Monday, May 30, 2011

Curvilinear Perspective

The quadratic fit for the distorted graph paper grid in the photograph is an example of a nonlinear transformation. But the formula is linear with respect to the coefficients which allows one to do a least squares fit. When both λ and μ are zero, the pixel function tells us that it is equal to the arbitrarily chosen zero point. One can then do ordinary least squares curve fits for both the horizontal and vertical lines through the zero point. This gives all of the coefficients except for the last in the quadratic function. In yesterday's post I estimated the the last coefficient by averaging the directions of a set of lines and then trying to improve on this by trial and error. Today I learned that the last coefficient can also be found by a least squares fit which involves all the points of the grid. There was a small deviation from yesterday's result. Residuals would probably show that this is a better fit.

The general problem is that of Curvilinear Perspective which was studied by Renaissance artists. Curve fits might be useful for approximating the transformations involved if they are not known in advance.

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