# Function Transfer

Intrinsic triangulations provide high quality function spaces, even on near-degenerate geometry, which dramatically improves the accuracy of PDE-based algorithms (e.g. the heat-based methods in geometry-central). However, these accurate solutions live on the intrinsic triangulation, and cannot be represented exactly as piecewise-linear functions on the original mesh.

Given an input mesh of some surface, and an intrinsic triangulations of that surface (which may have additional vertices inserted), these functions allow you to transfer values between the two domains via several different strategies.

## Pointwise Transfer at Vertices

The simplest method to transfer functions between two triangulations of the same surface is to simply copy values at vertices. The following methods, from the IntrinsicTriangulation class, implement this simple transfer scheme in both the forward and reverse direction.

VertexData<T> IntrinsicTriangulation::sampleFromInput(const VertexData<T>& dataOnInput)

Given data defined on the vertices of the input triangulation, samples it to the vertices of the intrinsic triangulation.

If the intrinsic triangulation contains new, inserted vertices which are not in common with the input triangulation, they sample a linearly-interpolated value from the input function.

VertexData<T> IntrinsicTriangulation::restrictToInput(const VertexData<T>& dataOnIntrinsic)

Given data defined on the vertices of the intrinsic triangulation, restrict it to the vertices of the input triangulation (that is, simply copy values from the shared vertices).

If the intrinsic triangulation contains new, inserted vertices which are not in common with the input triangulation, their values are ignored for the purposes of restriction.

## Transfer to the Common Subdivision

Another possibility is to transfer data to the common subdivison, a special triangulation which is a superset of both the input and the intrinsic triangulation. This transfer is easy to define, because the common subdivision is precisely the triangulation whose linear bases can exactly represent functions from either the input or intrinsic triangulation.

Of course, unlike the other methods described on this page, this strategy does not transfer functions directly between an input triangulation and an intrinsic triangulation, but rather transfers values from either the input or intrinsic triangulation to the common subdivision.

One common use for transferring values to the common subdivision is visualization, because the common subdivision is naturally embedded in space as an ordinary mesh.

VertexData<T> CommonSubdivision::interpolateAcrossA(const VertexData<T>& dataA)
VertexData<T> CommonSubdivision::interpolateAcrossB(const VertexData<T>& dataB)

Linearly interpolates data on meshA (resp. meshB) to the common subdivision.

FaceData<T> CommonSubdivision::copyFromA(const FaceData<T>& dataA)
FaceData<T> CommonSubdivision::copyFromB(const FaceData<T>& dataB)

Copy data at faces from one of the meshes to the common subdivision. Each face of the common subdivision gets the value from the face which contains it. The return value is defined per-face of the common subdivision mesh.

SparseMatrix<double> CommonSubdivison::interpolationMatrixA()
SparseMatrix<double> CommonSubdivision::interpolationMatrixB()

Yields a |V| x |V_A| matrix (resp. |V| x |V_B|) which linearly interpolates data on meshA (resp. meshB) to the common subdivision. Here |V| denotes the number of vertices in the common subdivision.

## L2-Optimal Transfer

#include "geometrycentral/surface/transfer_functions.h"

When transferring data directly between the input triangulation and intrinsic triangulation, simple sampling values is naive, and has no reason to be the “best” approach. Instead, one can directly compute the function on the other surface, in the sense of $L_2$-distance between functions—see Integer Coordinates for Intrinsic Geometry Processing for details.

Computationally, this amounts to solving a sparse linear least-squares problem defined via the common subdivision. Though somewhat more expensive, this approach can greatly improve accuracy.

### Single Transfer Functions

A one-off utility function is provided which transfers functions between different triangulations of the same surface. Repeated solves should use the stateful version below.

