### Create a 3D Translation Parametric Transformation in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Instantiates a 3D translation transformation using a specified Euclidean vector. This is an example of a parametric transformation, which is defined by a finite set of parameters.

```Scala
val translation = Translation3D(EuclideanVector3D(1, 3, 0.5))
translation.parameters
```

--------------------------------

### Get Inverse of Rigid Transformation in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Calculates the inverse of a rigid transformation, specifically demonstrating the inverse of a RotationThenTranslation transformation.

```Scala
val rinv : TranslationThenRotation[_3D] = RotationThenTranslation(rotation, translation).inverse
```

--------------------------------

### Get Derivative of a Differentiable 3D Transformation in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Calculates the derivative of a transformation with respect to its position. This is possible if the transformation implements the CanDifferentiateWRTPosition trait.

```Scala
val tDifferentiable : Transformation[_3D] with CanDifferentiateWRTPosition[_3D] = Translation3D(EuclideanVector3D(1, 0, 0))
val tDerivative : Point[_3D] => SquareMatrix[_3D] = tDifferentiable.derivativeWRTPosition
```

--------------------------------

### Get Inverse of an Invertible 3D Transformation in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Retrieves the inverse transformation for a transformation that supports inversion, such as a translation. This requires the transformation to have the CanInvert trait.

```Scala
val tInvertible : Transformation[_3D] with CanInvert[_3D, Transformation] = Translation3D(EuclideanVector3D(1, 0, 0))
val tinv : Translation[_3D] = tInvertible.inverse
```

--------------------------------

### Initialize Scalismo and Load Meshes

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Initializes Scalismo's native libraries and loads a 'noseless' mesh from an STL file. It then displays the mesh with a specific color in the UI.

```Scala
scalismo.initialize()
val ui = ScalismoUI()
val noseless = MeshIO.readMesh(new File("datasets/noseless.stl")).get
val targetGroup = ui.createGroup("noseless")
val targetView = ui.show(targetGroup, noseless, "noseless")
targetView.color = Color.PINK
```

--------------------------------

### Initialize Scalismo UI and Load Models

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Initializes the Scalismo environment and loads 3D mesh data and statistical models from files. It demonstrates how to organize objects into groups within the UI for visualization.

```scala
scalismo.initialize()
val ui = ScalismoUI()
val targetGroup = ui.createGroup("noseless")
val noseless = MeshIO.readMesh(new File("datasets/noseless.stl")).get
val targetView = ui.show(targetGroup, noseless, "noseless")
targetView.color = Color.PINK

val littleModelGroup = ui.createGroup("littleModel group")
val littleModel = StatismoIO.readStatismoMeshModel(new File("datasets/model.h5")).get
val littleModelView = ui.show(littleModelGroup, littleModel, "littleModel")
littleModelView.meshView.color = Color.ORANGE
```

--------------------------------

### Write 3D Triangle Mesh with Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Shows how to create a triangle mesh programmatically and save it to disk. The output format is automatically determined by the file extension provided in the File object.

```scala
import scalismo.io.MeshIO
import scalismo.mesh.{TriangleMesh3D, TriangleCell, TriangleList}
import scalismo.common.{PointId, UnstructuredPoints}
import scalismo.geometry.{Point, Point3D, _3D}
import java.io.File

val points = IndexedSeq(
  Point3D(0.0, 0.0, 0.0),
  Point3D(1.0, 0.0, 0.0),
  Point3D(0.5, 1.0, 0.0),
  Point3D(0.5, 0.5, 1.0)
)

val triangles = IndexedSeq(
  TriangleCell(PointId(0), PointId(1), PointId(2)),
  TriangleCell(PointId(0), PointId(1), PointId(3)),
  TriangleCell(PointId(1), PointId(2), PointId(3)),
  TriangleCell(PointId(0), PointId(2), PointId(3))
)

val mesh = TriangleMesh3D(points, TriangleList(triangles))

MeshIO.writeMesh(mesh, new File("output.vtk"))
MeshIO.writeMesh(mesh, new File("output.stl"))
MeshIO.writeMesh(mesh, new File("output.ply"))
```

--------------------------------

### Manage Scalismo UI Groups and Landmarks

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Demonstrates how to create UI groups and display models or landmarks within them. This replaces older global show methods with explicit group management.

```scala
val modelGroup = ui.createGroup("model group")
ui.show(modelGroup, model, "model")
val lms = LandmarkIO.readLandmarksJson[_3D](new File("datasets/noseFittingLandmarks.json")).get
ui.show(targetGroup, lms, "noseless")
```

--------------------------------

### Write Discrete Images to VTK and NIfTI

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Demonstrates creating a 3D discrete image domain and writing the resulting image data to VTK or NIfTI file formats. This process requires a defined image domain and voxel data.

```scala
import scalismo.io.ImageIO
import scalismo.image.{DiscreteImage, DiscreteImageDomain, DiscreteImageDomain3D}
import scalismo.geometry.{Point3D, EuclideanVector3D, IntVector3D, _3D}
import java.io.File

val origin = Point3D(0.0, 0.0, 0.0)
val spacing = EuclideanVector3D(1.0, 1.0, 1.0)
val size = IntVector3D(64, 64, 64)
val domain = DiscreteImageDomain3D(origin, spacing, size)

val image = DiscreteImage[_3D, Short](domain, pt => {
  val dist = math.sqrt(pt.x * pt.x + pt.y * pt.y + pt.z * pt.z)
  if (dist < 20) 1000.toShort else 0.toShort
})

ImageIO.writeVTK(image, new File("sphere.vtk"))
ImageIO.writeNifti(image, new File("sphere.nii"))
```

