The document discusses several papers related to quadric error metrics and mesh simplification. It summarizes techniques for using quadric error metrics to collapse vertices and preserve attributes, boundaries, and volumes during simplification. Methods include the memoryless approach of accumulating quadrics on local neighborhoods rather than the full mesh, and solving for optimal positions using constraints or optimization functions.

1. [Garland99] Thesis
[Garland98] Simplifying Surfaces with Color and Textures using Quadric Error Metrics
[Garland97] Surface Simplification using Quadric Error Metrics
Flip check – p.56
Homogeneous Variant – p.50
The homogeneous form of the quadrics can be transformed by a linear transform.
However this formulation is less convenient and less efficient (4x4 operations).
Vertex Placement Policies – p.51
Taking partial derivatives, we see that the gradient of Q is
and its error is

2. These are instances of the minimum of a positive definite quadratic form. Geometrically
it can be interpreted as the least-square optimal point which best fits the set of planes
represented by the quadric. It lies in the center of the ellipsoidal isosurfaces of Q.
Discontinuities and Constraints
The basic algorithm ignores boundary curves. We form a plane perpendicular to a
boundary; calculate a quadric add a large penalty and add it to both ends of the edge.
When using area-weighted quadrics, the constraints must be properly weighted. The
obvious choice of weighting by adjacent face’s area will led to dependency on the
tessellation next to the boundary. A better choice is to use the squared length of the
boundary edge. The same can be applied to ‘feature edges’ within a mesh.
We can also formulate a constraint for points too.
Volume Quadric and relation to Lindstrom
Using simple vector algebra we can get the squared volume of the tetrahedron as
m is the cross-prod of the edges. (4.9) is equivalent to scaling the fundamental quadric by
w2/9 where w is the area of the contributing face T.
Principal Components of Quadrics
We can regard A as the covariance of the normals with mean 0. The eigenvectors should
be roughly be in the direction of the average normal, direction of maximum curvature and
direction of minimum curvature.
Ambiguity of Sharp Angles

3. The initial quadric assigned to both of these cases are the same since its incident
segments lie along the same lines. In (b) the curve will become a single spike and quadric
will allow points to move freely along the line.
[Kho03] User-Guided Simplification
The user can adaptively weight areas of the models. The actual weight applied is wlogV
where V is the number of vertices in the input in order to get similar results of a feature
from two different scales of the model. Using weights leaves more polygons in areas
where user feel is important. Another tool is to use constraints – like the virtual plane in
the Garland’s original work. To avoid constraints unnecessary increase cost we keep the
constraint quadrics separate from the geometric ones and only add them when we are
computing the optimal positions. They propose 3 types of constraints – contour, plane
and point.
[Lindstrom98] Fast and Memory Efficient Polygonal Simplification
They pioneered the memoryless approach whereby the cost is computed wrt the current
simplified meshes and not the original mesh. C(v) as a sum of squared tetrahedral
volumes formed by ‘v’ and the faces around the vertex neighborhood of ‘v’ in the
simplified mesh.
The basic approach to finding the optimal position is to combine a number of linear
equality constraints aiTv = bi, i.e. v is the intersection of three non-parallel planes in 3.
If two of more of these planes are nearly parallel, minor perturbations to the plane
coefficients lead to large variations in the solution. So we add a constraint to a set of
existing constraints only if the plane normal does not fall within an angle of all linear
combinations of the plane normals of the previous constraints. See [Lindstrom00] for a
SVD based alternative.
Volume Preservation
By considering the signed changes in the volumes of the tetrahedrons formed by the new
vertex and the old triangle.
where ni is the outward normal vector of triangle ti, with magnitude twice the area of ti.
Boundary Preservation

4. For a planar boundary, we attempt to preserve the area enclosed by the boundary. We use
signed changes in area.
Volume Optimization
The volume preservation constraint leaves an entire plane of candidate vertices. To
further constrain the vertex position we minimize the unsigned volume of each individual
tetrahedron, which is a measure of the local surface error for each triangle in the edge’s
neighborhood. If all of these are coplanar the volume optimization has many solutions
Boundary Optimization
We minimize the sum of squared areas.
Triangle Shape Optimization
Which is the sum of square of the edge lengths incident upon v. This unsure the area to
perimeter ration is maximized. This is used as a last resort only when both fV and fB are
both close to zero.
Edge Costs
[Lindstrom00] Out-of-core simplification of large polygonal meshes
Points are binned into grid cells, within each cell qem is used to collapse vertices and to
calculate the new vertex. Only a single pass is needed through the input model. The qem
used is the memoryless version from [Lindstrom98]. A hash table based on the grid cell
index is used to store the input. The quadric for the triangle is computed and added to all
3 bins. Each bin represents one output vertex.
Where n is the 4-vector made up of area-weighted triangle normal and scalar triple
product of its vertices. Since Q is symmetric and c is not used, only 9 scalars are needed.
After adding all the quadric within a grid, we use the block decomposition above and
solve the Ax = b for the optimal position. They introduce a robust SVD based way to
invert the quadric matrix instead of the linear constraints of [Lindstrom98].

