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17th May 2009, 13:06  #21  Link 
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Hmm? The first tangent for the Akima spline constraints on page 4 of that pdf you linked was :
1/2*(z2z1/y2y1+z3z2/y3y2), so essentially the tangent between P1 and P3 with uniform control points (y2y1=y3y2). Last edited by MfA; 17th May 2009 at 13:11. 
17th May 2009, 13:38  #22  Link 
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BTW, I should add that as far as linear interpolators go this is all really old hat ... stuff like this is a little more fresh :
http://wwwist.massey.ac.nz/elai/Pub...eilei_2008.pdf This method is pretty cute and easy to implement (basically doing a little unsharp before interpolation) : http://www.ing.unibs.it/~marco.dalai/pub/DLM_VLBV05.pdf Last edited by MfA; 17th May 2009 at 15:39. 
19th May 2009, 19:37  #23  Link 
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Trying to demonstrate that some of the weights equal 1.
I assume that all the weights are polynomial order n1 and do not depends on the yi values. Will take then all the yi=1 and try to prove that the polynomial G(x)= (sum weights) 1 has n zeros and therefore is null. Based on this paper [http://online.redwoods.cc.ca.us/inst...kyMeg/Proj.PDF (equation 23), for i=1to n1: ai+bi+ci=0 di=1 (h=1 and all the yi=1) Si(xi+h)=ai(h)^3 +bi(h)^2+ci(h)+di=1 because h=1 and Si(xi+1)=sum of weigths(xi) because all the yi=1 so all the xi(for 1 to n1) are zero of G(x). Also S1(xn)=dn1=1 so xn is also a zero of G(x). so G(x) has n zeros (polynomial order n1) , and G(x)=0. Edit:different algorithm for Akima ( that is not a CatmullRom ) from the one from "fugroairbone": http://www.iue.tuwien.ac.at/phd/rottinger/node60.html They compute different derivatives at all points and interpolation is a parabole in the "endintervals. They use five points to compute derivatives. It is probably too hard compared to CatmullRom (only two points)). We probably could modify CatmullRom to make it harder . Tangent in 2 = (s3s1)/2.We could take (s3s1)/k or k*ln(s3/s1) . We couls also compute first and second derivative at all points and based on a "magic formula", decide to increase radius, to favor natural cubic spline or CatmullRom. Edit2: I read that pixels that are in a lowgradient environment should be given more weight. May be when we interpolate along the axis X, we should keep an "indicator" of the gradient along X of the interpolated "pixel" and before interpolating along Y, take into account those "gradient along X" and normalize. And if there is value to it, is there a difference (human perception) by starting the interpolation along the X or Y axis. Last edited by mikenadia; 21st May 2009 at 02:59. Reason: Local gradient interpolation 
19th May 2009, 20:54  #24  Link  
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If all the yi are equal (to k, say), then it is clear that the solution Si(x)=k (for all i,x) satisfies the initial constraints. (intuitively, interpolating a set of constant values gives that same value). Since S1(x) (= sum(Wn(x)*yn)) in this case reduces to k times the sum of the weights, it follows that the sum is 1. 

20th May 2009, 08:34  #25  Link  
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20th May 2009, 16:01  #27  Link  
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Thanks! 

20th May 2009, 16:22  #29  Link  
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For downscaling should I also use Bilinear as the interpolation filter (after having done Gaussian blur)? To be honest, I'm not sure about doing Gaussian blur + Bilinear. I have already tried using Gaussian resampling for downscaling and the results were much too soft. 

20th May 2009, 17:46  #31  Link  
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http://madshi.net/prefiltering.rar (25MB)  Bilinear.bmp  Prefiltering + Bilinear.bmp  Bicubic50.bmp  Bicubic75.bmp  Spline36.bmp  Lanczos4  madshi tweaked.bmp The "Prefiltering + Bilinear" is what MfA's 2nd "fresh" paper produces. It's a lot less blurry compared to simple Bilinear resampling, but it has quite visible ringing and isn't any better in terms of aliasing compared to simple Bilinear filtering. Spline36 has a similar amount of ringing, but has *MUCH* reduced aliasing.  Sooooo, back to Spline16/36/64/256? 

20th May 2009, 18:41  #34  Link 
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In my experience the more taps you use, the less obvious the difference is. For my eyes going from 3 to 4 taps is a bigger difference than going from 4 taps to 8 taps. Oh well, I'm talking about positive differences here. Negative differences (ringing) are quite obvious when going from 4 taps to 8 taps. Personally, I don't think more than 4 taps is really useful, because the positive aspects are extremely small while the negative aspects are quite noticeable. But that's just my personal opinion...

20th May 2009, 20:02  #38  Link  
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I see in the paper that they tested by rotating images. Maybe the results would be different in that case, I don't know. But I do find it kind of funny that both papers seem to use some simple cubic interpolation (probably bicubic 0.5) for the "conventional" interpolation comparison screenshots. And then they wonder why the results are so blurry?! Are they not aware that e.g. Lanczos exists? Which usually produces much better results when doing multiple operations after each other (like rotating multiple times or upscaling/downscaling multiple times)... 

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