466 lines
19 KiB
Markdown
466 lines
19 KiB
Markdown
# Chapter 3: Matrices and linear algebra
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Libigl relies heavily on the Eigen library for dense and sparse linear algebra
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routines. Besides geometry processing routines, libigl has linear algebra
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routines which bootstrap Eigen and make it feel even more similar to a high-level
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algebra library such as Matlab.
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## Slice
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A very familiar and powerful routine in Matlab is array slicing. This allows
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reading from or writing to a possibly non-contiguous sub-matrix. Let's consider
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the Matlab code:
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```matlab
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B = A(R,C);
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```
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If `A` is a $m \times n$ matrix and `R` is a $j$-long list of row-indices
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(between 1 and $m$) and `C` is a $k$-long list of column-indices, then as a
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result `B` will be a $j \times k$ matrix drawing elements from `A` according to
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`R` and `C`. In libigl, the same functionality is provided by the `slice`
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function ([Example 301]({{ repo_url }}/tutorial/301_Slice/main.cpp)):
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```cpp
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VectorXi R,C;
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MatrixXd A,B;
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...
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igl::slice(A,R,C,B);
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```
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Note that `A` and `B` could also be sparse matrices.
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Similarly, consider the Matlab code:
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```matlab
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A(R,C) = B;
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```
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Now, the selection is on the left-hand side so the $j \times k$ matrix `B` is
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being _written into_ the submatrix of `A` determined by `R` and `C`. This
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functionality is provided in libigl using `slice_into`:
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```cpp
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igl::slice_into(B,R,C,A);
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```
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## Sort
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Matlab and other higher-level languages make it very easy to extract indices of
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sorting and comparison routines. For example in Matlab, one can write:
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```matlab
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[Y,I] = sort(X,1,'ascend');
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```
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so if `X` is a $m \times n$ matrix then `Y` will also be an $m \times n$ matrix
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with entries sorted along dimension `1` in `'ascend'`ing order. The second
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output `I` is a $m \times n$ matrix of indices such that `Y(i,j) =
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X(I(i,j),j);`. That is, `I` reveals how `X` is sorted into `Y`.
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This same functionality is supported in libigl:
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```cpp
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igl::sort(X,1,true,Y,I);
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```
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Similarly, sorting entire rows can be accomplished in Matlab using:
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```matlab
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[Y,I] = sortrows(X,'ascend');
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```
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where now `I` is a $m$ vector of indices such that `Y = X(I,:)`.
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In libigl, this is supported with
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```cpp
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igl::sortrows(X,true,Y,I);
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```
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where again `I` reveals the index of sort so that it can be reproduced with
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`igl::slice(X,I,1,Y)`.
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Analogous functions are available in libigl for: `max`, `min`, and `unique`.
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).](images/decimated-knight-sort-color.jpg)
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### Other Matlab-style functions
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Libigl implements a variety of other routines with the same api and
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functionality as common Matlab functions.
