Model fitting using MCMC - Fitting a shape model

In this tutorial we show how the MCMC framework, which was introduced in the previous tutorial, can be used for shape model fitting.

We will illustrate it by computing a posterior of a shape model, given a set of corresponding landmark pairs. This is the same setup that we have discussed in the tutorial about Gaussian process regression. The difference is, that here we will also allow for rotation and translation of the model. In this setting, it is not possible anymore to compute the posterior analytically. Rather, our only hope are approximation methods, such as using Markov-chain monte carlo methods.

In this tutorial we show not only a working example, but also how to make it computationally efficient. Making the individual parts as efficient as possible is important in sampling approaches, as we need to produce many samples to get accurate estimates.

Related resources

Week 3 of our online course on shape model fitting may provide some helpful context for this tutorial.

To run the code from this tutorial, download the following Scala file:

• Tutorial15.scala

Preparation

As in the previous tutorials, we start by importing some commonly used objects and initializing the system.

import scalismo.io.StatisticalModelIO
import scalismo.io.LandmarkIO
import scalismo.ui.api.ScalismoUI
import scalismo.geometry._
import scalismo.common.PointId
import scalismo.common.interpolation.TriangleMeshInterpolator3D
import scalismo.common.UnstructuredPointsDomain
import scalismo.common.interpolation.NearestNeighborInterpolator3D
import scalismo.common.UnstructuredPointsDomain1D
import scalismo.common.UnstructuredPointsDomain3D
import scalismo.statisticalmodel.PointDistributionModel
import scalismo.statisticalmodel.MultivariateNormalDistribution

import scalismo.mesh.TriangleMesh
import scalismo.transformations._


import scalismo.sampling._
import scalismo.sampling.proposals._
import scalismo.sampling.parameters._
import scalismo.sampling.evaluators._
import scalismo.sampling.loggers.MHSampleLogger
import scalismo.sampling.algorithms.MetropolisHastings

import breeze.linalg.DenseVector
import breeze.linalg.DenseMatrix

implicit val rng: scalismo.utils.Random = scalismo.utils.Random(42)
implicit val randBasisBreeze : breeze.stats.distributions.RandBasis = rng.breezeRandBasis
scalismo.initialize()

val ui = ScalismoUI()

Loading and visualizing the data

In a first step, we load and visualize all the data that we need. First, we load the statistical model:

val model = StatisticalModelIO.readStatisticalTriangleMeshModel3D(new java.io.File("datasets/bfm.h5")).get

val modelGroup = ui.createGroup("model")
val modelView = ui.show(modelGroup, model, "model")
modelView.referenceView.opacity = 0.5

In this example, we will fit the model such that a set of model landmarks, coincide with a set of landmark points defined on a target face. We load and visualize the corresponding landmark data:

val modelLms = LandmarkIO.readLandmarksJson[_3D](new java.io.File("datasets/modelLM_mcmc.json")).get
val modelLmViews = ui.show(modelGroup, modelLms, "modelLandmarks")
modelLmViews.foreach(lmView => lmView.color = java.awt.Color.BLUE)

val targetGroup = ui.createGroup("target")

val targetLms = LandmarkIO.readLandmarksJson3D(new java.io.File("datasets/targetLM_mcmc.json")).get
val targetLmViews = ui.show(targetGroup, targetLms, "targetLandmarks")
modelLmViews.foreach(lmView => lmView.color = java.awt.Color.RED)

The modelPoints (which are actually points on the reference mesh defining the model) and the target points are assumed to be in correspondence. To highlight this, we zip them together, such that the corresponding points are stored as a tuple.

  val modelPoints = modelLms.map(l => l.point)
  val targetPoints = targetLms.map(l => l.point)
  val correspondences = modelPoints.zip(targetPoints)

In any shape modelling application, it is important to correctly set up the center of rotation. Usually we take this to be the center of mass of the model mean.

def computeCenterOfMass(mesh: TriangleMesh[_3D]): Point[_3D] = {
    val normFactor = 1.0 / mesh.pointSet.numberOfPoints
    mesh.pointSet.points.foldLeft(Point(0, 0, 0))((sum, point) => sum + point.toVector * normFactor)
  }

val rotationCenter = computeCenterOfMass(model.mean)

The parameter class

In this example, we want to model the posterior $p(\theta | D)$, where the parameters $\theta =( t, r, \alpha)$ consist of the translation parameters $t=(t_x, t_y, t_z)$, the rotation parameters $r = (\phi, \psi, \omega)$, represented as Euler angles as well a shape model coefficients $\alpha = (\alpha_1, \ldots, \alpha_n)$. As this combination of parameters is very common in shape modelling, Scalismo already provides a corresponding parameter class, called PoseAndShapeParameters. We need, however, to define any additional parameters that we want to model by ourselves.