--------------------------------

### Read Statistical Shape Models

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Illustrates loading a Point Distribution Model (PDM) from an HDF5 file. It covers accessing the mean shape, generating random samples, and creating specific instances using PCA coefficients.

```scala
import scalismo.io.StatisticalModelIO
import scalismo.statisticalmodel.PointDistributionModel
import scalismo.mesh.TriangleMesh
import scalismo.geometry._3D
import java.io.File
import scala.util.Success

val modelFile = new File("face_model.h5")
val modelResult = StatisticalModelIO.readStatisticalTriangleMeshModel3D(modelFile)

modelResult match {
  case Success(model) =>
    println(s"Model rank: ${model.rank}")
    val meanShape = model.mean
    implicit val rng = scalismo.utils.Random(42L)
    val randomSample = model.sample()
    import breeze.linalg.DenseVector
    val coeffs = DenseVector.zeros[Double](model.rank)
    coeffs(0) = 2.0
    val specificShape = model.instance(coeffs)
  case _ => println("Failed to load model")
}
```

--------------------------------

### Manage Discrete Images and Interpolation

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Explains how to create discrete 3D image domains, populate them with values, and perform interpolation to continuous representations. It also covers resampling and domain rotation.

```scala
import scalismo.image.{DiscreteImage, DiscreteImageDomain3D}
import scalismo.geometry.{Point3D, EuclideanVector3D, IntVector3D, IntVector, _3D}
import scalismo.common.Field
import scalismo.common.interpolation.BSplineImageInterpolator3D

val origin = Point3D(0.0, 0.0, 0.0)
val spacing = EuclideanVector3D(1.0, 1.0, 1.0)
val size = IntVector3D(100, 100, 100)
val domain = DiscreteImageDomain3D(origin, spacing, size)

val image = DiscreteImage[_3D, Float](domain, pt => (pt.x + pt.y + pt.z).toFloat)

val voxelValue = image(IntVector(50, 50, 50))
val continuousImage: Field[_3D, Float] = image.interpolate(BSplineImageInterpolator3D(degree = 3))
val valueAtPoint = continuousImage(Point3D(50.5, 50.5, 50.5))

val newSize = IntVector3D(50, 50, 50)
val newDomain = DiscreteImageDomain3D(domain.boundingBox, size = newSize)
val resampledImage = continuousImage.discretize(newDomain, outsideValue = 0.0f)
```

--------------------------------

### Read 3D Triangle Mesh with Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Demonstrates how to load a 3D triangle mesh from VTK, STL, or PLY files. The function returns a Try object to handle potential IO errors gracefully.

```scala
import scalismo.io.MeshIO
import scalismo.mesh.TriangleMesh
import scalismo.geometry._3D
import java.io.File
import scala.util.{Success, Failure}

val meshFile = new File("face.vtk")
val meshResult = MeshIO.readMesh(meshFile)

meshResult match {
  case Success(mesh) =>
    println(s"Loaded mesh with ${mesh.pointSet.numberOfPoints} points")
    println(s"Number of triangles: ${mesh.triangulation.triangles.size}")
    println(s"Bounding box: ${mesh.boundingBox}")
  case Failure(ex) =>
    println(s"Failed to load mesh: ${ex.getMessage}")
}

val stlMesh = MeshIO.readMesh(new File("model.stl"))
val plyMesh = MeshIO.readMesh(new File("scan.ply"))
```

--------------------------------

### Perform 3D Geometric Transformations in Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Demonstrates how to apply translation, rotation, and scaling transformations in 3D space. It also shows how to compose these transformations and use transformation spaces to generate them from parameters.

```scala
import scalismo.transformations._
import scalismo.geometry.{Point3D, EuclideanVector3D, _3D}
import breeze.linalg.DenseVector

val translation = Translation3D(EuclideanVector3D(10.0, 20.0, 30.0))
val translatedPoint = translation(Point3D(0.0, 0.0, 0.0))

val center = Point3D(0.0, 0.0, 0.0)
val rotation = Rotation3D(phi = Math.PI / 4, theta = 0.0, psi = 0.0, center = center)
val rotatedPoint = rotation(Point3D(0.0, 1.0, 0.0))

val scaling = Scaling3D(2.0)
val scaledPoint = scaling(Point3D(1.0, 1.0, 1.0))

val rigidTransform = TranslationAfterRotation3D(translation, rotation)
val similarityTransform = TranslationAfterScalingAfterRotation3D(translation, scaling, rotation)

val inverseRigid = rigidTransform.inverse

val translationSpace = TranslationSpace3D
val params = DenseVector(5.0, 10.0, 15.0)
val translationFromParams = translationSpace.transformationForParameters(params)

val rotationSpace = RotationSpace3D(center)
val compositeSpace = ProductTransformationSpace(translationSpace, rotationSpace)
```

