How performing fast Catmull-Rom texture tiltering by linear interpolations?

Question

I've read some articles which introduce fast third-order interpolation using GL_LINEAR.

• [1] http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter20.html
• [2] http://0xef.wordpress.com/2013/01/12/third-order-texture-filtering-using-linear-interpolation/

Because [1] contains lots of errata, I recommend reading [2] if you want to catch the formalism.

Both of them mention the restriction of this method. For filtered texture with GL_LINEAR, next relation holds only if 0 <= b/(a+b) <= 1

a*f(i, j) + b*f(i+1, j) = F(i+b/(a+b), j)

where f is original image data and F is linearly interpolated texture by OpenGL.

Here's the problem. [1] mentions that this method also can be applied to Catmull-Rom bicubic.

the method can also be adapted to interpolating filters such as Catmull-Rom splines

However, it's obvious that with Catmull-Rom weighting function which contains negative parts, the condition(0 <= b/(a+b) <= 1) cannot be fullfiled. In fact, I have tried to implement Catmull-Rom with same logic, it produces just blurry images.

Is there a special way to apply the method in [1] and [2] to Catmull-Rom interpolation? Or Do I have to fetch all 16 texels for Catmull-Rom?

Answer 1

I think that the paper by Sigg and Hadwiger is correct.

Catmull-Rom polynomial p(t) can be written as

p(t) = 0.5 [ w0(t) * p0 + w1(t) * p1 + w2(t) * p2 + w3(t) * p3]

where

w0(t) = -t + 2*t^2 - t^3;
w1(t) = 2 - 5*t^2 + 3*t^3;
w2(t) = t + 4*t^2 - 3*t^3;
w3(t) = -t^2 + t^3;

and p0, p1, p2, p3 are the sampled function values.

For the B-spline, you group together the first two terms, namely w0(t) * p0 and w1(t) * p1 and the second two terms, namely w2(t) * p2 and w3(t) * p3. For the Catmull-Rom spline, you group together w0(t) * p0 and w3(t) * p3 and w1(t) * p1 and w2(t) * p2. It is easy to verify, even by a Matlab plot, that the condition b/(a+b) is satisfied for this choice. So, the idea can be used also in Catmull-Rom interpolation. However, for the Catmull-Rom case, it seems to me that there is no possible extension to the last step, since neither (w0(t) + w3(t))/(w0(t) + w1(t) + w2(t) + w3(t)) nor (w1(t) + w2(t))/(w0(t) + w1(t) + w2(t) + w3(t)) meet the condition you specified.