Lecture 8: Mesh Representations and Geometry Processing (41)
Gabe-Mitnick
It looks like the subdivided object always falls within the convex hull of the original object, because of the averaging involved in the algorithm. A potential downside of this is that for a convex object, subdivision will shrink the objects volume. In this example, it seems like each object would fit inside of the previous object and the smooth object on the right is significantly smaller than the original object on the left.
Gabe-Mitnick
I guess that problem applies to any subdivision scheme that treats the vertices as spline control points, same with Catmull-Clark subdivision.
joeyzhao123
I'm also interested in the effect of why it keeps shrinking as we do more subdivisions. Would it be that bad if we kept the locations of some of the original vertices and "increased" the overall volume? It sort of still keeps the same dimensions.
Staffjamesfobrien
The surface is converging to a limit surface. The behavior is that the amount of shrinkage is getting exponentially smaller with each level of subdivision.
akshitdewan
Would anyone be able to provide intuition for why this subdivision leads to convergence to a surface?
Staffjamesfobrien
There are a couple ways to do this. One is the original derivation from box splines: the box splines define a surface and this subdivision provides an increasingly finer control mesh for that surface. Another is to look at the matrices: If P_new = M . P_old then we can write P_final = M . M . ... M . P_original which is Lim n->inf P_final = M^n . P_original. So from our linear algebra we know that M^n . P_original will converge if the eigen values <= 1.
Staffjamesfobrien
I guess an intuitive way to think about it would be if you took the first shape and used a sander to sand the points flat, and then sanded the new points flat, until it looked smooth.
It looks like the subdivided object always falls within the convex hull of the original object, because of the averaging involved in the algorithm. A potential downside of this is that for a convex object, subdivision will shrink the objects volume. In this example, it seems like each object would fit inside of the previous object and the smooth object on the right is significantly smaller than the original object on the left.
I guess that problem applies to any subdivision scheme that treats the vertices as spline control points, same with Catmull-Clark subdivision.
I'm also interested in the effect of why it keeps shrinking as we do more subdivisions. Would it be that bad if we kept the locations of some of the original vertices and "increased" the overall volume? It sort of still keeps the same dimensions.
The surface is converging to a limit surface. The behavior is that the amount of shrinkage is getting exponentially smaller with each level of subdivision.
Would anyone be able to provide intuition for why this subdivision leads to convergence to a surface?
There are a couple ways to do this. One is the original derivation from box splines: the box splines define a surface and this subdivision provides an increasingly finer control mesh for that surface. Another is to look at the matrices: If P_new = M . P_old then we can write P_final = M . M . ... M . P_original which is Lim n->inf P_final = M^n . P_original. So from our linear algebra we know that M^n . P_original will converge if the eigen values <= 1.
I guess an intuitive way to think about it would be if you took the first shape and used a sander to sand the points flat, and then sanded the new points flat, until it looked smooth.
thank you!