--------------------------------

### Geometric Primitives: Point and EuclideanVector in Scala

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Illustrates the usage of Point and EuclideanVector for 1D, 2D, and 3D geometric entities. It covers creation, arithmetic operations, vector manipulations like normalization and dot product, and conversions to/from breeze vectors.

```Scala
import scalismo.geometry._

// Create points in different dimensions
val p1d = Point(5.0)
val p2d = Point(3.0, 4.0)
val p3d = Point(1.0, 2.0, 3.0)

// Convenience constructors
val point2D = Point2D(10.0, 20.0)
val point3D = Point3D(1.0, 2.0, 3.0)

// Create vectors
val v2d = EuclideanVector2D(1.0, 0.0)
val v3d = EuclideanVector3D(1.0, 2.0, 3.0)

// Point-vector arithmetic
val translated = point3D + v3d           // Point3D(2.0, 4.0, 6.0)
val displacement = point3D - Point3D.origin  // EuclideanVector3D(1.0, 2.0, 3.0)

// Vector operations
val scaled = v3d * 2.0                   // EuclideanVector3D(2.0, 4.0, 6.0)
val normalized = v3d.normalize           // Unit vector
val length = v3d.norm                    // Euclidean norm
val dot = v3d.dot(EuclideanVector3D(1.0, 0.0, 0.0))  // Dot product

// 3D-specific operations
val cross = v3d.crossproduct(EuclideanVector3D(0.0, 1.0, 0.0))

// Create from polar/spherical coordinates
val fromPolar = Point.fromPolar(1.0, Math.PI / 4)  // 2D point
val fromSpherical = Point.fromSpherical(1.0, Math.PI / 4, Math.PI / 2)  // 3D point

// Convert to/from breeze vectors for linear algebra
val breezeVec = point3D.toBreezeVector
val backToPoint = Point.fromBreezeVector[_3D](breezeVec)
```

--------------------------------

### Manipulate PointDistributionModel in Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Demonstrates how to load a statistical shape model, sample random shapes, generate instances from coefficients, project target meshes, and compute posterior models based on observations. It requires a pre-trained model file and standard Scalismo geometry and IO imports.

```scala
import scalismo.statisticalmodel.PointDistributionModel
import scalismo.io.{StatisticalModelIO, MeshIO}
import scalismo.geometry.{Point3D, _3D}
import scalismo.common.PointId
import breeze.linalg.DenseVector
import java.io.File

val model = StatisticalModelIO.readStatisticalTriangleMeshModel3D(new File("face_model.h5")).get
val meanShape = model.mean

implicit val rng = scalismo.utils.Random(42L)
val randomShape1 = model.sample()
val randomShape2 = model.sample()

val coefficients = DenseVector.zeros[Double](model.rank)
coefficients(0) = 2.0
coefficients(1) = -1.0
val specificShape = model.instance(coefficients)

val targetMesh = MeshIO.readMesh(new File("target.vtk")).get
val projectedShape = model.project(targetMesh)
val reconstructionCoeffs = model.coefficients(targetMesh)

val observations = IndexedSeq((PointId(100), Point3D(10.0, 20.0, 30.0)), (PointId(200), Point3D(15.0, 25.0, 35.0)))
val posteriorModel = model.posterior(observations, sigma2 = 1.0)
val posteriorSample = posteriorModel.sample()
val reducedModel = model.truncate(newRank = 10)
```

--------------------------------

### Create Training Data from Paired Landmarks

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Combines paired model and noseless landmarks to create training data for a statistical model. It associates points and adds a small noise factor.

```Scala
val trainingData2 = (modelLandmarks zip noselessLandmarks).map { case (mLm, nLm) =>
  (model.mean.pointSet.findClosestPoint(mLm.landmark.point).id, nLm.landmark.point, littleNoise)
}
```

--------------------------------

### Write Statistical Shape Models

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Demonstrates how to save a trained Point Distribution Model to an HDF5 file. This assumes the model has been created from a data collection of registered meshes.

```scala
import scalismo.io.{StatisticalModelIO, MeshIO}
import scalismo.statisticalmodel.PointDistributionModel
import scalismo.statisticalmodel.dataset.DataCollection
import scalismo.mesh.TriangleMesh
import scalismo.geometry._3D
import java.io.File

val referenceMesh = MeshIO.readMesh(new File("reference.vtk")).get
val trainingMeshes = (1 to 10).map(i =>
  MeshIO.readMesh(new File(s"training_$i.vtk")).get
)

val dataCollection = DataCollection.fromTriangleMesh3DSequence(referenceMesh, trainingMeshes)
val model = PointDistributionModel.createUsingPCA(dataCollection)

StatisticalModelIO.writeStatisticalTriangleMeshModel3D(model, new File("trained_model.h5"))
```

--------------------------------

### Create Discrete Images in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/images.md

Instantiates a discrete image by providing an image domain and a function that maps each point in the domain to a value, or by supplying an array of values.

```scala
val image = DiscreteImage3D(imageDomain, p => 0.toShort)
```

```scala
val image2 = DiscreteImage3D(imageDomain, Array.fill(imageDomain.pointSet.numberOfPoints)(0.toShort))
```

```scala
val image3 = DiscreteField3D(imageDomain, p => 0.toShort)
```

