06-16-11 - Optimal Halve for Doubling Filter

I've touched on this topic several times in the past . I'm going to wrap up a loose end.

Say you have some given linear doubling filter (linear in the operator sense, not that it's a line). You wish to halve your image in the best way such that the round trip has minimum error.

For a given discrete doubling filter (non-interpolating) find the optimal halving filter that minimizes L2 error. I did it numerically, not analytically, and measured the actual error of down->up vs. original on a large test set.

I generated halving filters for half-widths of 3,4, and 5. Large filters always produce lower error, but also more ringing, so you may not want the largest width halving filter.

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upfilter :  linear  :
const float c_filter[4] = { 0.12500, 0.37500, 0.37500, 0.12500 };

 downFilter : 
const float c_filter[6] = { -0.15431, 0.00162, 0.65269, 0.65269, 0.00162, -0.15431 };
fit err = 17549.328

 downFilter : 
const float c_filter[8] = { 0.05429, -0.21038, -0.01115, 0.66724, 0.66724, -0.01115, -0.21038, 0.05429 };
fit err = 17238.310

 downFilter : 
const float c_filter[10] = { 0.05159, 0.00138, -0.21656, -0.00044, 0.66402, 0.66402, -0.00044, -0.21656, 0.00138, 0.05159 };
fit err = 16959.596

upfilter :  mitchell1  :
const float c_filter[8] = { -0.00738, -0.01172, 0.12804, 0.39106, 0.39106, 0.12804, -0.01172, -0.00738 };

 downFilter : 
const float c_filter[6] = { -0.13475, 0.02119, 0.61356, 0.61356, 0.02119, -0.13475 };
fit err = 17496.548

 downFilter : 
const float c_filter[8] = { 0.05595, -0.19268, 0.00985, 0.62688, 0.62688, 0.00985, -0.19268, 0.05595 };
fit err = 17131.069

 downFilter : 
const float c_filter[10] = { 0.05239, 0.00209, -0.19664, 0.01838, 0.62379, 0.62379, 0.01838, -0.19664, 0.00209, 0.05239 };
fit err = 16811.168

upfilter :  lanczos4  :
const float c_filter[8] = { -0.00886, -0.04194, 0.11650, 0.43430, 0.43430, 0.11650, -0.04194, -0.00886 };

 downFilter : 
const float c_filter[6] = { -0.09637, 0.05186, 0.54451, 0.54451, 0.05186, -0.09637 };
fit err = 17332.452

 downFilter : 
const float c_filter[8] = { 0.04290, -0.14122, 0.04980, 0.54852, 0.54852, 0.04980, -0.14122, 0.04290 };
fit err = 17054.006

 downFilter : 
const float c_filter[10] = { 0.03596, 0.00584, -0.13995, 0.05130, 0.54685, 0.54685, 0.05130, -0.13995, 0.00584, 0.03596 };
fit err = 16863.054

upfilter :  lanczos5  :
const float c_filter[10] = { 0.00551, -0.02384, -0.05777, 0.12982, 0.44628, 0.44628, 0.12982, -0.05777, -0.02384, 0.00551 };

 downFilter : 
const float c_filter[6] = { -0.08614, 0.07057, 0.51557, 0.51557, 0.07057, -0.08614 };
fit err = 17323.692

 downFilter : 
const float c_filter[8] = { 0.05112, -0.13959, 0.06782, 0.52065, 0.52065, 0.06782, -0.13959, 0.05112 };
fit err = 16899.712

 downFilter : 
const float c_filter[10] = { 0.04554, 0.00403, -0.13655, 0.06840, 0.51857, 0.51857, 0.06840, -0.13655, 0.00403, 0.04554 };
fit err = 16566.352

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