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Re: linear combination on p.49

Posted by Scott Schaefer on September 27, 2004 at 10:42:58:

In Reply to: Re: linear combination on p.49 posted by Daniel Suen on September 24, 2004 at 12:37:38:

: So, it is due to equation (2.9) of the text on p.36, right? The (i-2j) dictates the extraction of every other entries in the matrix, right? Without explicity derivation, it is not easy to sort this out.

Yes, that is correct. The extraction formula comes from page 36.

: Why did Joe call them rules? From the text, I do not see that they are applied separately. I am sure they are NOT the linear subdivision, followed by averaging in Chapter 7 of the text.

Well, they are rules. Usually in subdivision, we are given a set of rules that specify different topological configurations and show how new vertices are inserted or how old vertices are repositioned. While there are 4 "rules" on page 49, the first 3 specify the same topological configuration (although different orientations in the grid). So the "rules" say that if you want to insert a new vertex on an edge, put the vertex at the midpoint. The fourth "rule" says that old vertices should be left alone.

: My math is very restricted to algebra and basic real analysis. If I want to understand more math behind subdivision, which text can I get myself up to speed? Should I read some basic functional analysis, topology? or is there any text that you would recommend?

These analysis methods that Joe is using are all Real Analysis. Subdivision is parametric in nature. The first thing you do is define some grid (usually R^2) that your points live on. Then you define a refinement rule for that grid. Finally, you construct a set of rules over that grid and analyze the smoothness of the resulting scheme. Box-splines are, by construction, smooth since they are projections of areas of higher dimensional cubes.

However, the analysis techniques that Joe uses in the book are all things like Cauchy sequences, limits, continuity, ... just the bones of an analysis course (although he usually presents the material in a slightly different fashion). So I'd say that you should work on your analysis if you're weak there.

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: : So, it is due to equation (2.9) of the text on p.36, right? The (i-2j) dictates the extraction of every other entries in the matrix, right? Without explicity derivation, it is not easy to sort this out. : Yes, that is correct. The extraction formula comes from page 36. : : Why did Joe call them rules? From the text, I do not see that they are applied separately. I am sure they are NOT the linear subdivision, followed by averaging in Chapter 7 of the text. : Well, they are rules. Usually in subdivision, we are given a set of rules that specify different topological configurations and show how new vertices are inserted or how old vertices are repositioned. While there are 4 "rules" on page 49, the first 3 specify the same topological configuration (although different orientations in the grid). So the "rules" say that if you want to insert a new vertex on an edge, put the vertex at the midpoint. The fourth "rule" says that old vertices should be left alone. : : My math is very restricted to algebra and basic real analysis. If I want to understand more math behind subdivision, which text can I get myself up to speed? Should I read some basic functional analysis, topology? or is there any text that you would recommend? : These analysis methods that Joe is using are all Real Analysis. Subdivision is parametric in nature. The first thing you do is define some grid (usually R^2) that your points live on. Then you define a refinement rule for that grid. Finally, you construct a set of rules over that grid and analyze the smoothness of the resulting scheme. Box-splines are, by construction, smooth since they are projections of areas of higher dimensional cubes. : However, the analysis techniques that Joe uses in the book are all things like Cauchy sequences, limits, continuity, ... just the bones of an analysis course (although he usually presents the material in a slightly different fashion). So I'd say that you should work on your analysis if you're weak there.

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