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| Name | Description |
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| :----------------------- | :---------------------------------------------------------------------------------- |
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| `igl::all` | Whether all elements are non-zero (true) |
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| `igl::any` | Whether any elements are non-zero (true) |
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| `igl::cat` | Concatenate two matrices (especially useful for dealing with Eigen sparse matrices) |
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| `igl::ceil` | Round entries up to nearest integer |
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| `igl::cumsum` | Cumulative sum of matrix elements |
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| `igl::colon` | Act like Matlab's `:`, similar to Eigen's `LinSpaced` |
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| `igl::components` | Connected components of graph (cf. Matlab's `graphconncomp`) |
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| `igl::count` | Count non-zeros in rows or columns |
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| `igl::cross` | Cross product per-row |
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| `igl::cumsum` | Cumulative summation |
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| `igl::dot` | dot product per-row |
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| `igl::eigs` | Solve sparse eigen value problem |
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| `igl::find` | Find subscripts of non-zero entries |
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| `igl::floor` | Round entries down to nearest integer |
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| `igl::histc` | Counting occurrences for building a histogram |
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| `igl::hsv_to_rgb` | Convert HSV colors to RGB (cf. Matlab's `hsv2rgb`) |
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| `igl::intersect` | Set intersection of matrix elements. |
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| `igl::isdiag` | Determine whether matrix is diagonal |
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| `igl::ismember` | Determine whether elements in A occur in B |
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| `igl::jet` | Quantized colors along the rainbow. |
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| `igl::max` | Compute maximum entry per row or column |
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| `igl::median` | Compute the median per column |
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| `igl::min` | Compute minimum entry per row or column |
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| `igl::mod` | Compute per element modulo |
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| `igl::mode` | Compute the mode per column |
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| `igl::null` | Compute the null space basis of a matrix |
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| `igl::nchoosek` | Compute all k-size combinations of n-long vector |
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| `igl::orth` | Orthogonalization of a basis |
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| `igl::parula` | Generate a quantized colormap from blue to yellow |
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| `igl::pinv` | Compute Moore-Penrose pseudoinverse |
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| `igl::randperm` | Generate a random permutation of [0,...,n-1] |
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| `igl::rgb_to_hsv` | Convert RGB colors to HSV (cf. Matlab's `rgb2hsv`) |
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| `igl::repmat` | Repeat a matrix along columns and rows |
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| `igl::round` | Per-element round to whole number |
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| `igl::setdiff` | Set difference of matrix elements |
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| `igl::setunion` | Set union of matrix elements |
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| `igl::setxor` | Set exclusive "or" of matrix elements |
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| `igl::slice` | Slice parts of matrix using index lists: (cf. Matlab's `B = A(I,J)`)
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| `igl::slice_mask` | Slice parts of matrix using boolean masks: (cf. Matlab's `B = A(M,N)`)
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| `igl::slice_into` | Slice left-hand side of matrix assignment using index lists (cf. Matlab's `B(I,J) = A`)
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| `igl::sort` | Sort elements or rows of matrix |
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| `igl::speye` | Identity as sparse matrix |
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| `igl::sum` | Sum along columns or rows (of sparse matrix) |
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| `igl::unique` | Extract unique elements or rows of matrix |
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## Laplace equation
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A common linear system in geometry processing is the Laplace equation:
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$∆z = 0$
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subject to some boundary conditions, for example Dirichlet boundary conditions
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(fixed value):
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$\left.z\right|_{\partial{S}} = z_{bc}$
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In the discrete setting, the linear system can be written as:
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$\mathbf{L} \mathbf{z} = \mathbf{0}$
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where $\mathbf{L}$ is the $n \times n$ discrete Laplacian and $\mathbf{z}$ is a
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vector of per-vertex values. Most of $\mathbf{z}$ correspond to interior
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vertices and are unknown, but some of $\mathbf{z}$ represent values at boundary
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vertices. Their values are known so we may move their corresponding terms to
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the right-hand side.
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Conceptually, this is very easy if we have sorted $\mathbf{z}$ so that interior
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vertices come first and then boundary vertices:
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$$
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\left(\begin{array}{cc}
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\mathbf{L}_{in,in} & \mathbf{L}_{in,b}\\
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\mathbf{L}_{b,in} & \mathbf{L}_{b,b}\end{array}\right)
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\left(\begin{array}{c}
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\mathbf{z}_{in}\\
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\mathbf{z}_{b}\end{array}\right) =
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\left(\begin{array}{c}
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\mathbf{0}_{in}\\
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\mathbf{z}_{bc}\end{array}\right)
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$$
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The bottom block of equations is no longer meaningful so we'll only consider
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the top block:
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$$
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\left(\begin{array}{cc}
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\mathbf{L}_{in,in} & \mathbf{L}_{in,b}\end{array}\right)
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\left(\begin{array}{c}
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\mathbf{z}_{in}\\
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\mathbf{z}_{b}\end{array}\right) =
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\mathbf{0}_{in}
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$$
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We can move the known values to the right-hand side:
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$$
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\mathbf{L}_{in,in}
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\mathbf{z}_{in} = -
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\mathbf{L}_{in,b}
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\mathbf{z}_{b}
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$$
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Finally we can solve this equation for the unknown values at interior vertices
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$\mathbf{z}_{in}$.