case class Parameters(poseAndShapeParameters : PoseAndShapeParameters, 
                      noiseStddev : Double
                     )

The class PoseAndShapeParameters class is defined as part of Scalismo. In this tutorial, we add an other parameter, namely the noiseStddev, which regulates the noise that we expect on the observations. To be able to derive proposals for such a user defined parameter class, we need to provide conversion methods that tell scalismo how to convert the parameters to and from a tuple. We will also add a convenience method to extract the pose transformation from the parameters.

object Parameters {

    implicit object parameterConversion
        extends ParameterConversion[
          Tuple2[PoseAndShapeParameters, Double],
          Parameters
        ] {
      def from(p: Parameters): Tuple2[PoseAndShapeParameters, Double] =
        (p.poseAndShapeParameters, p.noiseStddev)
      def to(t: Tuple2[PoseAndShapeParameters, Double]): Parameters =
        Parameters(t._1, t._2)
    }

    def poseTransformationForParameters(
        translationParameters: TranslationParameters,
        rotationParameters: RotationParameters,
        centerOfRotation: Point[_3D]
    ): TranslationAfterRotation[_3D] = {
      TranslationAfterRotation3D(
        Translation3D(translationParameters.translationVector),
        Rotation3D(rotationParameters.angles, centerOfRotation)
      )
    }
  }

Evaluators: Modelling the target density

As in the previous tutorial, we represent the unnormalized posterior distribution as the product of prior and likelihood: $p(\theta | D) \propto p(\theta) p(D | \theta)$, where $D$ denotes the data (i.e. the corresponding landmark points) and $\theta$ are our parameters.

As a prior over the shape parameters is given by the shape model. For the translation and rotation, we assume a zero-mean normal distribution. As the standard deviation characterizing the noise needs to be positive, we use a lognormal distribution.:

  case class PriorEvaluator(model: PointDistributionModel[_3D, TriangleMesh])
      extends MHDistributionEvaluator[Parameters] {

    val translationPrior = breeze.stats.distributions.Gaussian(0.0, 5.0)
    val rotationPrior = breeze.stats.distributions.Gaussian(0, 0.1)
    val noisePrior = breeze.stats.distributions.LogNormal(0, 0.25)

    override def logValue(sample: MHSample[Parameters]): Double = {
      val poseAndShapeParameters = sample.parameters.poseAndShapeParameters
      val translationParameters = poseAndShapeParameters.translationParameters
      val rotationParameters = poseAndShapeParameters.rotationParameters

      model.gp.logpdf(poseAndShapeParameters.shapeParameters.coefficients) +
        translationPrior.logPdf(translationParameters.translationVector.x) +
        translationPrior.logPdf(translationParameters.translationVector.y) +
        translationPrior.logPdf(translationParameters.translationVector.z) +
        rotationPrior.logPdf(rotationParameters.angles._1) +
        rotationPrior.logPdf(rotationParameters.angles._2) +
        rotationPrior.logPdf(rotationParameters.angles._3) +
        noisePrior.logPdf(sample.parameters.noiseStddev)
    }
  }

To compute the likelihood $p(D | \theta)$ we need to determine where the landmark points on the model are mapped under the transformation given by the parameters $\theta$. The distance between these mapped model points and the target points determine the likelihood. As we only need to map the landmark points, it is computationally more efficient to restrict the model to these points first, and then only to deform these points rather than all the points of the model.

 
  case class CorrespondenceEvaluator(
      model: PointDistributionModel[_3D, TriangleMesh],
      rotationCenter: Point[_3D],
      correspondences: Seq[(Point[_3D], Point[_3D])]
  ) extends MHDistributionEvaluator[Parameters] {