5. [Hoppe] New Quadric Metric for Simplifying Meshes with Appearance Attributes
They confirmed the memoryless method of Lindstrom gives better quality. In the
standard QEM scheme the Qv are computed on the original mesh and subsequently
summed. As a result, the merging of the non-parallel ovals (corresponding to fine level
features) gives rise to tight spherical quadrics that lock vertices and prevent further
simplification, even though the resulting mesh is planar.
Garland metric is quadratic in space – i.e. for m attributes it need o(n^2) space.
The new quadric defines both the geometric error and attribute error based on geometric
correspondence in 3D. Rather than projecting the point p onto the mesh face in an

6. abstract higher-dimensional space R3+m. The error is defined as the sum of the geometric
error and the attribute error. The attribute error is the squared deviation between s and
the value s’ interpolated from face f at the projected point p’. The projection is done in
R3. The new metric takes 11 + 4m coefficients, which is linear in m.
In the memoryless scheme the Qf are computed on the face neighborhood of the
collapsing edge. The memoryless simplification makes storing QEM’s unnecessary. For
speed it is still good to stored the values (area(f).Qf(v)) on the faces.
The new QEM sometimes shrinks the model geometry in areas of high attribute gradient
– new vertex is pushed towards the center of curvature of the surface at sharp attribute
transitions (see fig.7 above).
Preserving volume is equivalent to a linear constraint gTvol p + dvol = 0. It is solved using a
Lagrange multiplier. The volumetric gradient gvol is the sum of the face normals of Fi+1
weighted by 1/3 of their face areas.
[Hoppe00] Efficient Minimizatio of New Quadric Metric for Simplifying Meshes with
Appearance Attributes
The minimization of the quadric metric with m appearance attribute involves solving a
linear system of size (3+m)x(3+m). The system has only O(m) nonzero entries, so it can
be solved in O(m2) using sparse solvers such as conjugate gradients. Here the authors
show the special structure of the sparsity permits the system to be solved in O(m) time.

7. [DeCoro07] Real Time Simplification using the GPU
They use [Lindstrom00] qem based vertex collapse as the base and add a probabilistic
octree instead of a grid. The basic alg is very simple – it use a user specified uniform grid
and use the vertex shader to compute the grid index. The index is used to accumulate a
per-cluster qem. In pass 2 the qem is solved using a matrix invert to get the optimal vertex
position. In pass 3, the input mesh is send through the a 2 nd time, geometry shader is used
to cull triangles are collapsed; otherwise the shader retrieve the positions from pass 2 and
streamed out into a GPU buffer.
Two warping function are introduced a) the world-view projective transform b) area-of-
interest by a Gaussian. It is inverted using the Gaussian error function. The inverse
warping function is used only when computing the cluster coordinates.
To support adaptive resolution a probabilistic octree using hash was introduced.
[Frey99] Surface Mesh Quality Evaluation
Measures the aspect ratio of the set of triangles to define the quality.
1 6 Sf
K Tri (M j ) Q f , where Q f . Sf is the area of the face f, pk is the half-
F f F 3 p k .hk
perimeter of f and hf is the longest edge.
[Southern00] Evaluation of Memoryless Simplification
They evaluated the common alternatives for vertex placement – a) 0 bit: mid-point b) 1
bit: half-edge collapse c) 2 bits: to encode the end-pts + mid-pt.
They defined a triangle quality metric that is faster than [Frey99]
3
F
e ij
1 j 1
K (M ) ( ) ; divide the edge length sum by 3 times the minimum edge.
F i 1 3(min 3j 1 e ij )
F
1
For volume, they use the Gaussian Divergence Formula K vol ( M ) (v1i v2 ) v3 ,
i i
6 i 1