--------------------------------

### Create and Manipulate TriangleMesh3D in Scala

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Demonstrates the creation of a 3D triangle mesh from points and triangles, accessing its properties like area and normals, and performing transformations. It utilizes the TriangleMesh3D class for representing triangulated surfaces.

```Scala
import scalismo.mesh.{TriangleMesh3D, TriangleCell, TriangleList}
import scalismo.common.{PointId, UnstructuredPoints}
import scalismo.geometry.{Point3D, EuclideanVector3D, _3D}

// Create a simple tetrahedron mesh
val points = IndexedSeq(
  Point3D(0.0, 0.0, 0.0),
  Point3D(1.0, 0.0, 0.0),
  Point3D(0.5, 0.866, 0.0),
  Point3D(0.5, 0.289, 0.816)
)

val triangles = IndexedSeq(
  TriangleCell(PointId(0), PointId(1), PointId(2)),
  TriangleCell(PointId(0), PointId(1), PointId(3)),
  TriangleCell(PointId(1), PointId(2), PointId(3)),
  TriangleCell(PointId(0), PointId(2), PointId(3))
)

val mesh = TriangleMesh3D(points, TriangleList(triangles))

// Access mesh properties
println(s"Number of points: ${mesh.pointSet.numberOfPoints}")
println(s"Number of triangles: ${mesh.triangulation.triangles.size}")
println(s"Total surface area: ${mesh.area}")
println(s"Bounding box: ${mesh.boundingBox}")

// Compute normals
val vertexNormals = mesh.vertexNormals
val cellNormals = mesh.cellNormals

// Transform the mesh
val translatedMesh = mesh.transform(p => p + EuclideanVector3D(10.0, 0.0, 0.0))

// Access mesh operations for advanced functionality
val ops = mesh.operations
val closestPoint = ops.closestPointOnSurface(Point3D(0.5, 0.5, 0.5))
println(s"Closest point on surface: ${closestPoint.point}")
```

--------------------------------

### Model Deformations with GaussianProcess in Scala

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Shows how to define and use Gaussian Processes for modeling continuous fields, such as deformation fields. It covers creating kernels, defining a GP with a mean and covariance, sampling from it, and performing regression by conditioning on observations.

```Scala
import scalismo.statisticalmodel.GaussianProcess
import scalismo.kernels.{GaussianKernel, DiagonalKernel}
import scalismo.geometry.{Point, Point3D, EuclideanVector, EuclideanVector3D, _3D}
import scalismo.common.{Field, EuclideanSpace}

// Create a Gaussian kernel for smooth deformations
val gaussianKernel = GaussianKernel3D(sigma = 50.0, scaleFactor = 10.0)

// Create a diagonal kernel for vector-valued GP (3D deformation field)
val vectorKernel = DiagonalKernel(gaussianKernel, 3)

// Create a zero-mean GP for deformation modeling
val gp = GaussianProcess3D[EuclideanVector[_3D]](vectorKernel)

// Sample from the GP at specific points
implicit val rng = scalismo.utils.Random(42L)
val samplePoints = IndexedSeq(
  Point3D(0.0, 0.0, 0.0),
  Point3D(10.0, 0.0, 0.0),
  Point3D(0.0, 10.0, 0.0)
)
val domain = scalismo.common.UnstructuredPointsDomain3D.Create.create(samplePoints)
val sampledField = gp.sampleAtPoints(domain)

// Compute marginal distribution at a point
val marginalDist = gp.marginal(Point3D(5.0, 5.0, 5.0))
println(s"Mean at point: ${marginalDist.mean}")
println(s"Covariance: ${marginalDist.cov}")

// Condition GP on observations (regression)
val observations = IndexedSeq(
  (Point3D(0.0, 0.0, 0.0), EuclideanVector3D(1.0, 0.0, 0.0)),
  (Point3D(20.0, 0.0, 0.0), EuclideanVector3D(-1.0, 0.0, 0.0))
)
val posteriorGP = gp.posterior(observations, sigma2 = 1.0)
```

--------------------------------

### Define Discrete Image Domains in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/images.md

Creates a domain for discrete images, representing a rectangular grid of points. It can be initialized using structured points, or directly with origin, spacing, size, and optional rotation.

```scala
val imageDomain = DiscreteImageDomain3D(StructuredPoints3D(origin, spacing, size))
```

```scala
val imageDomain2 = DiscreteImageDomain3D(origin, spacing, size)
```

```scala
val imageDomain3 = DiscreteImageDomain3D(origin, spacing, size, yaw, pitch, roll)
```

```scala
val boundingBox : BoxDomain[_3D] = imageDomain.boundingBox
val imageDomain4 = DiscreteImageDomain3D(boundingBox, spacing = spacing)
```

```scala
val imageDomain5 = DiscreteImageDomain3D(boundingBox, size = size)
```

--------------------------------

### Extract Landmark Points from UI

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Shows how to filter landmark views from a specific UI group and extract the underlying point data for model fitting.

```scala
val modelLandmarkViews = ui.filter(modelGroup, (lv: LandmarkView) => true)
val modelPts: Seq[Point[_3D]] = modelLandmarkViews.map(lv => lv.landmark.point)
```

--------------------------------

### Load and Display Statistical Mesh Model

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Loads a statistical mesh model from an H5 file and displays it in the Scalismo UI with a distinct color. This represents a base shape model.

```Scala
val littleModel = StatismoIO.readStatismoMeshModel(new File("datasets/model.h5")).get
val littleModelGroup = ui.createGroup("littleModel group")
val littleModelView = ui.show(littleModelGroup, littleModel, "littleModel")
littleModelView.meshView.color = Color.ORANGE
```