    // we extract the points and build a model from only the points
    val (refPoints, targetPoints) = correspondences.unzip

    val newDomain = UnstructuredPointsDomain3D(refPoints.toIndexedSeq)
    val modelOnLandmarkPoints = model.newReference(newDomain, NearestNeighborInterpolator3D())
    
    override def logValue(sample: MHSample[Parameters]): Double = {

      val poseTransformation = Parameters.poseTransformationForParameters(
        sample.parameters.poseAndShapeParameters.translationParameters,
        sample.parameters.poseAndShapeParameters.rotationParameters,
        rotationCenter
      )
      
      val modelCoefficients = sample.parameters.poseAndShapeParameters.shapeParameters.coefficients
      val currentModelInstance =
        modelOnLandmarkPoints.instance(modelCoefficients).transform(poseTransformation)

      
      val lmUncertainty = MultivariateNormalDistribution(
        DenseVector.zeros[Double](3),
        DenseMatrix.eye[Double](3) * sample.parameters.noiseStddev
      )

      val likelihoods = for ((pointOnInstance, targetPoint) <- currentModelInstance.pointSet.points.zip(targetPoints)) yield {

        val observedDeformation = targetPoint - pointOnInstance

        lmUncertainty.logpdf(observedDeformation.toBreezeVector)
      }

      val loglikelihood = likelihoods.sum
      loglikelihood
    }
  }

The posterior evaluator

Given these evaluators, we can now build the computationally efficient version of our target density $p(\theta | D)$

val likelihoodEvaluator = CorrespondenceEvaluator(model, rotationCenter, correspondences)
val priorEvaluator = PriorEvaluator(model).cached

val posteriorEvaluator = ProductEvaluator(priorEvaluator, likelihoodEvaluator)

The proposal generator

As in the previous tutorials, we will use simple random walk proposals. We will define separate proposals for shape, translation and rotation. On one hand, this lets us set the step length (i.e. stddev of the distribution from which we sample the next step) individually for each group, and thus to incorporate our knowledge that changes in rotation will be much smaller than the shape changes. On the other hand, splitting the parameter updates in blocks will increase our chance for the random updates to be accepted. The reason for this is that when many parameters are updated at one, chances are high that some of the proposed changes make the new state more unlikely, and hence increase the chance of the new state being rejected.

We start by defining the basic pose proposals

  val rotationProposal = MHProductProposal(
    GaussianRandomWalkProposal(0.01, "rx").forType[Double],
    GaussianRandomWalkProposal(0.01, "ry").forType[Double],
    GaussianRandomWalkProposal(0.01, "rz").forType[Double]
  ).forType[RotationParameters]

  val translationProposal = MHProductProposal(
    GaussianRandomWalkProposal(1.0, "tx").forType[Double],
    GaussianRandomWalkProposal(1.0, "ty").forType[Double],
    GaussianRandomWalkProposal(1.0, "tz").forType[Double]
  ).forType[TranslationParameters]

The next proposal is for updating the shape parameters. We define two shape proposals, one for the leading parameters, which adjusts the overall shape, and one for the remaining parameters. We can achieve this using the .partial method of the GaussianRandomWalkProposal, which will only update the components in a given range.

val shapeProposalLeading =
    GaussianRandomWalkProposal(0.01, "shape-0-5").partial(0 until 5)
      .forType[ShapeParameters]
  val shapeProposalRemaining =
    GaussianRandomWalkProposal(0.01, "shape-6-").partial(6 until model.rank)
      .forType[ShapeParameters]

From these building blocks, we can define different proposal for shape and pose, which always only update a part of the parameters. For better readability, we also give them simpler name, under which they later appear in the logs.

  val identTranslationProposal =
    MHIdentityProposal.forType[TranslationParameters]
  val identRotationProposal = MHIdentityProposal.forType[RotationParameters]
  val identShapeProposal = MHIdentityProposal.forType[ShapeParameters]

  val poseAndShapeTranslationOnlyProposal =
    MHProductProposal(
      translationProposal,
      identRotationProposal,
      identShapeProposal
    )
      .forType[PoseAndShapeParameters]
      .relabel("translation-only")
  val poseAndShapeRotationOnlyProposal =
    MHProductProposal(
      identTranslationProposal,
      rotationProposal,
      identShapeProposal
    )
      .forType[PoseAndShapeParameters]
      .relabel("rotation-only")
  val poseAndShapeLeadingShapeOnlyProposal =
    MHProductProposal(
      identTranslationProposal,
      identRotationProposal,
      shapeProposalLeading
    )
      .forType[PoseAndShapeParameters]
      .relabel("shape-leading-only")

  val poseAndShapeRemainingShapeOnlyProposal =
    MHProductProposal(
      identTranslationProposal,
      identRotationProposal,
      shapeProposalRemaining
    )
      .forType[PoseAndShapeParameters]
      .relabel("shape-trailing-only")

Using a mixture proposal, we can combine them to one proposal for pose and shape

  val mixturePoseAndShapeProposal = MHMixtureProposal(
    (0.2, poseAndShapeTranslationOnlyProposal),
    (0.2, poseAndShapeRotationOnlyProposal),
    (0.3, poseAndShapeLeadingShapeOnlyProposal),
    (0.3, poseAndShapeRemainingShapeOnlyProposal)
  )