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However, our vertices will often not be sorted in this way. One option would be to sort `V`,
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then proceed as above and then _unsort_ the solution `Z` to match `V`. However,
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this solution is not very general.
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With array slicing no explicit sort is needed. Instead we can _slice-out_
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submatrix blocks ($\mathbf{L}_{in,in}$, $\mathbf{L}_{in,b}$, etc.) and follow
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the linear algebra above directly. Then we can slice the solution _into_ the
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rows of `Z` corresponding to the interior vertices ([Example 303]({{ repo_url }}/tutorial/303_LaplaceEquation/main.cpp)).
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### Quadratic energy minimization
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The same Laplace equation may be equivalently derived by minimizing Dirichlet
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energy subject to the same boundary conditions:
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$\mathop{\text{minimize }}_z \frac{1}{2}\int\limits_S \|\nabla z\|^2 dA$
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On our discrete mesh, recall that this becomes
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$\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{G}^T \mathbf{D}
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\mathbf{G} \mathbf{z} \rightarrow \mathop{\text{minimize }}_\mathbf{z} \mathbf{z}^T \mathbf{L} \mathbf{z}$
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The general problem of minimizing some energy over a mesh subject to fixed
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value boundary conditions is so wide spread that libigl has a dedicated api for
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solving such systems.
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Let us consider a general quadratic minimization problem subject to different
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common constraints:
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$$
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\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} +
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\mathbf{z}^T \mathbf{B} + \text{constant},
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$$
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subject to
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$$
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\mathbf{z}_b = \mathbf{z}_{bc} \text{ and } \mathbf{A}_{eq} \mathbf{z} =
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\mathbf{B}_{eq},
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$$
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where
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- $\mathbf{Q}$ is a (usually sparse) $n \times n$ positive semi-definite
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matrix of quadratic coefficients (Hessian),
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- $\mathbf{B}$ is a $n \times 1$ vector of linear coefficients,
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- $\mathbf{z}_b$ is a $|b| \times 1$ portion of
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$\mathbf{z}$ corresponding to boundary or _fixed_ vertices,
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- $\mathbf{z}_{bc}$ is a $|b| \times 1$ vector of known values corresponding to
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$\mathbf{z}_b$,
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- $\mathbf{A}_{eq}$ is a (usually sparse) $m \times n$ matrix of linear
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equality constraint coefficients (one row per constraint), and
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- $\mathbf{B}_{eq}$ is a $m \times 1$ vector of linear equality constraint
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right-hand side values.
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This specification is overly general as we could write $\mathbf{z}_b =
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\mathbf{z}_{bc}$ as rows of $\mathbf{A}_{eq} \mathbf{z} =
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\mathbf{B}_{eq}$, but these fixed value constraints appear so often that they
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merit a dedicated place in the API.
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In libigl, solving such quadratic optimization problems is split into two
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routines: precomputation and solve. Precomputation only depends on the
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quadratic coefficients, known value indices and linear constraint coefficients:
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```cpp
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igl::min_quad_with_fixed_data mqwf;
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igl::min_quad_with_fixed_precompute(Q,b,Aeq,true,mqwf);
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```
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The output is a struct `mqwf` which contains the system matrix factorization
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and is used during solving with arbitrary linear terms, known values, and
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constraint in the right-hand sides:
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```cpp
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igl::min_quad_with_fixed_solve(mqwf,B,bc,Beq,Z);
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```
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The output `Z` is a $n \times 1$ vector of solutions with fixed values
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correctly placed to match the mesh vertices `V`.
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## Linear equality constraints
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We saw above that `min_quad_with_fixed_*` in libigl provides a compact way to
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solve general quadratic programs. Let's consider another example, this time
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with active linear equality constraints. Specifically let's solve the
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`bi-Laplace equation` or equivalently minimize the Laplace energy:
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$$
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\Delta^2 z = 0 \leftrightarrow \mathop{\text{minimize }}\limits_z \frac{1}{2}
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\int\limits_S (\Delta z)^2 dA
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$$
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subject to fixed value constraints and a linear equality constraint:
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$z_{a} = 1, z_{b} = -1$ and $z_{c} = z_{d}$.
