diff --git a/.gitmodules b/.gitmodules index 811ef2e..2f98bd2 100644 --- a/.gitmodules +++ b/.gitmodules @@ -1,3 +1,6 @@ [submodule "shared"] path = shared url = https://github.com/alecjacobson/geometry-processing-shared.git +[submodule "libigl"] + path = libigl + url = http://github.com/libigl/libigl diff --git a/CMakeLists.txt b/CMakeLists.txt index 4da1b30..938c669 100644 --- a/CMakeLists.txt +++ b/CMakeLists.txt @@ -1,4 +1,38 @@ cmake_minimum_required(VERSION 2.6) project(deformation) -set(CMAKE_MODULE_PATH ${CMAKE_MODULE_PATH} ${CMAKE_CURRENT_SOURCE_DIR}/shared/cmake) -include("${CMAKE_CURRENT_SOURCE_DIR}/shared/cmake/CMakeLists.txt") +set(CMAKE_MODULE_PATH ${CMAKE_MODULE_PATH} ${CMAKE_CURRENT_SOURCE_DIR}/cmake) + +### libIGL options: choose between header only and compiled static library +option(LIBIGL_USE_STATIC_LIBRARY "Use libigl as static library" OFF) +option(LIBIGL_WITH_OPENGL "Use OpenGL" ON) +option(LIBIGL_WITH_OPENGL_GLFW "Use GLFW" ON) + +find_package(LIBIGL REQUIRED) + +if((CMAKE_CXX_COMPILER_ID MATCHES "GNU") OR (CMAKE_CXX_COMPILER_ID MATCHES "Clang")) + add_compile_options(-mavx2) +elseif(CMAKE_CXX_COMPILER_ID MATCHES "Intel") + message(FATAL_ERROR "Intel not supported") +elseif((CMAKE_CXX_COMPILER_ID MATCHES "MSVC") AND NOT CMAKE_CL_64) + add_compile_options(/arch:AVX2) +endif() +set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -std=c++11") + +add_definitions(${CMAKE_CXX_FLAGS} "-mfpmath=sse") +add_definitions(${CMAKE_CXX_FLAGS} "-march=native") +add_definitions(${CMAKE_CXX_FLAGS} "-g") +add_definitions(${CMAKE_CXX_FLAGS} "-Wall") + +# Add your project files +include_directories("include/") +if(USE_SOLUTION) + file(GLOB SRCFILES solution/*.cpp) +else() + file(GLOB SRCFILES src/*.cpp) +endif() + +add_library(core ${SRCFILES}) +target_link_libraries(core igl::core igl::opengl igl::opengl_glfw ) + +add_executable(deformation "main.cpp") +target_link_libraries(deformation core igl::core igl::opengl igl::opengl_glfw ) diff --git a/README.html b/README.html new file mode 100644 index 0000000..e8959e4 --- /dev/null +++ b/README.html @@ -0,0 +1,807 @@ + + + + + + + + + + +
+\(\newcommand{\A}{\mat{A}}\) +\(\newcommand{\B}{\mat{B}}\) +\(\newcommand{\C}{\mat{C}}\) +\(\newcommand{\D}{\mat{D}}\) +\(\newcommand{\E}{\mat{E}}\) +\(\newcommand{\F}{\mat{F}}\) +\(\newcommand{\G}{\mat{G}}\) +\(\newcommand{\I}{\mat{I}}\) +\(\newcommand{\K}{\mat{K}}\) +\(\newcommand{\L}{\mat{L}}\) +\(\newcommand{\M}{\mat{M}}\) +\(\newcommand{\N}{\mat{N}}\) +\(\newcommand{\One}{\mathbf{1}}\) +\(\newcommand{\P}{\mat{P}}\) +\(\newcommand{\Q}{\mat{Q}}\) +\(\newcommand{\Rot}{\mat{R}}\) +\(\newcommand{\R}{\mathbb{R}}\) +\(\newcommand{\S}{\mathcal{S}}\) +\(\newcommand{\T}{\mat{T}}\) +\(\newcommand{\U}{\mat{U}}\) +\(\newcommand{\V}{\mat{V}}\) +\(\newcommand{\W}{\mat{W}}\) +\(\newcommand{\X}{\mat{X}}\) +\(\newcommand{\Y}{\mat{Y}}\) +\(\newcommand{\argmax}{\mathop{\text{argmax}}}\) +\(\newcommand{\argmin}{\mathop{\text{argmin}}}\) +\(\newcommand{\c}{\vec{c}}\) +\(\newcommand{\d}{\vec{d}}\) +\(\newcommand{\e}{\vec{e}}\) +\(\newcommand{\f}{\vec{f}}\) +\(\newcommand{\g}{\vec{g}}\) +\(\newcommand{\mat}[1]{\mathbf{#1}}\) +\(\newcommand{\min}{\mathop{\text{min}}}\) +\(\newcommand{\n}{\vec{n}}\) +\(\newcommand{\p}{\vec{p}}\) +\(\newcommand{\transpose}{{\mathsf T}}\) +\(\newcommand{\tr}[1]{\mathop{\text{tr}}{\left(#1\right)}}\) +\(\newcommand{\t}{\vec{t}}\) +\(\newcommand{\u}{\vec{u}}\) +\(\newcommand{\vec}[1]{\mathbf{#1}}\) +\(\newcommand{\x}{\vec{x}}\) +\(\newcommand{\y}{\vec{y}}\) +\(\newcommand{\z}{\vec{z}}\) +\(\renewcommand{\v}{\vec{v}}\) +
+ +

Geometry Processing – Deformation

+ +
+

To get started: Fork this repository then issue

+ +
git clone --recursive http://github.com/[username]/geometry-processing-deformation.git
+
+
+ +

Installation, Layout, and Compilation

+ +

See +introduction.

+ +

Execution

+ +

Once built, you can execute the assignment from inside the build/ by running +on a given mesh:

+ +
./deformation [path to mesh.obj]
+
+ +

When the mesh is blue, the system is in “place handles” mode. Click on the mesh +to select vertex locations for control point handles. After pressing space to switch to +deformation mode, drag the handles. Pressing m will switch between different +deformation methods.

+ +
+ +
+ +

Background

+ +

In this assignment we explore smooth deformation of an existing shape. Shape +deformation has many +applications in geometry process; we will focus on interactive handle-based +deformation. In this setup, the user repositions a sparse set of points and the +goal is to propagate the transformation at these “handles” to the rest of the +shape. To be interactive we should aim for computing the new deformation of the +shape at around 30 frames per +second.

+ +

Shape deformation is the transformation from its rest shape to a +new/current/deformed shape. If the position of a point on some 3D rest shape + given by \(\hat{\x} ∈ \R³\) then we will write that the unknown position +on the deformed shape is given by \(\x ∈ \R³\). We can write this point’s +displacement vector as \(\x - \hat{\x} =: \d ∈ \R³\).

+ +

The propagation of the handles’ deformation can be thought of in two +complementary ways:

+ +
    +
  1. as a scattered data + interpolation + problem, where handles provide sparse samples of an unknown displacement + field, or
    2. +
  2. as a shape optimization problem, where we try to define a new shape that + retains the details of the old shape but fulfills the handle constraints.
    3. +
+ +

In the following discussion we will take advantage of the ability to switch +between thinking of the unknowns as the positions of the deformed shape (\(\x\)) +and displacements (\(\d\)). These views are equivalent, but often one or the +other provides a better intuitive understanding.

+ +

Continuity

+ +

We will limit ourselves to continuous deformations of shapes. That is, the +shape will not tear, crack or change its topological features.

+ +

If we represent our shape discretely as a triangle +mesh (e.g., with rest vertices +in \(\hat{\V} +∈ \R^{n × 3}\) and faces in \(F ∈ \{1,…,n\}^{m × 3}\), then we can trivially +ensure a continuous deformation by determining new vertex positions \(\V\). The +topology (connectivity) of the mesh (\(F\)) will not change.

+ +

Generic Distortion Minimization

+ +

A rest surface \(\hat{S}\) +immersed in \(\R³\) can +be described as a mapping \(\hat{\x}\) from some 2D parametric domain \(Ω\). For any +parameters \(u\) and \(v\), \(\hat{\x}\) describes the 3D position:

+ +

\[ +\hat{\x}(u,v) ∈ \R³. +\]

+ +

Similarly the deformed surface can be represented as a position function \(\x: +Ω → \R³\). The displacement vector field is thus a function taking the +difference: \(\d(u,v) = \x(u,v) - \hat{\x}(u,v)\).

+ +

For the handle-based deformation problem we would like to find a new surface +(defined by \(\x\)) that:

+ +
    +
  1. adds as little distortion as possible to the shape, and
    2. +
  2. satisfies the users constraints at selected handle positions.
    3. +
+ +

We can cast this as an energy optimization problem. Suppose we have +energy functional +\(E(\x)\) that +measures the amount of distortion between the new shape (\(\x\)) and the rest +shape \(\hat{x}\), then we could optimize for the best possible shape \(\x\) by +minimizing \(E\):

+ +

\[ +\min_\x E(\x). +\]

+ +

To ensure that the user’s \(k\) handle points are interpolated we add the +constraints:

+ +

\[ +\text{ subject to } \x(u_i, v_i) = \g_i \ ∀ i = \{1, … , k\}, +\] +where \(\g_i\) is the position of the $i$th control point handle.

+ +

While the constraints are straightforward, we have many choices for how to +formulate the energy function \(E\). A natural choice is to measure distortion in +an egalitarian way by integrating a local measure of distortion at all points +on the surface:

+ +

\[ +\min_\x ∫_Ω ‖ e(\x) ‖² \ dA \quad \text{ subject to } \x(u_i, v_i) = \g_i \ ∀ i = \{1, … , k\}, +\]

+ +

where \(e\) is a vector- or scalar- valued function measuring local (unsquared) +distortion. We will now consider different choices for \(e\).

+ +

Linear Methods

+ +

If we assume that the deformation between the rest shape given by \(\hat{\x}\) +and the new shape given by \(\x\) is small then we can measure the distortion +of the deformation in terms of the smoothness of the displacement field. This +simplest methods will integrate the magnitude of derivatives of the +displacement field (\(\d\)): if the displacement field has large variations or +sudden changes then it is inducing a lot of distortion.

+ +

Gradient-based energy

+ +

Let us first consider minimizing the integral of squared variation of the +displacement field:

+ +

\[ +\min_\d ∫_Ω ‖ ∇\d ‖_F^2 \ dA \quad \text{ subject to } \d_i = +\g_i-\hat{\x}_i \ ∀ i = \{1, … , k\}, +\] +where +\(∇\d = \left(\begin{array}{ccc} +\frac{∂d^x}{∂u} & \frac{∂d^y}{∂u} & \frac{∂d^z}{∂u} \\ +\frac{∂d^x}{∂v} & \frac{∂d^y}{∂v} & \frac{∂d^z}{∂v} +\end{array}\right)\) +the Jacobian +matrix of the displacement field \(\d\)

+ +
+
Deformation Gradient
+ +

If \(\I ∈ \R^{3 × 3}\) is the identity matrix, then the quantity \(\F := \I + +∇\d\) is referred to as the deformation +gradient +in the mechanics +community.