Finally, we do the same for the noise proposal to obtain a proposal that updates the full parameter vector

val noiseProposal = GaussianRandomWalkProposal(0.1, "noise").forType[Double]
  val identNoiseProposal = MHIdentityProposal.forType[Double]
  val identPoseAndShapeProposal =
    MHIdentityProposal.forType[PoseAndShapeParameters]

  val noiseOnlyProposal =
    MHProductProposal(identPoseAndShapeProposal, noiseProposal)
      .forType[Parameters]
  val poseAndShapeOnlyProposal =
    MHProductProposal(mixturePoseAndShapeProposal, identNoiseProposal)
      .forType[Parameters]
  val fullproposal = MHMixtureProposal(
    (0.9, poseAndShapeOnlyProposal),
    (0.1, noiseOnlyProposal)
  )

Building the Markov Chain

For running the Markov Chain, we proceed exactly as in the previous tutorial.

val logger = MHSampleLogger[Parameters]()
  val chain = MetropolisHastings(fullproposal, posteriorEvaluator)

  val initialParameters = Parameters(
    PoseAndShapeParameters(TranslationParameters(EuclideanVector3D(0, 0, 0)),
                            RotationParameters((0.0, 0.0, 0.0)),
                            ShapeParameters(DenseVector.zeros[Double](model.rank))),
    noiseStddev = 1.0)

  val mhIterator = chain.iterator(MHSample(initialParameters, "inital"), logger)

In this example we are interested to visualize some samples from the posterior as we run the chain. This can be done by augmenting the iterator with visualization code:

val samplingIterator = for ((sample, iteration) <- mhIterator.zipWithIndex) yield {
    println("iteration " + iteration)
    if (iteration % 500 == 0) {
      val poseAndShapeParameters = sample.parameters.poseAndShapeParameters
      val poseTransformation = Parameters.poseTransformationForParameters(poseAndShapeParameters.translationParameters, poseAndShapeParameters.rotationParameters, rotationCenter)
      modelView.shapeModelTransformationView.shapeTransformationView.coefficients =
        poseAndShapeParameters.shapeParameters.coefficients
      modelView.shapeModelTransformationView.poseTransformationView.transformation = poseTransformation

    }
    sample
}

Finally, we draw the samples using the chain by consuming the iterator. We drop the first 1000 iterations, as the chain needs some burn-in time to converge to a equilibrium solution:

  val samples = samplingIterator.drop(1000).take(10000).toIndexedSeq

Before working with the results, we check the acceptance ratios to verify that all the proposals work as expected:

  println(logger.samples.acceptanceRatios)

Analyzing the results

Once we have the samples, we can now use them to analyze our fit. For example, we can select the best fit from these samples and visualize it

val bestSample = samples.maxBy(posteriorEvaluator.logValue)
  
val bestPoseAndShapeParameters = bestSample.parameters.poseAndShapeParameters
val bestPoseTransformation = Parameters.poseTransformationForParameters(
    bestPoseAndShapeParameters.translationParameters,
    bestPoseAndShapeParameters.rotationParameters,
    rotationCenter)    
  

  val bestFit = model.instance(bestPoseAndShapeParameters.shapeParameters.coefficients).transform(bestPoseTransformation)
  val resultGroup = ui.createGroup("result")

  ui.show(resultGroup, bestFit, "best fit")

The samples allow us to infer much more about the distribution. For example, we could use them to estimate the distribution of the length of the femur or any other measuremnt of the shape that we are interested.

Beyond landmark fitting

We have shown above how Scalismo can be used to perform Bayesian model fitting on the example of fitting 3D landmarks. This example can easily be extended to other fitting tasks, such as fitting the model to points with unkown correspondences, fitting shapes in surfaces of fitting a model to an image using an Active Shape Model as a likelihood function. In principle, all that is required is to change the likelihood function and rerun the fit. In practice, however, as a change in the likelihood function can dramatically change the posterior density, it is often required to tune the proposals, such that good convergence can be achieved. Indeed, finding good proposal distributions is the key to applying this method successfully. The more prior knowledge about the target distribution we can incorporate into the proposals, the faster will the chain converge to the equilibrium distribution.

For more complicated use-cases of this method in image analysis, the interested reader is referred to the paper by S. Schönborn et al. and references therein:

• Schönborn, Sandro, et al. "Markov chain monte carlo for automated face image analysis." International Journal of Computer Vision 123.2 (2017): 160-183.