of computing the convolution $\int f(\tau) g(t - \tau) dt$, I'm computing the least squares error $\int (f(\tau)-g(t - \tau))^2 dt$.
Question 1: What is this technique called?
Question 2: Can it be computed in frequency domain? I would like to speed-up this algorithm by transforming both image and filter to frequency domain using FFT and computing the least squares error in frequency domain.
The Laplace transform of the convolution operator is a simple multiplication in frequency domain. $$\mathcal{L}\biggl\{\int f(\tau) g(t - \tau) dt\biggr\} = F(s) \cdot G(s)$$ What is the Laplace transform of least squares? $$\mathcal{L}\biggl\{\int (f(\tau)-g(t - \tau))^2 dt\biggr\} = \, ?$$