+
+ +

This is simply the familiar Dirichlet +energy applied to each +coordinate function of the displacement field independently.

+ +

We can discretize this over our triangle mesh surface the same way we have in +smoothing and parameterization assignments:

+ +

\[ +\min_{\D} \tr{\D^\transpose \L \D} \quad \text{subject to } \D_\text{handles} = +\g_\text{handles} - \hat{\V}_\text{handles}, +\] +where the rows of \(\g_\text{handles} ∈ \R^{k × 3}\) contains the new positions +of the \(k\) control point handles.

+ +

While easy to implement, this method suffers from a couple immediate problems:

+ +
    +
  1. it is not smooth at constraints, and
    2. +
  2. then influence of handles dies off too quickly.
    3. +
+ +
+Two regions of selected points (purple) are constrained. The gradient-based +energy produces a continuous displacement but is not smooth at the inner +constrained points (sharp crease). The Laplacian-based energy produces a +displacement with continuous positions and +derivatives. +
Two regions of selected points (purple) are constrained. The gradient-based +energy produces a continuous displacement but is not smooth at the inner +constrained points (sharp crease). The Laplacian-based energy produces a +displacement with continuous positions and +derivatives.
+
+ +

By minimizing the Dirichlet energy, each coordinate of “displacement field” is +a harmonic function. +Intuitively (however abstractly) we can think of each function as diffusing +the user’s constraints as if they were +heat values. As such, each +function diffuses quickly to an average, slowly varying value over most of the +domain. As a displacement field a constant value for each coordinate function +would mean a translation: most of the shape is simply translated.

+ +
+ +
+ +

The gradient operator (\(∇\)) is a linear +operator. We can alternatively view +our minimization above in terms of the unknown positions \(\x\):

+ +

\[ +\min_\d ∫_Ω ‖ ∇\d ‖_F^2 \ dA ⇒ +\min_\x ∫_Ω ‖ ∇(\x - \hat{\x}) ‖_F^2 \ dA ⇒ +\min_\x ∫_Ω ‖ \underbrace{∇\x}_\text{after} - +\underbrace{∇\hat{\x}}_\text{before} ‖_F^2 \ dA. +\] +If we think of the gradient of the position function \(∇\x\) (with respect to the +underlying parameterization \(u,v\)) as a local geometric feature +descriptor then this +energy can be re-understood as measuring the difference in this feature before +and after the deformation. This is very sensible as we are trying to measure +distortion. We would expect that a low-distortion deformation would correspond +with a small change to local features.

+ +

Unfortunately, the gradient of the position function \(\x\) is a poor, +first-order local feature.

+ +

Laplacian-based energy

+ +

If we model distortion as the change in a local feature descriptor, then a +natural local and relative descriptor would be one that compared the position +of some point on the shape to the average of its local neighborhood. We have +studied an operator that computes this in the smoothing assignment. The +Laplace(-Beltrami) operator +can be derived as taking exactly the difference of a functions value at a point +and the average (i.e., centroid) of +an infinitesimal region around that point:

+ +

\[ +∆ f(\x) = \lim_{|B(\x)| → 0} \frac{1}{|B(\x))|} ∫_{B(\x)} f(\z) \;d\z - f(\x) +\]

+ +

(see, e.g., “Differential coordinates for local mesh morphing and deformation” +[Alexa et al. 2003] and expanded upon in “Laplacian Surface Editing” +[Sorkine et al. 2004]).

+ +

When applied to the embedding function \(\x\) the Laplace operator computes the +difference in position between a point and its local neighborhood. This +vector points in the +normal direction and its +magnitude corresponds inversely with how flat the surface is +locally.

+ +

Let’s replace the gradient feature above with this second-order feature +descriptor and massage our optimization problem back in terms of +displacements:

+ +

\[ +\min_\x ∫_Ω ‖ \underbrace{∆\x}_\text{after} - +\underbrace{∆\hat{\x}}_\text{before} ‖^2 \ dA ⇒ +\min_\x ∫_Ω ‖ ∆(\x - \hat{\x}) ‖^2 \ dA ⇒ +\min_\d ∫_Ω ‖ ∆\d ‖^2 \ dA. +\]

+ +

Just as we can show that harmonic functions (\(∆\d = 0\)) minimize the Dirichlet +energy, we can use calculus of +variations apply +Green’s identity twice to +show that minimizers of the squared-Laplacian energy are _bi-_harmonic +functions (\(∆∆\d = 0\) or \(∆²\d = 0\)). Obviously all harmonic functions are also +biharmonic functions. This implies that the space of biharmonic functions is +strictly larger. In particular, this will allow use to ensure continuity of +first derivatives across constrained values at handles.

+ +
+ +
+ +

The fact that each coordinate of the displacement field \(\d\) will be a +bi-harmonic function is not so elucidating, but by minimizing the +squared-Laplacian energy integrated over the domain, we can say that it is +as-harmonic-as-possible. Harmonic functions include constant functions (i.e., +translations) but also any displacement that bends in one direction by an +equal and opposite amount as it bends in the other direction: \(∆\d = 0 → ∂\d/∂u ++ ∂\d/∂v = 0 → ∂\d/∂u = -∂\d/∂v\).

+ +

To discretize this energy we can make use of our discrete Laplacian operator +\(\L\). This matrix computes the locally integrated Laplacian of a given +function specified by per-vertex values \(\f\). Now we would like to integrate +the square of the point-wise Laplacian. We can approximate the point-wise +Laplacian by the local integral +average of the Laplacian +\(\M^{-1}\L \f\). Integrating this over the mesh we have our complete approximate +of the energy:

+ +

\[ +\begin{align} +∫_Ω ‖∆\d‖² \;dA &≈ \tr{ \D^\transpose \L^\transpose \M^{-\transpose} \M \M^{-1} \L +\D}\\ +&= \tr{ \D^\transpose \underbrace{\L^\transpose \M^{-1} \L}_{\Q} \D }, +\end{align} +\] +where \(\M ∈ \R^{n × n}\) is the mass-matrix for the given mesh and \(\Q ∈ +\R^{n×n}\) can be thought of as the bi-Laplacian matrix.

+ +
+

\(k\)-harmonic

+ +

The logical continuation of harmonic and biharmonic deformation is to consider +triharmonic (\(∆³\d = 0\)) and tetraharmonic (\(∆⁴\d = 0\)) and so on. It’s +straightforward to implement these, though there are diminishing returns and +increasing costs.

+
+ +
Precomputation
+ +

With out loss of generality, assume that the rows of the unknown displacements +\(\D\) have been sorted so that displacements corresponding to handle vertices +are in the bottom part:

+ +

\[ +\D = \left(\begin{array}{c} +\D_\text{u} \\ +\D_\text{h} +\end{array} +\right) +\]

+ +

Since the displacements at handles are known before the optimization, we can +separate the knowns and unknowns in the energy:

+ +

\[ +\min_{\D_\text{u}} +\tr{(\D_\text{u}^\transpose \ \D_\text{h}^\transpose) +\left(\begin{array}{cc} +\Q_\text{u,u} & \Q_\text{u,h} \\ +\Q_\text{h,u} & \Q_\text{h,h} +\end{array}\right) +\left(\begin{array}{c} + \D_\text{u} \\ + \D_\text{h} +\end{array} +\right)} \\ +\min_{\D_\text{u}} +\tr{\D_\text{u}^\transpose \Q_\text{u,u} \D_\text{u} + +2 \D_\text{u}^\transpose \Q_\text{u,h} \D_\text{h} + +\underbrace{\D_\text{h}^\transpose \Q_\text{h,h} +\D_\text{h}}_\text{constant}} \\ +\min_{\D_\text{u}} +\tr{ +\D_\text{u}^\transpose \Q_\text{u,u} \D_\text{u} + +2 \D_\text{u}^\transpose \Q_\text{u,h} \D_\text{h}} +\] +where \(\Q_\text{u,u} ∈ \R^{(n-k) × (n-k)}\) is the quadratic coefficients matrix +corresponding to the unknown displacements.

+ +

This quadratic optimization problem may solved by setting all partial +derivatives with respect to degrees of freedom in \(\D_\text{u}\) to zero:

+ +

\[ +2 \Q_\text{u,u} \D_\text{u} + 2 \Q_\text{u,h} \D_\text{h} += 0 → \D_\text{u} = \Q_\text{u,u}^{-1} \Q_\text{u,h} \D_\text{h} +\]

+ +

If we don’t change which vertices are handles, but only change the positions +of the selected handles, then only \(\D_\text{h}\) changes above. In particular, +the matrix \(\Q_\text{u,u}\) is unchanged. Therefore, we can +prefactorize it so that +applying its inverse is fast (igl::min_quad_with_fixed does this for you).

+ +
+

Actually the entire term \(\Q_\text{u,u}^{-1} \Q_\text{u,h} =: \W ∈ \R^{(n-k) +× k}\) does not change. The columns of \(\W\) reveal how unknown displacements +respond to each handle point’s displacement (see also “An intuitive framework +for real-time freeform modeling” [Botsch & Kobbelt 2004]). This provides a +gateway to the relationship with linear blend skinning and automatic +(biharmonic) weighting functions (see “Bounded Biharmonic Weights for +Real-Time Deformation” [Jacobson et al. 2011]).

+
+ +
Trouble in paradise
+ +

Biharmonic displacements work well for small deformations and deformations that +do not imply a large rotation. However, the Laplacian of the position function +\(∆\x\) as a feature descriptor is not rotation invariant. This problem is true +for all linear differential features including the gradient of the embedding +function \(∇\x\) considered above (see “On linear variational surface deformation +methods” [Botsch & Sorkine 2008]).

+ +
+ +
+ +

This means that if the user transforms all of the handle locations by a rigid +transformation \(\T\) these energies will not measure zero for a displacement +equivalent to applying the rigid transformation \(\T\) to the entire shape. We +would like global rotation invariance, but we would also like this property +to apply locally so that parts of the shape can easily rotate.

+ +

As-rigid-as-possible

+ +

In the scenario where each handle +\(i\) are perfectly transformed by a rigid +transformation \(\x_i = +\Rot \hat{\x}_i + \t\), where \(\Rot ∈ SO(3) ⊂ \R^{3×3}\) is a rotation matrix and +\(\t∈\R^3\) is a translation vector. If an +oracle could only tell us this +particular rigid transformation then we could repair +the gradient-based energy above by pre-rotating the rest shape by this +transformation:

+ +

\[ +\begin{align} +∫_Ω ‖ ∇ \x - ∇(\Rot \hat{\x} + \t) ‖² \;dA + &= ∫_Ω ‖ ∇ \x - ∇(\Rot \hat{\x}) - ∇\t‖² \;dA \\ + &= ∫_Ω ‖ ∇ \x - \Rot ∇\hat{\x} ‖² \;dA, +\end{align} +\]

+ +

where the translation vector \(\t\) falls out because a translation has constant +gradient.