--------------------------------

### MeshIO.writeMesh API

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Writes a 3D triangle mesh to disk in various formats (VTK, STL, PLY, H5). The format is automatically determined by the file extension.

```APIDOC
## MeshIO.writeMesh

### Description
Writes a 3D triangle mesh to disk in various formats (VTK, STL, PLY, H5). The format is automatically determined by the file extension.

### Method
POST (conceptual, as it writes to a file)

### Endpoint
N/A (File-based operation)

### Parameters
#### Path Parameters
- **mesh** (TriangleMesh[_3D]) - Required - The 3D triangle mesh to write.
- **file** (File) - Required - The file object representing the destination path and filename.

### Request Example
```scala
import scalismo.io.MeshIO
import scalismo.mesh.{TriangleMesh3D, TriangleCell, TriangleList}
import scalismo.common.{PointId, UnstructuredPoints}
import scalismo.geometry.{Point, Point3D, _3D}
import java.io.File

// Create a simple triangle mesh programmatically
val points = IndexedSeq(
  Point3D(0.0, 0.0, 0.0),
  Point3D(1.0, 0.0, 0.0),
  Point3D(0.5, 1.0, 0.0),
  Point3D(0.5, 0.5, 1.0)
)

val triangles = IndexedSeq(
  TriangleCell(PointId(0), PointId(1), PointId(2)),
  TriangleCell(PointId(0), PointId(1), PointId(3)),
  TriangleCell(PointId(1), PointId(2), PointId(3)),
  TriangleCell(PointId(0), PointId(2), PointId(3))
)

val mesh = TriangleMesh3D(points, TriangleList(triangles))

// Write to different formats
MeshIO.writeMesh(mesh, new File("output.vtk"))  // VTK format
MeshIO.writeMesh(mesh, new File("output.stl"))  // STL format
MeshIO.writeMesh(mesh, new File("output.ply"))  // PLY format
```

### Response
#### Success Response (200)
Indicates that the mesh was successfully written to the specified file.

#### Response Example
N/A (Operation is side-effecting, no explicit return value for success other than completion.)
```

--------------------------------

### Metropolis-Hastings Sampling with Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Implements a Metropolis-Hastings sampling algorithm for shape parameters. It defines a proposal distribution (Gaussian) and a prior (Gaussian prior on coefficients), then sets up and runs the Markov chain to generate samples. The analysis includes calculating the mean of the sampled coefficients. Dependencies include Scalismo, Breeze for linear algebra, and a Random number generator.

```Scala
import scalismo.sampling._
import scalismo.sampling.proposals._
import scalismo.sampling.evaluators._
import scalismo.utils.Random

// Define a simple state type
case class ShapeParameters(coefficients: breeze.linalg.DenseVector[Double])

// Create a proposal generator
class GaussianProposal(stddev: Double)(implicit rng: Random)
  extends ProposalGenerator[ShapeParameters] with SymmetricTransitionRatio[ShapeParameters] {

  override def propose(current: ShapeParameters): ShapeParameters = {
    val noise = breeze.linalg.DenseVector.rand(current.coefficients.length,
      breeze.stats.distributions.Gaussian(0, stddev))
    ShapeParameters(current.coefficients + noise)
  }
}

// Create a likelihood evaluator
class GaussianPriorEvaluator extends DistributionEvaluator[ShapeParameters] {
  override def logValue(sample: ShapeParameters): Double = {
    // Standard normal prior on coefficients
    -0.5 * sample.coefficients.dot(sample.coefficients)
  }
}

// Set up Metropolis-Hastings chain
implicit val rng = Random(42L)
val proposal = new GaussianProposal(0.1)
val prior = new GaussianPriorEvaluator()

// Create MH proposal generator
val mhGenerator = MHProposalGenerator(proposal, prior)

// Create Markov chain
val chain = new MarkovChain[ShapeParameters] {
  override def next(current: ShapeParameters): ShapeParameters = {
    val proposed = proposal.propose(current)
    val acceptRatio = math.exp(prior.logValue(proposed) - prior.logValue(current))
    if (rng.scalaRandom.nextDouble() < acceptRatio) proposed else current
  }
}

// Run sampling
val initial = ShapeParameters(breeze.linalg.DenseVector.zeros[Double](10))
val samples = chain.iterator(initial).take(1000).toIndexedSeq

// Analyze samples
val meanCoeffs = samples.map(_.coefficients).reduce(_ + _) / samples.length.toDouble

