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@@ -37,7 +37,7 @@ of these lecture notes links to a cross-platform example application.
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* [204 Laplacian](#laplacian)
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* [Mass matrix](#massmatrix)
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* [Alternative construction of
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- Laplacian](#alternativeconstuctionoflaplacian)
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+ Laplacian](#alternativeconstructionoflaplacian)
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* [Chapter 3: Matrices and Linear Algebra](#chapter3:matricesandlinearalgebra)
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* [301 Slice](#slice)
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* [302 Sort](#sort)
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@@ -45,6 +45,15 @@ of these lecture notes links to a cross-platform example application.
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* [303 Laplace Equation](#laplaceequation)
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* [Quadratic energy minimization](#quadraticenergyminimization)
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* [304 Linear Equality Constraints](#linearequalityconstraints)
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+ * [305 Quadratic Programming](#quadraticprogramming)
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+* [Chapter 4: Shape Deformation](#chapter4:shapedeformation)
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+ * [401 Biharmonic Deformation](#biharmonicdeformation)
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+ * [402 Bounded Biharmonic Weights](#boundedbiharmonicweights)
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+ * [403 Dual Quaternion Skinning](#dualquaternionskinning)
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+ * [404 As-rigid-as-possible](#as-rigid-as-possible)
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+ * [405 Fast automatic skinning
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+ transformations](#fastautomaticskinningtransformations)
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+
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# Compilation Instructions
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@@ -370,7 +379,7 @@ An alternative construction of the discrete cotangent Laplacian is by
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"squaring" the discrete gradient operator. This may be derived by applying
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Green's identity (ignoring boundary conditions for the moment):
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- $\int_S \nabla f \nabla f dA = \int_S f \Delta f dA$
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+ $\int_S \|\nabla f\|^2 dA = \int_S f \Delta f dA$
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Or in matrix form which is immediately translatable to code:
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@@ -647,8 +656,9 @@ 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 1
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-in the column corresponding to vertex $c$ and a -1 at $d$. The right-hand side
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+Now we can assembly `Aeq` as a $1 \times n$ sparse matrix with a coefficient
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+$1$
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+in the column corresponding to vertex $c$ and a $-1$ at $d$. The right-hand side
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`Beq` is simply zero.
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Internally, `min_quad_with_fixed_*` solves using the Lagrange Multiplier
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@@ -710,6 +720,48 @@ hand and foot constrained to be equal).](images/cheburashka-biharmonic-leq.jpg)
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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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+
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+ $\mathbf{l} \le \mathbf{z} \le \mathbf{u},$
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+
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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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+
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+ $\mathbf{A}_{ieq} \mathbf{z} \le \mathbf{B}_{ieq},$
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+
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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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+
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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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+
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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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+
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+After deciding which constraints are active each iteration simple reduces to a
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+quadratic minimization subject to linear _equality_ constraints, and the method
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+from the previous section is invoked. This is repeated until convergence.
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+
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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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+
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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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+
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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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+
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+![The example `QuadraticProgramming` uses an active set solver to optimize
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+discrete biharmonic kernels at multiple scales [#rustamov_2011][].](images/cheburashka-multiscale-biharmonic-kernels.jpg)
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[#meyer_2003]: Mark Meyer and Mathieu Desbrun and Peter Schröder and Alan H. Barr,
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"Discrete Differential-Geometry Operators for Triangulated
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@@ -721,3 +773,4 @@ _Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes_,
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2013.
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[#kazhdan_2012]: Michael Kazhdan, Jake Solomon, Mirela Ben-Chen,
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"Can Mean-Curvature Flow Be Made Non-Singular," 2012.
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+[#rustamov_2011]: Raid M. Rustamov, "Multiscale Biharmonic Kernels", 2011.
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Alec Jacobson