+ +

We do not know the rotation \(\Rot\) ahead of time, but we could be as generous +as possible and use the “best” rotation \(\Rot ← \argmin_\Rot ∫_Ω ‖ ∇\x - \Rot +∇\hat{\x} ‖² \;dA\):

+ +

\[ +∫_Ω \left\|∇\x - \left( \argmin_\Rot ∫_Ω ‖ ∇\x - \Rot ∇\hat{\x} ‖² \;dA \right)∇\hat{\x}\right\|² \;dA. +\]

+ +

If we treat \(\Rot\) as a degree of freedom along with the unknown positions +\(\x\), we can unify this into an optimization over \(\x\) and \(\Rot\):

+ +

\[ +\min_{\x,\Rot∈SO(3)} ∫_Ω \left\|∇\x - \Rot ∇\hat{\x}\right\|² \;dA. +\]

+ +

Optimizing this energy will ensure global rotation invariance. To ensure +local rotation invariance, we can replace \(\Rot ∈ SO(3)\) with a spatially +varying function \(\Rot : Ω → SO(3)\) that outputs a “best” rotation for any +point on the shape (see “A simple geometric model for elastic deformations” +[Chao et al. 2010]). In this way, the optimal rotation will be locally rigid +everywhere, or as-rigid-as-possible (ARAP).

+ +
+ +
+ +
+

For embedded solid shapes, we can take the rest shape given by \(\hat{\x}\) as +the parameterization \(Ω\), so that \(∇\hat{\x} = \I\). This allows us to rewrite +the as-rigid-as-possible energy as the square of the difference between the +deformation gradient and the closest rotation:

+ +

\[ +∫_Ω \left\|∇\x - \Rot ∇\hat{\x}\right\|² \;dA \\ +∫_Ω \left\|(∇\x + \I - \I) - \Rot \I \right\|² \;dA \\ +∫_Ω \left\|(\I + ∇\x - ∇\hat{x}) - \Rot \right\|² \;dA \\ +∫_Ω \left\|(\I + ∇\d) - \Rot \right\|² \;dA \\ +∫_Ω \left\|\F - \Rot \right\|² \;dA \\ +\]

+ +

This form provides a bridge between the as-rigid-as-possible energy common in +geometry processing to corotated linear elasticity used in +mechanics/physically-based simulation (made explicit in “A simple geometric +model for elastic deformations” [Chao et al. 2010]). See Section 3.4 of “FEM +Simulation of 3D Deformable Solids” [Sifakis 2012] for a graphics-mechanics +perspective.

+
+ +

Discrete as-rigid-as-possible energy

+ +

For a triangle mesh with displacing vertices, the gradient of the embedding +function is constant inside each triangle. In this way we can write the raw +gradient energy above as a double sum over all half-edges \(ij\) of all faces \(f\) +in the mesh:

+ +

\[ +½ ∫_Ω ‖ ∇ \x - ∇\hat{\x}‖² \;dA = ½ ∑\limits_{f ∈ F} ∑\limits_{ ij ∈ f} c_{ij} ‖ +(\v_i-\v_j) - (\hat{\v}_i-\hat{\v}_j)‖², +\] +where \(c_{ij}\) is cotangent of the angle opposite half-edge \(ij\).

+ +

To inject localized best fit rotations, we will assign an unknown rotation +matrix \(\Rot_k\) to each vertex \(k\) of the mesh and accounts for a third of the +energy integrated over incident triangles: +\[ +½ ∫_Ω ‖ ∇ \x - \Rot ∇\hat{\x}‖² \;dA = +⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} +c_{ij} ‖ (\v_i-\v_j) - \Rot_k (\hat{\v}_i-\hat{\v}_j)‖², +\] +where \(F(k)\) is the set of all faces incident on the \(k\)-th vertex.

+ +
+
Where do rotations live?
+ +

We have assigned a rotation for each vertex: there are \(n\) rotation matrices +as auxiliary degrees of freedom. This is in contrast to assigning +rotations per face (as in “A Local/Global Approach to Mesh Parameterization” +[Liu et al. 2008]). Per-face–or more generally per-element–rotations work +well for codimension zero +objects (triangle meshes in \(\R²\) or tetrahedral meshes in \(\R³\)). But for +triangle mesh surfaces in \(\R³\) (i.e., codimension 1), per-face rotations +would lead to “crumpling”) because bending along edges would not be +measured.

+
+ +

Optimization

+ +

The simplest method for optimizing the ARAP energy is by alternating between

+ +
    +
  1. finding the optimal rotations \(\Rot_k\) assuming the vertex positions \(\V\) are + fixed, and
    2. +
  2. finding the optimal vertex positions \(\V\) assuming all rotations \(\Rot_k\) are + fixed.
    3. +
+ +

Each rotation \(\Rot_k\) only affects the local energy and doesn’t interact with +the other rotations. So each can be optimized locally. In contrast, the +mesh vertex positions \(\V\) depend on each other requiring a global solve. In +the geometry processing literature, this is known as a local-global +optimization (see “As-rigid-as-possible surface modeling” [Sorkine & Alexa +2007]). It is also known as “alternating block coordinate descent” because we +have separated the variables into disjoint sets (\(\V,\Rot_1,…,\Rot_n\)) and taking +the optimal descent direction for each independently.

+ +

Observing the discrete energy above we can see that the energy is quadratic in +\(\V\) and quadratic in each \(\Rot_k\). Let’s start by separating the terms that are +quadratic and linear in \(\V\):

+ +

\[ +⅙ \underbrace{∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} c_{ij} (\v_i-\v_j)^\transpose(\v_i-\v_j)}_\text{quadratic} ++ +⅙ \underbrace{∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} c_{ij} (\v_i-\v_j)^\transpose \Rot_k (\hat{\v}_i-\hat{\v}_j)}_\text{linear} +\]

+ +

if we stack the rotation matrices \(\Rot_k\) into large matrix \(\Rot ∈ \R^{3n × +3}\) then we can write this energy in matrix form as:

+ +

\[ +\tr{ \V^\transpose \L \V } + \tr{ \V^\transpose \K \Rot }, +\] +where \(\L ∈ \R^{n × n}\) is the familiar cotangent discrete Laplacian matrix and +\(\K ∈ \R^{n × 3n}\) sparse matrix containing cotangents multiplied against +differences across edges in the rest mesh (e.g., \(\hat{\v}_i - \hat{\v}_j\)).

+ +
+
I’m so confused. What’s in the \(\K\) matrix?
+ +

Let’s take it slow. The \(\K\) matrix is represents the +bilinear form that combines unknown +vertex positions and unknown rotations. We have identified above that we can +write this in summation or matrix form: +\[ +⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} c_{ij} (\v_i-\v_j)^\transpose \Rot_k (\hat{\v}_i-\hat{\v}_j) += \tr{ \V^\transpose \K \Rot }, +\] +but how did we get here?

+ +

Let’s start with the summation form. The constants of this formula are the +\(c_{ij}\) terms and the \((\hat{\v}_i-\hat{\v}_j)\) terms. Since these always +appear together, let us merge them into weighted edge difference vectors +\(c_{ij} (\hat{\v}_i-\hat{\v}_j) =: \hat{\e}_{ij} ∈ \R³\):

+ +

\[ +⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} \underbrace{(\v_i-\v_j)^\transpose +\Rot_k \hat{\e}_{ij}}_{∈\R}, +\] +the inner term in the summation is an inner +product; that is, a +scalar. Let’s expose this +by expanding the matrix-vector products of the inner-product: +\[ +⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} ∑\limits_{α=1}^3 ∑\limits_{β=1}^3 + (v_i^α - v_j^α)R_k^{αβ}\hat{e}_{ij}^β. +\]

+ +

If our mesh is stored as a vertex list and face list, it’s not easy/efficient +to loop over per-vertex rotations (outer sum) and then over all half-edges of +incident faces (second sum). Instead, let’s rearrange these sums to loop over +all faces first, then the half-edges of that face, and then over all +per-vertex rotations that involve this half-edge: +\[ +⅙ ∑\limits_{f=1}^m ∑\limits_{ ij ∈ E(f)} \quad ∑\limits_{k | ij ∈F(k)} \quad ∑\limits_{α=1}^3 ∑\limits_{β=1}^3 + (v_i^α - v_j^α)R_k^{αβ}\hat{e}_{ij}^β, +\] +where the third sum is over all rotations \(k\) such that the half-edge \(ij\) +belongs to the half-edges of the faces incident on the \(k\)-th vertex: \(k | ij +∈F(k)\). Well, this means \(k\) can either be \(i\) or \(j\) or the third vertex of +the \(f\)-th face.

+ +

Now let’s turn our attention back to the +summand. The terms indexed by \(α\) +never mix. That is, we never add/multiply \(v_i^α\), \(v_j^γ\), and \(R_k^{δβ}\) +unless \(α=γ=δ\). This implies that we can write this summation in matrix form +as: +\[ +\V₁^\transpose \K₁ \Rot₁ + +\V₂^\transpose \K₂ \Rot₂ + +\V₃^\transpose \K₃ \Rot₃, +\] +where \(\V_α ∈ \R^n\) is \(α\)-th column of \(\V\), \(\Rot_α ∈ \R^{3n}\) is the \(α\)-th column of +\(\Rot\) and \(\K₁,\K₂,\K₃ ∈ \R^{n × 3n}\) are sparse matrices.

+ +

Further, the constant term \(\hat{e}^β_{ij}\) in the summation acts the same on +\((v_i^α - v_j^α)R_k^{αβ}\) for any \(α\) value. This implies that \(\K₁=\K₂=\K₃\), +so we can reduce the matrix form to: +\[ +\tr{\V \K \Rot}. +\]

+ +

Finally, we can answer what is in each entry \(K_{vw}\), where \(v\) chooses the +row and \(w=3k+β\) chooses the column of \(\K\). Treating our nested summation as +nested for-loops, we can increment entries in \(\K\) +\[ +\begin{align} +K_{i\ 3k+β} & \mathrel{+}= \hat{e}_{ij}^β, \\ +K_{j\ 3k+β} & \mathrel{+}= -\hat{e}_{ij}^β, \\ +\end{align} +\] +for each half-edge encountered.

+
+ +
Local step
+ +

Minimizing this energy with respect \(\Rot\) corresponds to minimizing:

+ +

\[ +\tr{ \underbrace{\V^\transpose \K}_{\C^\transpose} \Rot }, +\] +where \(\C ∈ \R^{3n × 3}\) stacks weighted covariance matrices \(\C_k ∈ \R^{3 × +3}\) for each region covered by the corresponding rotation \(\Rot_k\). We have +seen this problem before in the registration assignment. For each \(\C_k\), +\(\Rot_k\) will be the closest rotation matrix solved via singular value +decomposition.

+ +
Global step
+ +

Minimizing the energy above with respect to \(\V\) corresponds to solving a +Dirichlet energy-like minimization problem:

+ +

\[ +\min_\V \tr{ \V^\transpose \L \V } + \tr{ \V^\transpose \B } +\]

+ +

where \(\K \Rot =: \B ∈ \R^{n × 3}\) is a matrix of rotated vertex gradients. +Adding the handle constraints to the corresponding rows of \(\V\) this is easily +minimized by setting all partial derivatives with respect to the unknowns in +\(\V\) equal to zero (as in the linear methods above).

+ +
Implementation
+ +

In order to facilitate interactive deformation we would like our local and +global iterations to be computed as quickly as possible. Since the quadratic +form in the global step is the same regardless of the current rotations or +current handle positions, we can +prefactorize it (again, +as above). The matrix \(\K\) also does not depend on the rotations, current +positions or handle positions, so we can pre-build this matrix. This way +computing \(\C\) and \(\B\) is a simple matrix-matrix multiplication.