```

--------------------------------

### Perform Landmark Registration in Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Shows how to compute rigid and similarity transformations between sets of corresponding 3D landmarks using SVD-based least squares estimation. The registration results can be applied to points or meshes to align source data with target coordinates.

```scala
import scalismo.registration.LandmarkRegistration
import scalismo.geometry.{Point, Point3D, Landmark, _3D}
import scalismo.transformations.{TranslationAfterRotation, TranslationAfterScalingAfterRotation}

val sourceLandmarks = Seq(Landmark[_3D]("p1", Point3D(0.0, 0.0, 0.0)), Landmark[_3D]("p2", Point3D(1.0, 0.0, 0.0)), Landmark[_3D]("p3", Point3D(0.0, 1.0, 0.0)), Landmark[_3D]("p4", Point3D(0.0, 0.0, 1.0)))
val targetLandmarks = Seq(Landmark[_3D]("p1", Point3D(10.0, 10.0, 10.0)), Landmark[_3D]("p2", Point3D(11.0, 10.0, 10.0)), Landmark[_3D]("p3", Point3D(10.0, 11.0, 10.0)), Landmark[_3D]("p4", Point3D(10.0, 10.0, 11.0)))

val center = Point3D(0.0, 0.0, 0.0)
val rigidTransform: TranslationAfterRotation[_3D] = LandmarkRegistration.rigid3DLandmarkRegistration(sourceLandmarks, targetLandmarks, center)
val transformedPoint = rigidTransform(Point3D(0.5, 0.5, 0.5))

val similarityTransform: TranslationAfterScalingAfterRotation[_3D] = LandmarkRegistration.similarity3DLandmarkRegistration(sourceLandmarks, targetLandmarks, center)

val pointPairs = Seq((Point3D(0.0, 0.0, 0.0), Point3D(10.0, 10.0, 10.0)), (Point3D(1.0, 0.0, 0.0), Point3D(11.0, 10.0, 10.0)), (Point3D(0.0, 1.0, 0.0), Point3D(10.0, 11.0, 10.0)))
val rigidFromPairs = LandmarkRegistration.rigid3DLandmarkRegistration(pointPairs, center)
```

--------------------------------

### Load and Display 3D Landmarks from JSON

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Loads 3D landmarks from a JSON file and displays them using the Scalismo UI. This is useful for preparing landmark data for statistical shape analysis.

```Scala
val modelLMs = LandmarkIO.readLandmarksJson[_3D](new File("datasets/modelLandmarks.json")).get
ui.show(modelGroup, modelLMs, "model")

val noselessLMs = LandmarkIO.readLandmarksJson[_3D](new File("datasets/noselessLandmarks.json")).get
ui.show(targetGroup, noselessLMs, "noseless")
```

--------------------------------

### Read and Write Anatomical Landmarks

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Shows how to read landmarks from JSON files and write them back to disk in both JSON and CSV formats. Includes handling of uncertainty information and descriptive metadata.

```scala
import scalismo.io.LandmarkIO
import scalismo.geometry.{Landmark, Point, Point3D, _3D}
import java.io.File
import scala.util.Success

val landmarksResult = LandmarkIO.readLandmarksJson3D(new File("landmarks.json"))

landmarksResult match {
  case Success(landmarks) =>
    landmarks.foreach { lm =>
      println(s"${lm.id}: ${lm.point}")
      lm.description.foreach(d => println(s"  Description: $d"))
      lm.uncertainty.foreach(u => println(s"  Has uncertainty information"))
    }
  case _ => println("Failed to load landmarks")
}

val landmarks = Seq(
  Landmark[_3D]("nasion", Point3D(0.0, 50.0, 30.0), Some("Nose bridge")),
  Landmark[_3D]("left_eye", Point3D(-15.0, 45.0, 25.0), Some("Left eye center")),
  Landmark[_3D]("right_eye", Point3D(15.0, 45.0, 25.0), Some("Right eye center")),
  Landmark[_3D]("chin", Point3D(0.0, 0.0, 20.0), Some("Chin point"))
)

LandmarkIO.writeLandmarksJson(landmarks, new File("face_landmarks.json"))
LandmarkIO.writeLandmarksCsv(landmarks, new File("face_landmarks.csv"))
```

--------------------------------

### Define Gaussian Process Covariance Kernels

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Shows how to define and manipulate covariance kernels for Gaussian processes, including RBF kernels, diagonal kernels for vector fields, and kernel composition.

```scala
import scalismo.kernels._
import scalismo.geometry.{Point3D, _3D}

val gaussianKernel = GaussianKernel3D(sigma = 50.0, scaleFactor = 10.0)
val point1 = Point3D(0.0, 0.0, 0.0)
val point2 = Point3D(10.0, 0.0, 0.0)
val kernelValue = gaussianKernel(point1, point2)

val vectorKernel = DiagonalKernel(gaussianKernel, outputDim = 3)
val matrixValue = vectorKernel(point1, point2)

val fineKernel = GaussianKernel3D(sigma = 10.0, scaleFactor = 5.0)
val coarseKernel = GaussianKernel3D(sigma = 100.0, scaleFactor = 20.0)
val multiScaleKernel = fineKernel + coarseKernel

val scaledKernel = gaussianKernel * 2.0
val bsplineKernel = BSplineKernel[_3D](order = 3, scale = 2)
```

--------------------------------

### Create a 3D Translation Space and Transformations in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Initializes a TranslationSpace3D object, which acts as a factory for creating translation transformations. It can generate transformations from parameters or provide the identity transformation.

```Scala
import breeze.linalg.DenseVector
val translationSpace = TranslationSpace3D
val parameters = DenseVector(1.0, 0.0, 3.0)
val translationFromParameters = translationSpace.transformationForParameters(parameters)
val ident = translationSpace.identityTransformation
```