Example

#include "geometrycentral/surface/meshio.h"
#include "geometrycentral/surface/signpost_intrinsic_triangulation.h"
#include "geometrycentral/surface/transfer_functions.h"

std::unique_ptr<SurfaceMesh> mesh;
std::unique_ptr<VertexPositionGeometry> geometry;

// Create an intrinsic triangulation
SignpostIntrinsicTriangulation intTri(*mesh, *geometry);

// Change the intrinsic triangulation
intTri.delaunayRefine();

// Compute something useful on the intrinsic triangulation
VertexData<double> f_intrinsic = /* some function */

// Transfer function back to extrinsic mesh
VertexData<double> f_extrinsic = transferBtoA(intTri, f_intrinsic, TransferMethod::L2);


VertexData<double> transferAtoB(IntrinsicTriangulation& intTri, const VertexData<double>& valuesOnA, TransferMethod method)

Transfers a scalar function from intTri.inputMesh to intTri.intrinsicMesh

• valuesOnA : the data on intTri.inputMesh to be transferred.

• method : either TransferMethod::Pointwise for pointwise transfer of TransferMethod::L2 for $L^2$-optimal transfer.

VertexData<double> transferBtoA(IntrinsicTriangulation& intTri, const VertexData<double>& valuesOnB, TransferMethod method)

Transfers a scalar function from intTri.intrinsicMesh to intTri.inputMesh

• valuesOnB : the data on intTri.intrinsicMesh to be transferred.

• method : either TransferMethod::Pointwise for pointwise transfer of TransferMethod::L2 for $L^2$-optimal transfer.

### Transfer Method

The method argument is an enum:

enum class TransferMethod { Pointwise = 0, L2 };


for completeness, theTransferMethod::Pointwise option implements the pointwise transfer described above, while TransferMethod::L2 is the optimal scheme described here.

### Repeated Transfer Functions

If many functions are to be transferred, pre-factoring the linear least-squares problem can greatly improve performance. The stateful class AttributeTransfer facilitates this precomputation.

#include "geometrycentral/surface/meshio.h"
#include "geometrycentral/surface/signpost_intrinsic_triangulation.h"
#include "geometrycentral/surface/transfer_functions.h"

std::unique_ptr<SurfaceMesh> mesh;
std::unique_ptr<VertexPositionGeometry> geometry;

// Create an intrinsic triangulation
SignpostIntrinsicTriangulation intTri(*mesh, *geometry);

// Change the intrinsic triangulation
intTri.delaunayRefine();

// Create the AttributeTransfer object
AttributeTransfer transfer(intTri);

// Compute several functions on the intrinsic triangulation
std::vector<VertexData<double>> intrinsicFunctions = /* some functions */

// Transfer functions back to extrinsic mesh
for (VertexData<double> f_intrinsic : intrinsicFunctions) {
VertexData<double> f_extrinsic = transfer.transferBtoA(f_intrinsic, TransferMethod::L2);
/* do something with f_extrinsic */
}


#### Constructors

AttributeTransfer::AttributeTransfer(CommonSubdivision& cs, VertexPositionGoemetry& geomA)

Create a new solver for attribute transfer. Precomputation is performed lazily as needed.

• cs is the common subdivision of the two triangulations between which the solver will transfer data.

• geomA is the geometry of meshA.

AttributeTransfer::AttributeTransfer(CommonSubdivision& cs, IntrinsicTriangulation& intTri)

Create a new solver for attribute transfer. Precomputation is performed lazily as needed.

• cs is the common subdivision of the two triangulations between which the solver will transfer data.

• intTri is the IntrinsicTriangulation object representing the two triangulations.

#### Methods

VertexData<double> AttributeTransfer::transferAtoB(const VertexData<double>& valuesOnA, TransferMethod method)

Transfers a scalar function from meshA to meshB

• valuesOnA : the data on meshA to be transferred.

• method : either TransferMethod::Pointwise for pointwise transfer of TransferMethod::L2 for $L^2$-optimal transfer.

VertexData<double> AttributeTransfer::transferBtoA(const VertexData<double>& valuesOnB, TransferMethod method)

Transfers a scalar function from meshB to meshA

• valuesOnB : the data on meshB to be transferred.

• method : either TransferMethod::Pointwise for pointwise transfer of TransferMethod::L2 for $L^2$-optimal transfer.