+ +
+

Note: When constructing \(\K\) it’s easiest to iterate over all +half-edges in the mesh (by iterating over all faces and then each of the +three edges). Each half-edge \(ij\) contributes terms tying \(\v_i,\v_j\) to +each of the (three) rotations \(\Rot_k\) that apply against their difference +(see “Fast Automatic Skinning Transformations” [Jacobson et al. 2012]).

+
+ +
Still trouble in paradise
+ +
+ +
+ +

The as-rigid-as-possible deformation method for surfaces described above has a +number of remaining problems:

+ +
    +
  1. like its gradient-based energy ancestor the deformation is not smooth at + constraints;
    2. +
  2. the energy punishes bending of the surface–which is good–but does so in + a way that diminishes as the mesh becomes higher and higher resolution, in + otherwords, the discrete energy is mesh-resolution dependent; and
    3. +
  3. the energy is biased by the original combinatorics of the mesh (even in + flat regions).
    4. +
+ +

Tasks

+ +

Blacklist

+ + + +

Whitelist

+ + + +

src/biharmonic_precompute.cpp

+ +

Precompute data needed to efficiently solve for a biharmonic deformation given +a mesh with vertices V and faces F and a list of selected vertices as +indices b into V. The output should be a prefacorized system using the +data struct employed by igl::min_quad_with_fixed.

+ +

src/biharmonic_solve.cpp

+ +

Given precomputation data and a list of handle displacements determine +displacements for all vertices in the mesh.

+ +

src/arap_precompute.cpp

+ +

Precompute data needed to efficiently conduct local-global iterations for an +arap deformation. This includes the data struct employed by +igl::min_quad_with_fixed to solve the global step" and constructing the +bilinear form K that mixes rotation degrees of freedom with unknown +positions for preparing the covariance matrices of the local step and the +linear term of the global step.

+ +

src/arap_single_iteration.cpp

+ +

Given precomputed data (data and K), handle positions bc and current +positions of all vertices U, conduct a single iteration of the local-global +solver for minimizing the as-rigid-as-possible energy. Output the positions +of all vertices of the mesh (by overwriting U).