--------------------------------

### Update and Display Posterior Model

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Calculates a new posterior model using the generated training data and displays it in the Scalismo UI. This refines the statistical model based on new data.

```Scala
val betterPosterior = model.posterior(trainingData2.toIndexedSeq)
val betterPosteriorGroup = ui.createGroup("better posterior")
ui.show(betterPosteriorGroup, betterPosterior, "betterPosterior")
```

--------------------------------

### Create Structured Points for Image Grids in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/images.md

Defines a grid of points in 3D space, used as the basis for discrete images. It allows specifying the origin, spacing, size, and optional rotation of the grid.

```scala
val origin = Point3D(10.0, 15.0, 7.0)
val spacing = EuclideanVector3D(0.1, 0.1, 0.1)
val size = IntVector3D(32, 64, 92)

val points = StructuredPoints3D(origin, spacing, size)
```

```scala
val yaw = Math.PI / 2
val pitch = 0.0
val roll = Math.PI

val points2 = StructuredPoints3D(origin, spacing, size, yaw, pitch, roll)
```

--------------------------------

### Filter Landmarks using Scalismo UI

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Filters landmarks displayed in the Scalismo UI based on a given predicate. This allows for selective visualization and processing of landmark data.

```Scala
val modelLandmarks = ui.filter(modelGroup, (lv: LandmarkView) => true)
val noselessLandmarks = ui.filter(targetGroup, (lv: LandmarkView) => true)
```

--------------------------------

### Interpolate and Discretize Images in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/images.md

Converts a discrete image to a continuous representation using interpolation (e.g., BSpline) and then back to a discrete image with a potentially different domain and resolution.

```scala
val continuousImage : Field[_3D, Short] = image.interpolate(BSplineImageInterpolator3D(degree = 3))
```

```scala
val newDomain = DiscreteImageDomain3D(image.domain.boundingBox, size=IntVector(32, 32, 32))
val resampledImage : DiscreteImage[_3D, Short] = continuousImage.discretize(newDomain, outsideValue = 0.toShort)
```

--------------------------------

### Create a Product Transformation Space in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Defines a composite transformation space by combining individual transformation spaces, such as TranslationSpace3D and RotationSpace3D. It can generate a composite transformation from a combined set of parameters.

```Scala
import breeze.linalg.DenseVector
val compositeSpace = ProductTransformationSpace(TranslationSpace3D, RotationSpace3D(Point3D(0, 0, 0)))
val ct : CompositeTransformation[_3D, Translation, Rotation] = compositeSpace.transformationForParameters(DenseVector(0.0 ,0.0 ,0.0, 0.0, 0.0, 0.0))
```

--------------------------------

### MeshIO.readMesh API

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Reads a 3D triangle mesh from various file formats including VTK, STL, and PLY. The function automatically detects the file format based on the extension and returns a TriangleMesh[_3D] wrapped in a Try for error handling.

```APIDOC
## MeshIO.readMesh

### Description
Reads a 3D triangle mesh from various file formats including VTK, STL, and PLY. The function automatically detects the file format based on the extension and returns a `TriangleMesh[_3D]` wrapped in a `Try` for error handling.

### Method
GET (conceptual, as it reads from a file)

### Endpoint
N/A (File-based operation)

### Parameters
#### Path Parameters
- **file** (File) - Required - The file object representing the mesh file to read.

### Request Example
```scala
import scalismo.io.MeshIO
import scalismo.mesh.TriangleMesh
import scalismo.geometry._3D
import java.io.File
import scala.util.{Success, Failure}

// Read a mesh from a VTK file
val meshFile = new File("face.vtk")
val meshResult = MeshIO.readMesh(meshFile)

meshResult match {
  case Success(mesh) =>
    println(s"Loaded mesh with ${mesh.pointSet.numberOfPoints} points")
  case Failure(ex) =>
    println(s"Failed to load mesh: ${ex.getMessage}")
}
```

### Response
#### Success Response (200)
- **TriangleMesh[_3D]** - The loaded 3D triangle mesh.
- **Try[TriangleMesh[_3D]]** - A Try monad wrapping the mesh, allowing for error handling.

#### Response Example
```scala
// Example of a successful result
Success(mesh: TriangleMesh[_3D])

// Example of a failure
Failure(new Exception("File not found"))
```
```

--------------------------------

### Read 3D Scalar Image with Scalismo

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Reads 3D scalar images from NIfTI or VTK files. It supports generic voxel types and handles coordinate system transformations internally.

```scala
import scalismo.io.ImageIO
import scalismo.image.DiscreteImage
import scalismo.geometry._3D
import java.io.File
import scala.util.{Success, Failure}

val imageFile = new File("ct_scan.nii")
val imageResult = ImageIO.read3DScalarImage[Short](imageFile)

imageResult match {
  case Success(image) =>
    println(s"Image dimensions: ${image.domain.pointSet.size}")
    println(s"Spacing: ${image.domain.pointSet.spacing}")
    println(s"Origin: ${image.domain.pointSet.origin}")

    val centerVoxel = image.domain.pointSet.size.map(_ / 2)
    val value = image(scalismo.geometry.IntVector(centerVoxel(0), centerVoxel(1), centerVoxel(2)))
    println(s"Center voxel value: $value")
  case Failure(ex) =>
    println(s"Failed to load image: ${ex.getMessage}")
}

val floatImage = ImageIO.read3DScalarImageAsType[Float](imageFile)
```