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Notice that we can rewrite the last constraint in the familiar form from above:
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$z_{c} - z_{d} = 0.$
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Now we can assembly `Aeq` as a $1 \times n$ sparse matrix with a coefficient
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$1$ in the column corresponding to vertex $c$ and a $-1$ at $d$. The right-hand
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side `Beq` is simply zero.
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Internally, `min_quad_with_fixed_*` solves using the Lagrange Multiplier
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method. This method adds additional variables for each linear constraint (in
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general a $m \times 1$ vector of variables $\lambda$) and then solves the
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saddle problem:
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$$
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\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\, \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} +
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\mathbf{z}^T \mathbf{B} + \text{constant} + \lambda^T\left(\mathbf{A}_{eq}
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\mathbf{z} - \mathbf{B}_{eq}\right)
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$$
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This can be rewritten in a more familiar form by stacking $\mathbf{z}$ and
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$\lambda$ into one $(m+n) \times 1$ vector of unknowns:
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$$
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\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\,
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\frac{1}{2}
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\left(
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\mathbf{z}^T
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\lambda^T
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\right)
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\left(
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\begin{array}{cc}
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\mathbf{Q} & \mathbf{A}_{eq}^T\\
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\mathbf{A}_{eq} & 0
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\end{array}
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\right)
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\left(
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\begin{array}{c}
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\mathbf{z}\\
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\lambda
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\end{array}
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\right) +
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\left(
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\mathbf{z}^T
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\lambda^T
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\right)
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\left(
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\begin{array}{c}
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\mathbf{B}\\
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-\mathbf{B}_{eq}
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\end{array}
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\right)
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+ \text{constant}
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$$
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Differentiating with respect to $\left( \mathbf{z}^T \lambda^T \right)$ reveals
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a linear system and we can solve for $\mathbf{z}$ and $\lambda$. The only
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difference from the straight quadratic _minimization_ system, is that this
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saddle problem system will not be positive definite. Thus, we must use a
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different factorization technique (LDLT rather than LLT): libigl's
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`min_quad_with_fixed_precompute` automatically chooses the correct solver in
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the presence of linear equality constraints ([Example 304]({{ repo_url }}/tutorial/304_LinearEqualityConstraints/main.cpp)).
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## Quadratic programming
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We can generalize the quadratic optimization in the previous section even more
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by allowing inequality constraints. Specifically box constraints (lower and
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upper bounds):
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$\mathbf{l} \le \mathbf{z} \le \mathbf{u},$
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where $\mathbf{l},\mathbf{u}$ are $n \times 1$ vectors of lower and upper
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bounds
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and general linear inequality constraints:
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$\mathbf{A}_{ieq} \mathbf{z} \le \mathbf{B}_{ieq},$
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where $\mathbf{A}_{ieq}$ is a $k \times n$ matrix of linear coefficients and
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$\mathbf{B}_{ieq}$ is a $k \times 1$ matrix of constraint right-hand sides.
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Again, we are overly general as the box constraints could be written as
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rows of the linear inequality constraints, but bounds appear frequently enough
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to merit a dedicated api.
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Libigl implements its own active set routine for solving _quadratric programs_
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(QPs). This algorithm works by iteratively "activating" violated inequality
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constraints by enforcing them as equalities and "deactivating" constraints
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which are no longer needed.
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After deciding which constraints are active at each iteration, the problem
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reduces to a quadratic minimization subject to linear _equality_ constraints,
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and the method from the previous section is invoked. This is repeated until convergence.
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Currently the implementation is efficient for box constraints and sparse
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non-overlapping linear inequality constraints.
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Unlike alternative interior-point methods, the active set method benefits from
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a warm-start (initial guess for the solution vector $\mathbf{z}$).