+ + + diff --git a/README.md b/README.md index 5222e7c..6a673e3 100644 --- a/README.md +++ b/README.md @@ -17,80 +17,682 @@ on a given mesh: ./deformation [path to mesh.obj] +When the mesh is blue, the system is in "place handles" mode. Click on the mesh +to select vertex locations for control point handles. After pressing space to switch to +deformation mode, drag the handles. Pressing `m` will switch between different +deformation methods. + +![](images/knight-deformation.gif) + ## Background +In this assignment we explore smooth deformation of an existing shape. [Shape +deformation](https://en.wikipedia.org/wiki/Deformation_(mechanics)) has many +applications in geometry process; we will focus on interactive **_handle-based +deformation_**. In this setup, the user repositions a sparse set of points and the +goal is to propagate the transformation at these "handles" to the rest of the +shape. To be interactive we should aim for computing the new deformation of the +shape at around 30 [frames per +second](https://en.wikipedia.org/wiki/Frame_rate). + +Shape deformation is the transformation from its _rest shape_ to a +new/current/deformed shape. If the position of a point on some 3D rest shape + given by $\hat{\x} ∈ \R³$ then we will write that the unknown position +on the deformed shape is given by $\x ∈ \R³$. We can write this point's +displacement vector as $\x - \hat{\x} =: \d ∈ \R³$. + +The propagation of the handles' deformation can be thought of in two +complementary ways: + + 1. as a [scattered data + interpolation](https://en.wikipedia.org/wiki/Multivariate_interpolation#Irregular_grid_.28scattered_data.29) + problem, where handles provide sparse samples of an unknown [displacement + field](https://en.wikipedia.org/wiki/Displacement_field_(mechanics)), or + 2. as a shape optimization problem, where we try to define a _new_ shape that + retains the details of the old shape but fulfills the handle constraints. + +In the following discussion we will take advantage of the ability to switch +between thinking of the unknowns as the positions of the deformed shape ($\x$) +and displacements ($\d$). These views are equivalent, but often one or the +other provides a better intuitive understanding. + +#### Continuity + +We will limit ourselves to _continuous_ deformations of shapes. That is, the +shape will not tear, crack or change its topological features. + +If we represent our shape discretely as a [triangle +mesh](https://en.wikipedia.org/wiki/Triangle_mesh) (e.g., with _rest_ vertices +in $\hat{\V} +∈ \R^{n × 3}$ and faces in $F ∈ \{1,…,n\}^{m × 3}$, then we can _trivially_ +ensure a continuous deformation by determining new vertex positions $\V$. The +topology (connectivity) of the mesh ($F$) will not change. + +#### Generic Distortion Minimization + +A rest surface $\hat{S}$ +[immersed](https://en.wikipedia.org/wiki/Immersion_(mathematics)) in $\R³$ can +be described as a mapping $\hat{\x}$ from _some_ 2D parametric domain $Ω$. For any +parameters $u$ and $v$, $\hat{\x}$ describes the 3D position: + +\\[ +\hat{\x}(u,v) ∈ \R³. +\\] + +Similarly the deformed surface can be represented as a position function $\x: +Ω → \R³$. The displacement vector field is thus a function taking the +difference: $\d(u,v) = \x(u,v) - \hat{\x}(u,v)$. + +For the handle-based deformation problem we would like to find a new surface +(defined by $\x$) that: + + 1. adds as little _distortion_ as possible to the shape, and + 2. satisfies the users constraints at selected handle positions. + +We can cast this as an energy optimization problem. Suppose we have +energy [functional](https://en.wikipedia.org/wiki/Functional_(mathematics)) +$E(\x)$ that +measures the amount of distortion between the new shape ($\x$) and the rest +shape $\hat{x}$, then we could optimize for the best possible shape $\x$ by +minimizing $E$: + +\\[ +\min_\x E(\x). +\\] + +To ensure that the user's $k$ handle points are interpolated we add the +constraints: + +\\[ +\text{ subject to } \x(u_i, v_i) = \g_i \ ∀ i = \{1, … , k\}, +\\] +where $\g_i$ is the position of the $i$th control point handle. + +While the constraints are straightforward, we have many choices for how to +formulate the energy function $E$. A natural choice is to measure distortion in +an egalitarian way by integrating a _local_ measure of distortion at all points +on the surface: + +\\[ +\min_\x ∫_Ω ‖ e(\x) ‖² \ dA \quad \text{ subject to } \x(u_i, v_i) = \g_i \ ∀ i = \{1, … , k\}, +\\] + +where $e$ is a vector- or scalar- valued function measuring local (unsquared) +distortion. We will now consider different choices for $e$. + ### Linear Methods -Measure distortion with respect to small displacements +If we assume that the deformation between the rest shape given by $\hat{\x}$ +and the new shape given by $\x$ is _small_ then we can measure the distortion +of the deformation in terms of the smoothness of the displacement field. This +simplest methods will integrate the magnitude of derivatives of the +displacement field ($\d$): if the displacement field has large variations or +sudden changes then it is inducing a lot of distortion. #### Gradient-based energy -- Dependent on global coordinate system +Let us first consider minimizing the integral of squared variation of the +displacement field: + +\\[ +\min_\d ∫_Ω ‖ ∇\d ‖_F^2 \ dA \quad \text{ subject to } \d_i = +\g_i-\hat{\x}_i \ ∀ i = \{1, … , k\}, +\\] +where +$∇\d = \left(\begin{array}{ccc} +\frac{∂d^x}{∂u} & \frac{∂d^y}{∂u} & \frac{∂d^z}{∂u} \\ +\frac{∂d^x}{∂v} & \frac{∂d^y}{∂v} & \frac{∂d^z}{∂v} +\end{array}\right)$ +the [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant) +matrix of the displacement field $\d$ + +> ##### Deformation Gradient +> +> If $\I ∈ \R^{3 × 3}$ is the identity matrix, then the quantity $\F := \I + +> ∇\d$ is referred to as the [deformation +> gradient](https://en.wikipedia.org/wiki/Finite_strain_theory#Deformation_gradient_tensor) +> in the [mechanics](https://en.wikipedia.org/wiki/Continuum_mechanics) +> community. + +This is simply the familiar [Dirichlet +energy](https://en.wikipedia.org/wiki/Dirichlet's_energy) applied to each +coordinate function of the displacement field _independently_. + +We can discretize this over our triangle mesh surface the same way we have in +smoothing and parameterization assignments: + +\\[ +\min_{\D} \tr{\D^\transpose \L \D} \quad \text{subject to } \D_\text{handles} = +\g_\text{handles} - \hat{\V}_\text{handles}, +\\] +where the rows of $\g_\text{handles} ∈ \R^{k × 3}$ contains the new positions +of the $k$ control point handles. + +While easy to implement, this method suffers from a couple immediate problems: + + 1. it is not smooth at constraints, and + 2. then _influence_ of handles dies off too quickly. + +![Two regions of selected points (purple) are constrained. The gradient-based +energy produces a _continuous_ displacement but is not _smooth_ at the inner +constrained points (sharp crease). The Laplacian-based energy produces a +displacement with continuous positions and +derivatives.](images/bump-k-harmonic.jpg) + +By minimizing the Dirichlet energy, each coordinate of "displacement field" is +a [harmonic function](https://en.wikipedia.org/wiki/Harmonic_function). +Intuitively (however abstractly) we can think of each function as _diffusing_ +the user's constraints as if they were +[heat](https://en.wikipedia.org/wiki/Heat_equation) values. As such, each +function diffuses quickly to an average, slowly varying value over most of the +domain. As a displacement field a constant value for each coordinate function +would mean a translation: most of the shape is simply translated. + +![](images/bunny-harmonic.gif) + +The gradient operator ($∇$) is a [_linear_ +operator](https://en.wikipedia.org/wiki/Linear_map). We can alternatively view +our minimization above in terms of the unknown positions $\x$: + +\\[ +\min_\d ∫_Ω ‖ ∇\d ‖_F^2 \ dA ⇒ +\min_\x ∫_Ω ‖ ∇(\x - \hat{\x}) ‖_F^2 \ dA ⇒ +\min_\x ∫_Ω ‖ \underbrace{∇\x}_\text{after} - +\underbrace{∇\hat{\x}}_\text{before} ‖_F^2 \ dA. +\\] +If we think of the gradient of the position function $∇\x$ (with respect to the +underlying parameterization $u,v$) as a local geometric [feature +descriptor](https://en.wikipedia.org/wiki/Feature_(computer_vision)) then this +energy can be re-understood as measuring the difference in this feature before +and after the deformation. This is very sensible as we are trying to measure +distortion. We would expect that a low-distortion deformation would correspond +with a small change to local features. + +Unfortunately, the gradient of the position function $\x$ is a _poor_, +first-order local feature. + +#### Laplacian-based energy + +If we model distortion as the change in a local feature descriptor, then a +natural local and _relative_ descriptor would be one that compared the position +of some point on the shape to the average of its local neighborhood. We have +studied an operator that computes this in the smoothing assignment. The +[Laplace(-Beltrami) operator](https://en.wikipedia.org/wiki/Laplace_operator) +can be derived as taking exactly the difference of a functions value at a point +and the average (i.e., [centroid](https://en.wikipedia.org/wiki/Centroid)) of +an infinitesimal region around that point: + +\\[ +∆ f(\x) = \lim_{|B(\x)| → 0} \frac{1}{|B(\x))|} ∫_{B(\x)} f(\z) \;d\z - f(\x) +\\] + +(see, e.g., "Differential coordinates for local mesh morphing and deformation" +[Alexa et al. 2003] and expanded upon in "Laplacian Surface Editing" +[Sorkine et al. 2004]). + +When applied to the embedding function $\x$ the Laplace operator computes the +difference in _position_ between a point and its local neighborhood. This +_vector_ points in the +[normal](https://en.wikipedia.org/wiki/Normal_(geometry)) direction and its +magnitude corresponds inversely with [how flat the surface is +locally](https://en.wikipedia.org/wiki/Mean_curvature). + +Let's replace the gradient feature above with this _second-order_ feature +descriptor and massage our optimization problem back in terms of +displacements: + +\\[ +\min_\x ∫_Ω ‖ \underbrace{∆\x}_\text{after} - +\underbrace{∆\hat{\x}}_\text{before} ‖^2 \ dA ⇒ +\min_\x ∫_Ω ‖ ∆(\x - \hat{\x}) ‖^2 \ dA ⇒ +\min_\d ∫_Ω ‖ ∆\d ‖^2 \ dA. +\\] + +Just as we can show that harmonic functions ($∆\d = 0$) minimize the Dirichlet +energy, we can use [calculus of +variations](https://en.wikipedia.org/wiki/Calculus_of_variations) apply +[Green's identity](https://en.wikipedia.org/wiki/Green's_identities) _twice_ to +show that minimizers of the squared-Laplacian energy are _bi-_harmonic +functions ($∆∆\d = 0$ or $∆²\d = 0$). Obviously all harmonic functions are also +biharmonic functions. This implies that the space of biharmonic functions is +strictly _larger_. In particular, this will allow use to ensure continuity of +first derivatives across constrained values at handles. + +![](images/bunny-biharmonic.gif) + +The fact that each coordinate of the displacement field $\d$ will be a +bi-harmonic function is not so elucidating, but by minimizing the +squared-Laplacian energy integrated over the domain, we can say that it is +as-_harmonic_-as-possible. Harmonic functions include constant functions (i.e., +translations) but also any displacement that _bends_ in one direction by an +equal and opposite amount as it bends in the other direction: $∆\d = 0 → ∂\d/∂u ++ ∂\d/∂v = 0 → ∂\d/∂u = -∂\d/∂v$. + +To discretize this energy we can make use of our discrete Laplacian operator +$\L$. This matrix computes the locally _integrated_ Laplacian of a given +function specified by per-vertex values $\f$. Now we would like to integrate +the square of the _point-wise_ Laplacian. We can approximate the point-wise +Laplacian by the local [integral +average](https://en.wikipedia.org/wiki/Mean_of_a_function) of the Laplacian +$\M^{-1}\L \f$. Integrating this over the mesh we have our complete approximate +of the energy: + +\\[ +\begin{align} +∫_Ω ‖∆\d‖² \;dA &≈ \tr{ \D^\transpose \L^\transpose \M^{-\transpose} \M \M^{-1} \L +\D}\\ +&= \tr{ \D^\transpose \underbrace{\L^\transpose \M^{-1} \L}_{\Q} \D }, +\end{align} +\\] +where $\M ∈ \R^{n × n}$ is the mass-matrix for the given mesh and $\Q ∈ +\R^{n×n}$ can be thought of as the bi-Laplacian matrix. + +> #### $k$-harmonic +> +> The logical continuation of harmonic and biharmonic deformation is to consider +> _triharmonic_ ($∆³\d = 0$) and _tetraharmonic_ ($∆⁴\d = 0$) and so on. It's +> straightforward to implement these, though there are diminishing returns and +> increasing costs. + +##### Precomputation + +With out loss of generality, assume that the rows of the unknown displacements +$\D$ have been sorted so that displacements corresponding to handle vertices +are in the bottom part: + +\\[ +\D = \left(\begin{array}{c} +\D_\text{u} \\ +\D_\text{h} +\end{array} +\right) +\\] + +Since the displacements at handles are _known_ before the optimization, we can +separate the knowns and unknowns in the energy: + +\\[ +\min_{\D_\text{u}} +\tr{(\D_\text{u}^\transpose \ \D_\text{h}^\transpose) +\left(\begin{array}{cc} +\Q_\text{u,u} & \Q_\text{u,h} \\ +\Q_\text{h,u} & \Q_\text{h,h} +\end{array}\right) +\left(\begin{array}{c} + \D_\text{u} \\ + \D_\text{h} +\end{array} +\right)} \\ +\min_{\D_\text{u}} +\tr{\D_\text{u}^\transpose \Q_\text{u,u} \D_\text{u} + +2 \D_\text{u}^\transpose \Q_\text{u,h} \D_\text{h} + +\underbrace{\D_\text{h}^\transpose \Q_\text{h,h} +\D_\text{h}}_\text{constant}} \\ +\min_{\D_\text{u}} +\tr{ +\D_\text{u}^\transpose \Q_\text{u,u} \D_\text{u} + +2 \D_\text{u}^\transpose \Q_\text{u,h} \D_\text{h}} +\\] +where $\Q_\text{u,u} ∈ \R^{(n-k) × (n-k)}$ is the quadratic coefficients matrix +corresponding to the unknown displacements. + +This quadratic optimization problem may solved by setting all partial +derivatives with respect to degrees of freedom in $\D_\text{u}$ to zero: + +\\[ +2 \Q_\text{u,u} \D_\text{u} + 2 \Q_\text{u,h} \D_\text{h} += 0 → \D_\text{u} = \Q_\text{u,u}^{-1} \Q_\text{u,h} \D_\text{h} +\\] + +If we don't change _which_ vertices are handles, but only change the positions +of the selected handles, then only $\D_\text{h}$ changes above. In particular, +the matrix $\Q_\text{u,u}$ is unchanged. Therefore, we can +[prefactorize](https://en.wikipedia.org/wiki/Cholesky_decomposition) it so that +_applying its inverse_ is fast (`igl::min_quad_with_fixed` does this for you). + +> Actually the entire term $\Q_\text{u,u}^{-1} \Q_\text{u,h} =: \W ∈ \R^{(n-k) +> × k}$ does not change. The columns of $\W$ reveal how unknown displacements +> respond to each handle point's displacement (see also "An intuitive framework +> for real-time freeform modeling" [Botsch & Kobbelt 2004]). This provides a +> gateway to the relationship with linear blend skinning and automatic +> (biharmonic) weighting functions (see "Bounded Biharmonic Weights for +> Real-Time Deformation" [Jacobson et al. 2011]). + +##### Trouble in paradise + +Biharmonic displacements work well for small deformations and deformations that +do not imply a large rotation. However, the Laplacian of the position function +$∆\x$ as a feature descriptor is _not_ rotation invariant. This problem is true +for all linear differential features including the gradient of the embedding +function $∇\x$ considered above (see "On linear variational surface deformation +methods" [Botsch & Sorkine 2008]). + +![](images/knight-biharmonic-large-rotation.gif) + +This means that if the user transforms all of the handle locations by a rigid +transformation $\T$ these energies will _not_ measure zero for a displacement +equivalent to applying the rigid transformation $\T$ to the entire shape. We +would like _global_ rotation invariance, but we would also like this property +to apply _locally_ so that parts of the shape can easily rotate. + +### As-rigid-as-possible + +In the scenario where each handle +$i$ are perfectly transformed by a [rigid +transformation](https://en.wikipedia.org/wiki/Rigid_transformation) $\x_i = +\Rot \hat{\x}_i + \t$, where $\Rot ∈ SO(3) ⊂ \R^{3×3}$ is a rotation matrix and +$\t∈\R^3$ is a translation vector. If an +[oracle](https://en.wikipedia.org/wiki/Oracle) could only tell us this +particular rigid transformation then we could repair +the gradient-based energy above by pre-rotating the rest shape by this +transformation: + +\\[ +\begin{align} +∫_Ω ‖ ∇ \x - ∇(\Rot \hat{\x} + \t) ‖² \;dA + &= ∫_Ω ‖ ∇ \x - ∇(\Rot \hat{\x}) - ∇\t‖² \;dA \\ + &= ∫_Ω ‖ ∇ \x - \Rot ∇\hat{\x} ‖² \;dA, +\end{align} +\\] + +where the translation vector $\t$ falls out because a translation has constant +gradient. + +We do not know the rotation $\Rot$ ahead of time, but we could be as generous +as possible and use the "best" rotation $\Rot ← \argmin_\Rot ∫_Ω ‖ ∇\x - \Rot +∇\hat{\x} ‖² \;dA$: + +\\[ +∫_Ω \left\|∇\x - \left( \argmin_\Rot ∫_Ω ‖ ∇\x - \Rot ∇\hat{\x} ‖² \;dA \right)∇\hat{\x}\right\|² \;dA. +\\] + +If we treat $\Rot$ as a degree of freedom along with the unknown positions +$\x$, we can unify this into an optimization over $\x$ and $\Rot$: + +\\[ +\min_{\x,\Rot∈SO(3)} ∫_Ω \left\|∇\x - \Rot ∇\hat{\x}\right\|² \;dA. +\\] + +Optimizing this energy will ensure _global_ rotation invariance. To ensure +_local_ rotation invariance, we can replace $\Rot ∈ SO(3)$ with a spatially +varying _function_ $\Rot : Ω → SO(3)$ that outputs a "best" rotation for any +point on the shape (see "A simple geometric model for elastic deformations" +[Chao et al. 2010]). In this way, the optimal rotation will be locally rigid +everywhere, or _as-rigid-as-possible_ (ARAP). + +![](images/knight-arap-large-rotation.gif) + +> For embedded solid shapes, we can take the rest shape given by $\hat{\x}$ as +> the parameterization $Ω$, so that $∇\hat{\x} = \I$. This allows us to rewrite +> the as-rigid-as-possible energy as the square of the difference between the +> [deformation gradient](#deformationgradient) and the closest rotation: +> +> \\[ +> ∫_Ω \left\|∇\x - \Rot ∇\hat{\x}\right\|² \;dA \\ +> ∫_Ω \left\|(∇\x + \I - \I) - \Rot \I \right\|² \;dA \\ +> ∫_Ω \left\|(\I + ∇\x - ∇\hat{x}) - \Rot \right\|² \;dA \\ +> ∫_Ω \left\|(\I + ∇\d) - \Rot \right\|² \;dA \\ +> ∫_Ω \left\|\F - \Rot \right\|² \;dA \\ +> \\] +> +> This form provides a bridge between the as-rigid-as-possible energy common in +> geometry processing to _corotated linear elasticity_ used in +> mechanics/physically-based simulation (made explicit in "A simple geometric +> model for elastic deformations" [Chao et al. 2010]). See Section 3.4 of "FEM +> Simulation of 3D Deformable Solids" [Sifakis 2012] for a graphics-mechanics +> perspective. +> + +#### Discrete as-rigid-as-possible energy + +For a triangle mesh with displacing vertices, the gradient of the embedding +function is constant inside each triangle. In this way we can write the raw +gradient energy above as a double sum over all half-edges $ij$ of all faces $f$ +in the mesh: + +\\[ +½ ∫_Ω ‖ ∇ \x - ∇\hat{\x}‖² \;dA = ½ ∑\limits_{f ∈ F} ∑\limits_{ ij ∈ f} c_{ij} ‖ +(\v_i-\v_j) - (\hat{\v}_i-\hat{\v}_j)‖², +\\] +where $c_{ij}$ is cotangent of the angle opposite half-edge $ij$. + +To inject localized best fit rotations, we will assign an unknown rotation +matrix $\Rot_k$ to each vertex $k$ of the mesh and accounts for a third of the +energy integrated over incident triangles: +\\[ +½ ∫_Ω ‖ ∇ \x - \Rot ∇\hat{\x}‖² \;dA = +⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} +c_{ij} ‖ (\v_i-\v_j) - \Rot_k (\hat{\v}_i-\hat{\v}_j)‖², +\\] +where $F(k)$ is the set of all faces incident on the $k$-th vertex. + +> ##### Where do rotations _live_? +> We have assigned a rotation for each vertex: there are $n$ rotation matrices +> as auxiliary degrees of freedom. This is in contrast to assigning +> rotations per face (as in "A Local/Global Approach to Mesh Parameterization" +> [Liu et al. 2008]). Per-face--or more generally per-element--rotations work +> well for [codimension](https://en.wikipedia.org/wiki/Codimension) zero +> objects (triangle meshes in $\R²$ or tetrahedral meshes in $\R³$). But for +> triangle mesh surfaces in $\R³$ (i.e., codimension 1), per-face rotations +> would lead to "crumpling") because _bending_ along edges would not be +> measured. + +#### Optimization + +The simplest method for optimizing the ARAP energy is by alternating between + + 1. finding the optimal rotations $\Rot_k$ assuming the vertex positions $\V$ are + fixed, and + 2. finding the optimal vertex positions $\V$ assuming all rotations $\Rot_k$ are + fixed. + +Each rotation $\Rot_k$ only affects the local energy and doesn't interact with +the _other_ rotations. So each can be optimized _locally_. In contrast, the +mesh vertex positions $\V$ depend on each other requiring a _global_ solve. In +the geometry processing literature, this is known as a local-global +optimization (see "As-rigid-as-possible surface modeling" [Sorkine & Alexa +2007]). It is also known as "alternating block coordinate descent" because we +have separated the variables into disjoint sets ($\V,\Rot_1,…,\Rot_n$) and taking +the optimal descent direction for each independently. + +Observing the discrete energy above we can see that the energy is quadratic in +$\V$ and quadratic in each $\Rot_k$. Let's start by separating the terms that are +quadratic and linear in $\V$: + +\\[ +⅙ \underbrace{∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} c_{ij} (\v_i-\v_j)^\transpose(\v_i-\v_j)}_\text{quadratic} ++ +⅙ \underbrace{∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} c_{ij} (\v_i-\v_j)^\transpose \Rot_k (\hat{\v}_i-\hat{\v}_j)}_\text{linear} +\\] + +if we stack the rotation matrices $\Rot_k$ into large matrix $\Rot ∈ \R^{3n × +3}$ then we can write this energy in matrix form as: + +\\[ +\tr{ \V^\transpose \L \V } + \tr{ \V^\transpose \K \Rot }, +\\] +where $\L ∈ \R^{n × n}$ is the familiar cotangent discrete Laplacian matrix and +$\K ∈ \R^{n × 3n}$ sparse matrix containing cotangents multiplied against +differences across edges in the rest mesh (e.g., $\hat{\v}_i - \hat{\v}_j$). + +> ###### I'm so confused. What's in the $\K$ matrix? +> +> Let's take it slow. The $\K$ matrix is represents the +> [bilinear form](https://en.wikipedia.org/wiki/Bilinear_form) that combines unknown +> vertex positions and unknown rotations. We have identified above that we can +> write this in summation or matrix form: +> \\[ +> ⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} c_{ij} (\v_i-\v_j)^\transpose \Rot_k (\hat{\v}_i-\hat{\v}_j) +> = \tr{ \V^\transpose \K \Rot }, +> \\] +> but how did we get here? +> +> Let's start with the summation form. The _constants_ of this formula are the +> $c_{ij}$ terms and the $(\hat{\v}_i-\hat{\v}_j)$ terms. Since these always +> appear together, let us merge them into weighted edge difference vectors +> $c_{ij} (\hat{\v}_i-\hat{\v}_j) =: \hat{\e}_{ij} ∈ \R³$: +> +> \\[ +> ⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} \underbrace{(\v_i-\v_j)^\transpose +> \Rot_k \hat{\e}_{ij}}_{∈\R}, +> \\] +> the inner term in the summation is an [inner +> product](https://en.wikipedia.org/wiki/Inner_product_space); that is, a +> [scalar](https://en.wikipedia.org/wiki/Scalar_(mathematics)). Let's expose this +> by expanding the matrix-vector products of the inner-product: +> \\[ +> ⅙ ∑\limits_{k=1}^n ∑\limits_{ ij ∈ F(k)} ∑\limits_{α=1}^3 ∑\limits_{β=1}^3 +> (v_i^α - v_j^α)R_k^{αβ}\hat{e}_{ij}^β. +> \\] +> +> If our mesh is stored as a vertex list and face list, it's not easy/efficient +> to loop