--------------------------------

### Augment Statistical Model with Low-Rank GP

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Augments a statistical mesh model with a low-rank Gaussian process approximation. This enhances the model's ability to capture variations by using a sampler and approximating the GP.

```Scala
val sampler = UniformMeshSampler3D(littleModel.referenceMesh, 500)
val lowRankGP = LowRankGaussianProcess.approximateGP(gp, sampler, 50)
val model = StatisticalMeshModel.augmentModel(littleModel, lowRankGP)
```

--------------------------------

### Define Custom X-Mirrored Kernel

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Creates a custom kernel that mirrors the input kernel across the X-axis. This is used in constructing symmetric kernels for shape models.

```Scala
case class XmirroredKernel(ker: PDKernel[_3D]) extends PDKernel[_3D] {
  override def k(x: Point[_3D], y: Point[_3D]): Double = ker(Point(x(0) * -1f ,x(1), x(2)), y)
}
```

--------------------------------

### ImageIO.read3DScalarImage API

Source: https://context7.com/unibas-gravis/scalismo/llms.txt

Reads a 3D scalar image from VTK or NIfTI file formats. Supports various voxel types including Byte, Short, Int, Float, and Double. For NIfTI files, the library handles coordinate system transformations between RAS and LPS automatically.

```APIDOC
## ImageIO.read3DScalarImage

### Description
Reads a 3D scalar image from VTK or NIfTI file formats. Supports various voxel types including Byte, Short, Int, Float, and Double. For NIfTI files, the library handles coordinate system transformations between RAS and LPS automatically.

### Method
GET (conceptual, as it reads from a file)

### Endpoint
N/A (File-based operation)

### Parameters
#### Path Parameters
- **file** (File) - Required - The file object representing the image file to read.
- **Implicit Type Parameter** (VoxelType) - Required - The type of the scalar values in the image (e.g., `Short`, `Float`).

### Request Example
```scala
import scalismo.io.ImageIO
import scalismo.image.DiscreteImage
import scalismo.geometry._3D
import java.io.File
import scala.util.{Success, Failure}

// Read a 3D image with Short voxel type
val imageFile = new File("ct_scan.nii")
val imageResult = ImageIO.read3DScalarImage[Short](imageFile)

imageResult match {
  case Success(image) =>
    println(s"Image dimensions: ${image.domain.pointSet.size}")
    // Access voxel values
    val centerVoxel = image.domain.pointSet.size.map(_ / 2)
    val value = image(scalismo.geometry.IntVector(centerVoxel(0), centerVoxel(1), centerVoxel(2)))
    println(s"Center voxel value: $value")
  case Failure(ex) =>
    println(s"Failed to load image: ${ex.getMessage}")
}

// Read with automatic type conversion to Float
val floatImage = ImageIO.read3DScalarImageAsType[Float](imageFile)
```

### Response
#### Success Response (200)
- **DiscreteImage[_3D, VoxelType]** - The loaded 3D scalar image.
- **Try[DiscreteImage[_3D, VoxelType]]** - A Try monad wrapping the image, allowing for error handling.

#### Response Example
```scala
// Example of a successful result
Success(image: DiscreteImage[_3D, Short])

// Example of a failure
Failure(new Exception("Invalid file format"))
```
```

--------------------------------

### Define a 3D Transformation from a Function in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Creates a 3D transformation by providing a function that maps a 3D point to another 3D point and optionally specifying its domain. If the domain is not specified, it defaults to the entire Euclidean space.

```Scala
import scalismo.common._
import scalismo.geometry._
import scalismo.transformations._
val domain : Domain[_3D] = EuclideanSpace[_3D]
Transformation3D(domain, p => p + EuclideanVector3D(1, 0, 0))
```

--------------------------------

### Compute Posterior Model

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Approximates a Gaussian Process, augments a statistical model, and computes the posterior based on landmark correspondences between the model and the target mesh.

```scala
val sampler = UniformMeshSampler3D(littleModel.referenceMesh, 500)
val lowRankGP = LowRankGaussianProcess.approximateGP(gp, sampler, 50)
val model = StatisticalMeshModel.augmentModel(littleModel, lowRankGP)

val trainingData = (modelPts zip noselessPts).map{ case (mPt, nPt) =>
  (model.mean.pointSet.findClosestPoint(mPt).id, nPt, littleNoise)
}
val posterior = model.posterior(trainingData.toIndexedSeq)
```

--------------------------------

### Create a 3D Rotation Parametric Transformation in Scala

Source: https://github.com/unibas-gravis/scalismo/blob/main/docs/transformations.md

Defines a 3D rotation transformation by specifying Euler angles and a rotation center. This demonstrates another type of parametric transformation.

```Scala
val rotation = Rotation3D(0, 0.1, 0.3, Point3D(0, 0, 0))
```

--------------------------------

### Marginalize Posterior Model

Source: https://github.com/unibas-gravis/scalismo/wiki/revisions.html

Filters points based on distance to a reference point and creates a marginal model from the posterior.

```scala
val nosePtIDs = model.referenceMesh.pointSet.pointIds.filter { id => (model.referenceMesh.pointSet.point(id) - model.referenceMesh.pointSet.point(PointId(8152))).norm <= 42 }
val posteriorNoseModel = betterPosterior.marginal(nosePtIDs.toIndexedSeq)
```