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```cpp
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igl::active_set_params as;
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// Z is optional initial guess and output
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igl::active_set(Q,B,b,bc,Aeq,Beq,Aieq,Bieq,lx,ux,as,Z);
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```
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 uses an active set solver to optimize discrete biharmonic kernels [^rustamov_2011] at multiple scales .](images/cheburashka-multiscale-biharmonic-kernels.jpg)
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## Eigen Decomposition
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Libigl has rudimentary support for extracting eigen pairs of a generalized
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eigen value problem:
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$Ax = \lambda B x$
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where $A$ is a sparse symmetric matrix and $B$ is a sparse positive definite
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matrix. Most commonly in geometry processing, we let $A=L$ the cotangent
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Laplacian and $B=M$ the per-vertex mass matrix (e.g. [^vallet_2008]).
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Typically applications will make use of the _low frequency_ eigen modes.
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Analogous to the Fourier decomposition, a function $f$ on a surface can be
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represented via its spectral decomposition of the eigen modes of the
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Laplace-Beltrami:
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$f = \sum\limits_{i=1}^\infty a_i \phi_i$
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where each $\phi_i$ is an eigen function satisfying: $\Delta \phi_i = \lambda_i
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\phi_i$ and $a_i$ are scalar coefficients. For a discrete triangle mesh, a
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completely analogous decomposition exists, albeit with finite sum:
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$\mathbf{f} = \sum\limits_{i=1}^n a_i \phi_i$
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where now a column vector of values at vertices $\mathbf{f} \in \mathcal{R}^n$
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specifies a piecewise linear function and $\phi_i \in \mathcal{R}^n$ is an
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eigen vector satisfying:
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$\mathbf{L} \phi_i = \lambda_i \mathbf{M} \phi_i$.
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Note that Vallet & Levy [^vallet_2008] propose solving a symmetrized
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_standard_ eigen problem $\mathbf{M}^{-1/2}\mathbf{L}\mathbf{M}^{-1/2} \phi_i
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= \lambda_i \phi_i$. Libigl implements a generalized eigen problem solver so
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this unnecessary symmetrization can be avoided.
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Often the sum above is _truncated_ to the first $k$ eigen vectors. If the low
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frequency modes are chosen, i.e. those corresponding to small $\lambda_i$
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values, then this truncation effectively _regularizes_ $\mathbf{f}$ to smooth,
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slowly changing functions over the mesh (e.g. [^hildebrandt_2011]). Modal
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analysis and model subspaces have been used frequently in real-time deformation
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(e.g. [^barbic_2005]).
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In [Example 306]({{ repo_url }}/tutorial/306_EigenDecomposition/main.cpp)), the first 5 eigen vectors
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of the discrete Laplace-Beltrami operator are computed and displayed in
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pseudo-color atop the beetle. Eigen vectors are computed using `igl::eigs`
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(mirroring MATLAB's `eigs`). The 5 eigen vectors are placed into the columns
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of `U` and the eigen values are placed into the entries of `S`:
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```cpp
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SparseMatrix<double> L,M;
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igl::cotmatrix(V,F,L);
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igl::massmatrix(V,F,igl::MASSMATRIX_TYPE_DEFAULT,M);
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Eigen::MatrixXd U;
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Eigen::VectorXd S;
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igl::eigs(L,M,5,igl::EIGS_TYPE_SM,U,S);
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```
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) Low frequency eigen vectors of the discrete Laplace-Beltrami operator vary smoothly and slowly over the _Beetle_.](images/beetle-eigen-decomposition.gif)
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## References
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[^barbic_2005]: Jernej Barbic and Doug James. [Real-Time Subspace Integration for St.Venant-Kirchhoff Deformable Models](https://www.google.com/search?q=Real-Time+Subspace+Integration+for+St.Venant-Kirchhoff+Deformable+Models), 2005.
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[^hildebrandt_2011]: Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier. [Interactive Surface Modeling using Modal Analysis](https://www.google.com/search?q=Interactive+Surface+Modeling+using+Modal+Analysis), 2011.
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[^rustamov_2011]: Raid M. Rustamov, [Multiscale Biharmonic Kernels](https://www.google.com/search?q=Multiscale+Biharmonic+Kernels), 2011.
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[^vallet_2008]: Bruno Vallet and Bruno Lévy. [Spectral Geometry Processing with Manifold Harmonics](https://www.google.com/search?q=Spectral+Geometry+Processing+with+Manifold+Harmonics), 2008.
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