over per-vertex rotations (outer sum) and then over all half-edges of +> incident faces (second sum). Instead, let's rearrange these sums to loop over +> all faces first, then the half-edges of that face, and then over all +> per-vertex rotations that involve this half-edge: +> \\[ +> ⅙ ∑\limits_{f=1}^m ∑\limits_{ ij ∈ E(f)} \quad ∑\limits_{k | ij ∈F(k)} \quad ∑\limits_{α=1}^3 ∑\limits_{β=1}^3 +> (v_i^α - v_j^α)R_k^{αβ}\hat{e}_{ij}^β, +> \\] +> where the third sum is over all rotations $k$ such that the half-edge $ij$ +> belongs to the half-edges of the faces incident on the $k$-th vertex: $k | ij +> ∈F(k)$. Well, this means $k$ can either be $i$ or $j$ or the third vertex of +> the $f$-th face. +> +> Now let's turn our attention back to the +> [summand](https://en.wiktionary.org/wiki/summand). The terms indexed by $α$ +> never _mix_. That is, we never add/multiply $v_i^α$, $v_j^γ$, and $R_k^{δβ}$ +> unless $α=γ=δ$. This implies that we can write this summation in matrix form +> as: +> \\[ +> \V₁^\transpose \K₁ \Rot₁ + +> \V₂^\transpose \K₂ \Rot₂ + +> \V₃^\transpose \K₃ \Rot₃, +> \\] +> where $\V_α ∈ \R^n$ is $α$-th column of $\V$, $\Rot_α ∈ \R^{3n}$ is the $α$-th column of +> $\Rot$ and $\K₁,\K₂,\K₃ ∈ \R^{n × 3n}$ are sparse matrices. +> +> _Further_, the constant term $\hat{e}^β_{ij}$ in the summation acts the same on +> $(v_i^α - v_j^α)R_k^{αβ}$ for any $α$ value. This implies that $\K₁=\K₂=\K₃$, +> so we can reduce the matrix form to: +> \\[ +> \tr{\V \K \Rot}. +> \\] +> +> Finally, we can answer what is in each entry $K_{vw}$, where $v$ chooses the +> row and $w=3k+β$ chooses the column of $\K$. Treating our nested summation as +> nested for-loops, we can increment entries in $\K$ +> \\[ +> \begin{align} +> K_{i\ 3k+β} & \mathrel{+}= \hat{e}_{ij}^β, \\ +> K_{j\ 3k+β} & \mathrel{+}= -\hat{e}_{ij}^β, \\ +> \end{align} +> \\] +> for each half-edge encountered. -- Not smooth at constraints -- Each coordinate of "displacement field" is a harmonic function (picture of - arrows and picture of x-coordinate) - - Each coordinate is a heat diffusion toward some average - - As-_constant_-as-possible -#### Laplacian-based energy -- Each coordinate of "displacement field" is a biharmonic function - - larger space of functions than harmonic - - includes functions that are smooth at the constraints - - As-_harmonic_-as-possible (not simply harmonic since we have to satisfy more - constraints): locally harmonic roughly corresponds to locally - "fair"/non-ripply (technically, harmonic means bending in one direction - equals opposite amount of bending in the other direction. [Saddle]() - shaped) +##### Local step -- Corresponds to higher-order interpolation à la Hermite splines +Minimizing this energy with respect $\Rot$ corresponds to minimizing: -Dependent on local coordinate system, but still confused by large rotations. +\\[ +\tr{ \underbrace{\V^\transpose \K}_{\C^\transpose} \Rot }, +\\] +where $\C ∈ \R^{3n × 3}$ stacks weighted covariance matrices $\C_k ∈ \R^{3 × +3}$ for each region _covered_ by the corresponding rotation $\Rot_k$. We have +seen this problem before in the registration assignment. For each $\C_k$, +$\Rot_k$ will be the closest rotation matrix solved via [singular value +decomposition](https://en.wikipedia.org/wiki/Singular_value_decomposition). -#### $k$-harmonic +##### Global step -The logical continuation of harmonic and biharmonic deformation is to consider -triharmonic and tetraharmonic and so on. It's straightforward to extend these, -though there are diminishing returns and increasing costs. +Minimizing the energy above with respect to $\V$ corresponds to solving a +Dirichlet energy-like minimization problem: -### Non-Linear Methods +\\[ +\min_\V \tr{ \V^\transpose \L \V } + \tr{ \V^\transpose \B } +\\] -Take a step back. Why didn't we like the gradient based energy? Global -coordinate frame, non-smoothness at constraints. Put aside non-smoothness at -constraints. Let's try to tackle large rotations. +where $\K \Rot =: \B ∈ \R^{n × 3}$ is a matrix of rotated vertex gradients. +Adding the handle constraints to the corresponding rows of $\V$ this is easily +minimized by setting all partial derivatives with respect to the unknowns in +$\V$ equal to zero (as in the linear methods above). -**Deformation gradient** for 2D shapes +##### Implementation -#### As-rigid-as-possible energy +In order to facilitate interactive deformation we would like our local and +global iterations to be computed as quickly as possible. Since the quadratic +form in the global step is the _same_ regardless of the current rotations or +current handle positions, we can +[prefactorize](https://en.wikipedia.org/wiki/Cholesky_decomposition) it (again, +as above). The matrix $\K$ also does not depend on the rotations, current +positions or handle positions, so we can pre-build this matrix. This way +computing $\C$ and $\B$ is a simple matrix-matrix multiplication. -Warm-up: For 2D shapes +> **Note:** When constructing $\K$ it's easiest to iterate over _all_ +> half-edges in the mesh (by iterating over all faces and then each of the +> three edges). Each half-edge $ij$ _contributes_ terms tying $\v_i,\v_j$ to +> _each_ of the (three) rotations $\Rot_k$ that apply against their difference +> (see "Fast Automatic Skinning Transformations" [Jacobson et al. 2012]). -> ##### Volumes +##### Still trouble in paradise -##### Surfaces +![](images/knight-arap-high-vs-low-resolution.gif) -Overlapping integration regions +The as-rigid-as-possible deformation method for surfaces described above has a +number of remaining problems: + + 1. like its gradient-based energy ancestor the deformation is not smooth at + constraints; + 2. the energy punishes _bending_ of the surface--which is good--but does so in + a way that diminishes as the mesh becomes higher and higher resolution, in + otherwords, the discrete energy is mesh-resolution dependent; and + 3. the energy is biased by the original combinatorics of the mesh (even in + flat regions). ## Tasks ### Blacklist - - `igl::harmonic` - `igl::arap` + - `igl::arap_linear_block` - `igl::covariance_scatter_matrix` + - `igl::harmonic` ### Whitelist - - `igl::fit_rigid` (or your previous assignment's `closest_rotation`) + - `igl::polar_svd3x3` (or your previous assignment's `closest_rotation`) + - `igl::min_quad_with_fixed` + - `igl::cotmatrix_entries` + - `igl::cotmatrix` (or your previous implementation) + - `igl::massmatrix` (or your previous implementation) + +### `src/biharmonic_precompute.cpp` + +Precompute data needed to efficiently solve for a biharmonic deformation given +a mesh with vertices `V` and faces `F` and a list of selected vertices as +indices `b` into `V`. The output should be a prefacorized system using the +`data` struct employed by `igl::min_quad_with_fixed`. + +### `src/biharmonic_solve.cpp` -### k-harmonic precomputation +Given precomputation `data` and a list of handle _displacements_ determine +_displacements_ for all vertices in the mesh. -### k-harmonic deformation +### `src/arap_precompute.cpp` -### covariance scatter matrix +Precompute data needed to efficiently conduct local-global iterations for an +arap deformation. This includes the `data` struct employed by +`igl::min_quad_with_fixed` to solve the global step" and constructing the +bilinear form `K` that mixes rotation degrees of freedom with unknown +positions for preparing the covariance matrices of the local step and the +linear term of the global step. -### arap precompuation +### `src/arap_single_iteration.cpp` -### arap deformation +Given precomputed data (`data` and `K`), handle _positions_ `bc` and current +positions of all vertices `U`, conduct a _single_ iteration of the local-global +solver for minimizing the as-rigid-as-possible energy. Output the _positions_ +of all vertices of the mesh (by overwriting `U`). diff --git a/cmake/FindLIBIGL.cmake b/cmake/FindLIBIGL.cmake new file mode 100644 index 0000000..4ccc906 --- /dev/null +++ b/cmake/FindLIBIGL.cmake @@ -0,0 +1,36 @@ +# - Try to find the LIBIGL library +# Once done this will define +# +# LIBIGL_FOUND - system has LIBIGL +# LIBIGL_INCLUDE_DIR - **the** LIBIGL include directory +if(LIBIGL_FOUND) + return() +endif() + +find_path(LIBIGL_INCLUDE_DIR igl/readOBJ.h + HINTS + ENV LIBIGL + ENV LIBIGLROOT + ENV LIBIGL_ROOT + ENV LIBIGL_DIR + PATHS + ${CMAKE_SOURCE_DIR}/../.. + ${CMAKE_SOURCE_DIR}/.. + ${CMAKE_SOURCE_DIR} + ${CMAKE_SOURCE_DIR}/libigl + ${CMAKE_SOURCE_DIR}/../libigl + ${CMAKE_SOURCE_DIR}/../../libigl + /usr + /usr/local + /usr/local/igl/libigl + PATH_SUFFIXES include +) + +include(FindPackageHandleStandardArgs) +find_package_handle_standard_args(LIBIGL + "\nlibigl not found --- You can download it using:\n\tgit clone --recursive https://github.com/libigl/libigl.git ${CMAKE_SOURCE_DIR}/../libigl" + LIBIGL_INCLUDE_DIR) +mark_as_advanced(LIBIGL_INCLUDE_DIR) + +list(APPEND CMAKE_MODULE_PATH "${LIBIGL_INCLUDE_DIR}/../cmake") +include(libigl) diff --git a/data/decimated-knight.off b/data/decimated-knight.off new file mode 100644 index 0000000..7f3d2d8 --- /dev/null +++ b/data/decimated-knight.off @@ -0,0 +1,1504 @@ +OFF +502 1000 1500 +0.603331983089447 0.647517025470734 0.474732011556625 +0.474687993526459 0.39112401008606 0.514289021492004 +0.283832997083664 0.602063000202179 0.444357991218567 +0.531203985214233 0.294308006763458 0.534126996994019 +0.716175019741058 0.601866006851196 0.444018006324768 +0.70169997215271 0.554913997650146 0.494749993085861 +0.510098993778229 0.517956018447876 0.403840005397797 +0.705868005752563 0.622152984142303 0.423402011394501 +0.428317993879318 0.0994016975164413 0.422246009111404 +0.502596020698547 0.778217971324921 0.422021001577377 +0.58931702375412 0.315189987421036 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new file mode 100644 index 0000000..93950c8 --- /dev/null +++ b/include/arap_precompute.h @@ -0,0 +1,34 @@ +#ifndef ARAP_PRECOMPUTE_H +#define ARAP_PRECOMPUTE_H +#include +#include + + +namespace igl +{ + template struct min_quad_with_fixed_data; +} + +// Precompute data needed to efficiently conduct local-global iterations for an +// arap deformation. This includes the `data` struct employed by +// `igl::min_quad_with_fixed` to solve the global step" and constructing the +// bi-linear form `K` that mixes rotation degrees of freedom with unknown +// positions for preparing the covariance matrices of the local step and the +// linear term of the global step. +// +// Inputs: +// V #V by dim vertex positions +// F #F by simplex-size list of element indices +// b #b indices into V of handle vertices +// Outputs: +// data pre-factorized system matrix etc. (see `igl::min_quad_with_fixed`) +// K #R*dim by #V +void arap_precompute( + const Eigen::MatrixXd & V, + const Eigen::MatrixXi & F, + const Eigen::VectorXi & b, + igl::min_quad_with_fixed_data & data, + Eigen::SparseMatrix & K); + +#endif + diff --git a/include/arap_single_iteration.h b/include/arap_single_iteration.h new file mode 100644 index 0000000..21a68ff --- /dev/null +++ b/include/arap_single_iteration.h @@ -0,0 +1,38 @@ +#ifndef ARAP_SINGLE_ITERATION_H +#define ARAP_SINGLE_ITERATION_H +#include +#include + +#include "fit_rotation_avx.h" + + +using namespace Eigen; +namespace igl +{ + template struct min_quad_with_fixed_data; +} + +// Given precomputed data (`data` and `K`), handle _positions_ `bc` and current +// positions of all vertices `U`, conduct a _single_ iteration of the +// local-global solver for minimizing the as-rigid-as-possible energy. Output +// the _positions_ of all vertices of the mesh (by overwriting `U`). +// +// Inputs: +// data pre-factorized system matrix etc. (see `arap_precompute` +// K pre-constructed bi-linear term of energy combining rotations and +// positions +// U #V by dim list of current positions (see output) +// Outputs: +// U #V by dim list of new positions (see input) +Vector3d arap_single_iteration( + const igl::min_quad_with_fixed_data & data, + const Eigen::SparseMatrix & K, + const Eigen::MatrixXd & bc, + std::ostream & stream, + Eigen::MatrixXd & R_last, + Eigen::MatrixXd & U, + float* Rf, + float* Mf, + int num_of_group); + +#endif diff --git a/libigl b/libigl new file mode 160000 index 0000000..062e382 --- /dev/null +++ b/libigl @@ -0,0 +1 @@ +Subproject commit 062e3823c35cb868bf51cc8c247b04b7fa91c30f diff --git a/main.cpp b/main.cpp index 5d6487a..16a6591 100644 --- a/main.cpp +++ b/main.cpp @@ -1,25 +1,43 @@ +#include "arap_precompute.h" +#include "arap_single_iteration.h" +#include #include -#include +#include #include #include #include -#include -#include #include #include -#include #include +#include #include +#include + + +using namespace Eigen; +using namespace std; + + +std::ofstream outputFile; + + +Vector3d count_time; + + +using namespace Eigen; + +// Undoable struct State { - Eigen::MatrixXd CV; - Eigen::MatrixXd CU; + // Rest and transformed control points + Eigen::MatrixXd CV, CU; bool placing_handles = true; } s; int main(int argc, char *argv[]) { + // Undo Management std::stack undo_stack,redo_stack; const auto push_undo = [&](State & _s=s) @@ -47,88 +65,147 @@ int main(int argc, char *argv[]) } }; + + Eigen::MatrixXd R_last; Eigen::MatrixXd V,U; Eigen::MatrixXi F; long sel = -1; - Eigen::RowVector3f last; + Eigen::RowVector3f last_mouse; + igl::min_quad_with_fixed_data arap_data; + Eigen::SparseMatrix arap_K; + + // test IO + outputFile.open("./axis_angle_decimated_knight.txt", std::ios_base::app); + // Load input meshes igl::read_triangle_mesh( - (argc>1?argv[1]:"../shared/data/bunny.off"),V,F); + (argc>1?argv[1]:"../data/decimated-knight.off"),V,F); + + // init R_last + R_last.resize(3*V.rows(), 3); + for (int i = 0; i < V.rows(); i++) { + R_last.block(3 * i, 0, 3, 3) = MatrixXd::Identity(3, 3); + } + + int num_of_group = V.rows()/8+1; + int num_of_ele = num_of_group*72; + + + // initialize rotation matrices + void *R = NULL; + posix_memalign(&R, 32, num_of_ele * sizeof(float)); + float *Rf = (float *)R; + for (int i = 0; i < num_of_group; i++) { + for (int j = 0; j < 8; j++) { + Rf[i*72+0*8+j] = 1.0; // R(0,0) + Rf[i*72+1*8+j] = 0.0; // R(0,1) + Rf[i*72+2*8+j] = 0.0; // R(0,2) + Rf[i*72+3*8+j] = 0.0; // R(1,0) + Rf[i*72+4*8+j] = 1.0; // R(1,1) + Rf[i*72+5*8+j] = 0.0; // R(1,2) + Rf[i*72+6*8+j] = 0.0; // R(2,0) + Rf[i*72+7*8+j] = 0.0; // R(2,1) + Rf[i*72+8*8+j] = 1.0; // R(2,2) + } + } + + + void *M = NULL; + posix_memalign(&M, 32, num_of_ele * sizeof(float)); + float *Mf = (float *)M; + + U = V; - // Load data into MatrixXd rather than VectorXd for simpler `smooth` API - // Just use y-coordinates as data to be smoothed - // Create a libigl Viewer object to toggle between point cloud and mesh - igl::viewer::Viewer viewer; + igl::opengl::glfw::Viewer viewer; std::cout<bool + [&](igl::opengl::glfw::Viewer&, int, int)->bool { - last = Eigen::RowVector3f( - viewer.current_mouse_x, - viewer.core.viewport(3) - viewer.current_mouse_y, - 0); + last_mouse = Eigen::RowVector3f( + viewer.current_mouse_x,viewer.core().viewport(3)-viewer.current_mouse_y,0); if(s.placing_handles) { + // Find closest point on mesh to mouse position int fid; - Eigen::Vector3f bc; - // Cast a ray in the view direction starting from the mouse position + Eigen::Vector3f bary; if(igl::unproject_onto_mesh( - last.head(2), - viewer.core.view * viewer.core.model, - viewer.core.proj, - viewer.core.viewport, + last_mouse.head(2), + viewer.core().view, + viewer.core().proj, + viewer.core().viewport, V, F, - fid, bc)) + fid, bary)) { - push_undo(); - s.CV.conservativeResize(s.CV.rows()+1,3); - s.CV.row(s.CV.rows()-1) = - V.row(F(fid,0))*bc(0)+ V.row(F(fid,1))*bc(1)+ V.row(F(fid,2))*bc(2); - update(); - return true; + long c; + bary.maxCoeff(&c); + Eigen::RowVector3d new_c = V.row(F(fid,c)); + if(s.CV.size()==0 || (s.CV.rowwise()-new_c).rowwise().norm().minCoeff() > 0) + { + push_undo(); + s.CV.conservativeResize(s.CV.rows()+1,3); + // Snap to closest vertex on hit face + s.CV.row(s.CV.rows()-1) = new_c; + update(); + return true; + } } }else { - // Get closest control point + // Move closest control point Eigen::MatrixXf CP; igl::project( Eigen::MatrixXf(s.CU.cast()), - viewer.core.view * viewer.core.model, - viewer.core.proj, viewer.core.viewport, CP); - Eigen::VectorXf D = (CP.rowwise()-last).rowwise().norm(); + viewer.core().view, + viewer.core().proj, viewer.core().viewport, CP); + Eigen::VectorXf D = (CP.rowwise()-last_mouse).rowwise().norm(); sel = (D.minCoeff(&sel) < 30)?sel:-1; if(sel != -1) { - last(2) = CP(sel,2); + last_mouse(2) = CP(sel,2); push_undo(); update(); return true; @@ -137,32 +214,41 @@ int main(int argc, char *argv[]) return false; }; - viewer.callback_mouse_move = [&](igl::viewer::Viewer &, int,int)->bool + viewer.callback_mouse_move = [&](igl::opengl::glfw::Viewer &, int,int)->bool { if(sel!=-1) { - Eigen::RowVector3f drag( + Eigen::RowVector3f drag_mouse( viewer.current_mouse_x, - viewer.core.viewport(3) - viewer.current_mouse_y, - last(2)); + viewer.core().viewport(3) - viewer.current_mouse_y, + last_mouse(2)); Eigen::RowVector3f drag_scene,last_scene; - igl::unproject(drag, viewer.core.view * viewer.core.model, viewer.core.proj,viewer.core.viewport,drag_scene); - igl::unproject(last, viewer.core.view * viewer.core.model, viewer.core.proj,viewer.core.viewport,last_scene); + igl::unproject( + drag_mouse, + viewer.core().view, + viewer.core().proj, + viewer.core().viewport, + drag_scene); + igl::unproject( + last_mouse, + viewer.core().view, + viewer.core().proj, + viewer.core().viewport, + last_scene); s.CU.row(sel) += (drag_scene-last_scene).cast(); - last = drag; - + last_mouse = drag_mouse; update(); return true; } return false; }; - viewer.callback_mouse_up = [&](igl::viewer::Viewer&, int, int)->bool + viewer.callback_mouse_up = [&](igl::opengl::glfw::Viewer&, int, int)->bool { sel = -1; return false; }; viewer.callback_key_pressed = - [&](igl::viewer::Viewer &, unsigned int key, int mod) + [&](igl::opengl::glfw::Viewer &, unsigned int key, int mod) { switch(key) { @@ -173,12 +259,23 @@ int main(int argc, char *argv[]) s.CU = s.CV; break; } + case 'U': + case 'u': + { + // Just trigger an update + break; + } case ' ': push_undo(); s.placing_handles ^= 1; - if(!s.placing_handles) + if(!s.placing_handles && s.CV.rows()>0) { + // Switching to deformation mode s.CU = s.CV; + Eigen::VectorXi b; + igl::snap_points(s.CV,V,b); + // PRECOMPUTATION FOR DEFORMATION + arap_precompute(V,F,b,arap_data,arap_K); } break; default: @@ -188,32 +285,44 @@ int main(int argc, char *argv[]) return true; }; + // Special callback for handling undo viewer.callback_key_down = - [&](igl::viewer::Viewer &, unsigned char key, int mod)->bool + [&](igl::opengl::glfw::Viewer &, unsigned char key, int mod)->bool { - switch(key) + if(key == 'Z' && (mod & GLFW_MOD_SUPER)) { - case 'Z': - if(mod & GLFW_MOD_SUPER) - { - if(mod & GLFW_MOD_SHIFT) - { - redo(); - }else - { - undo(); - } - } - update(); - break; + (mod & GLFW_MOD_SHIFT) ? redo() : undo(); + update(); + return true; + } + return false; + }; + viewer.callback_pre_draw = + [&](igl::opengl::glfw::Viewer &)->bool + { + if(viewer.core().is_animating && !s.placing_handles && method == ARAP) + { + count_time = arap_single_iteration(arap_data,arap_K,s.CU,outputFile,R_last,U,Rf,Mf,num_of_group); + update(); } return false; }; - viewer.data.set_mesh(V,F); - viewer.core.show_lines = false; - viewer.core.is_animating = true; + + + viewer.core().background_color = Eigen::Vector4f(1,1,1,0); + + viewer.data().set_mesh(V,F); + viewer.data().show_lines = false; + viewer.core().is_animating = true; + viewer.data().face_based = true; update(); viewer.launch(); + + + outputFile.close(); + return EXIT_SUCCESS; + + } diff --git a/shared b/shared deleted file mode 160000 index 8e5599c..0000000 --- a/shared +++ /dev/null @@ -1 +0,0 @@ -Subproject commit 8e5599cfa7eb07492bb17682971373a85ff88c01 diff --git a/src/arap_precompute.cpp b/src/arap_precompute.cpp new file mode 100644 index 0000000..795595e --- /dev/null +++ b/src/arap_precompute.cpp @@ -0,0 +1,46 @@ +#include "arap_precompute.h" +#include +#include + +using namespace Eigen; +using namespace std; + +void arap_precompute( + const Eigen::MatrixXd & V, + const Eigen::MatrixXi & F, + const Eigen::VectorXi & b, + igl::min_quad_with_fixed_data & data, + Eigen::SparseMatrix & K) +{ + // prefactorize L first + SparseMatrix L, Aeq; + igl::cotmatrix(V, F, L); + igl::min_quad_with_fixed_precompute(L, b, Aeq, false, data); + + // construct K + vector> triplets; + triplets.reserve(F.rows() * 3 * 3 * 3 * 2); + + for (int m = 0; m < F.rows(); m++) { + for (int n = 0; n < 3; n++) { + int i = F(m, n % 3); + int j = F(m, (n + 1) % 3); + RowVector3d eij = (V.row(i) - V.row(j)) * L.coeff(i, j) / 6.0; + // iterate all three vertices + for (int a = 0; a < 3; a++) { + int k = F(m, (n + a) % 3); + // iterate all three coordinates + for (int b = 0; b < 3; b++) { + triplets.push_back(Triplet(i, 3 * k + b, eij(b))); + triplets.push_back(Triplet(j, 3 * k + b, -eij(b))); + } + } + } + } + + K.resize(V.rows(), V.rows() * 3); + + K.setFromTriplets(triplets.begin(), triplets.end()); + + +} diff --git a/src/arap_single_iteration.cpp b/src/arap_single_iteration.cpp new file mode 100644 index 0000000..2085c3a --- /dev/null +++ b/src/arap_single_iteration.cpp @@ -0,0 +1,91 @@ +#include "arap_single_iteration.h" +#include +#include +#include +#include +#include + + +using namespace Eigen; + + +Vector3d arap_single_iteration( + const igl::min_quad_with_fixed_data & data, + const Eigen::SparseMatrix & K, + const Eigen::MatrixXd & bc, + std::ostream & stream, + Eigen::MatrixXd & R_last, + Eigen::MatrixXd & U, + float* Rf, + float* Mf, + int num_of_group) +{ + + // two level optimization: local and global + // construct C first + MatrixXd C = K.transpose().eval() * U; + MatrixXd R(3 * data.n, 3); // a stack of rotation 3x3 matrix for each vertex + Matrix3d Ck, Rk, T, R_lastk; + Matrix3d Q = MatrixXd::Identity(3, 3); + + + // new version + for (int i = 0; i < num_of_group-1; i++) { + for (int j = 0; j < 9; j++) { + Mf[i*72+j*8+0] = (float)C(i*24+0*3+j/3, j%3); + Mf[i*72+j*8+1] = (float)C(i*24+1*3+j/3, j%3); + Mf[i*72+j*8+2] = (float)C(i*24+2*3+j/3, j%3); + Mf[i*72+j*8+3] = (float)C(i*24+3*3+j/3, j%3); + Mf[i*72+j*8+4] = (float)C(i*24+4*3+j/3, j%3); + Mf[i*72+j*8+5] = (float)C(i*24+5*3+j/3, j%3); + Mf[i*72+j*8+6] = (float)C(i*24+6*3+j/3, j%3); + Mf[i*72+j*8+7] = (float)C(i*24+7*3+j/3, j%3); + } + } + + fit_rotation_small_avx(Mf, Rf, num_of_group-1); + + + // new version + for (int i = 0; i < num_of_group-1; i++) { + for (int j = 0; j < 9; j++) { + R(i*24+0*3+j/3, j%3) = (double)Rf[i*72+j*8+0]; + R(i*24+1*3+j/3, j%3) = (double)Rf[i*72+j*8+1]; + R(i*24+2*3+j/3, j%3) = (double)Rf[i*72+j*8+2]; + R(i*24+3*3+j/3, j%3) = (double)Rf[i*72+j*8+3]; + R(i*24+4*3+j/3, j%3) = (double)Rf[i*72+j*8+4]; + R(i*24+5*3+j/3, j%3) = (double)Rf[i*72+j*8+5]; + R(i*24+6*3+j/3, j%3) = (double)Rf[i*72+j*8+6]; + R(i*24+7*3+j/3, j%3) = (double)Rf[i*72+j*8+7]; + } + } + + + // // old version + // for (int k = 0; k < data.n; k++) { + + // Ck = C.block(3*k, 0, 3, 3); + + // // fit rotation + // R_lastk = R_last.block(3*k, 0, 3, 3); + + // Ck = R_lastk.transpose().eval() * Ck; // warm start + + // igl::polar_svd3x3(Ck, Rk); + + // R_lastk = R_lastk * Rk; + // R.block(3 * k, 0, 3, 3) = R_lastk.eval(); + // R_last.block(3 * k, 0, 3, 3) = R_lastk.eval(); + // } + + + // Global Step + igl::min_quad_with_fixed_solve(data, K * R, bc, MatrixXd(), U); + + return Vector3d(0., 0., 0.); + +} + + + +