a bayesian approach for transformation estimation

A Bayesian Approachfor Transformation Estimation

Camille Izard and Bruno Jedynak

Landmark Detection in brain MRI

Laboratoire Paul Painlevé

Université des Sciences et Technologies de Lille

Center for Imaging Science

Johns Hopkins University

Image Registration

• Comparing structures– Time evolution – Between patients

• Comparing different image modalities– MRI, CT

• General Approach for registration– Define the mean image– Define the norms– Different types of

• Affine transformation• Diffeomorphisms

• Use of landmarks– Characterize the underlying shape– Rough analysis of the shape (Bookstein, 1991)– Corresponding point for registration algorithm

• Manual Landmarking

Image Registration

SCC

HoH

HT

Image Model

Generating an imageFor all u,

Let’s denote v 2 I the voxels of an imageGraylevels modeled with a mixture of Gaussian,Zv the matter at voxel v, unknown random variable. We define : RR3 R3.Matter in the new coordinate system:The template:

Matter Distribution

Template obtained when is a translation, considering the landmark SCC

CSF

GM WM

With a new image

-Contains the geometry of the images

-Includes the variation of geometry

-Learned offline on a training set

-Estimating the transformation = locating the landmarks

-Caracterize the photometry

-Learned for each image by EM algorithm

Unkonwn :

Comparison

• Data term– No needs to define the mean image– Adjustable weight depending on the law distribution– Use of the matter and not gray level

• Regularity constraints– Prior on the transformation parameters

Estimating Photometry distributions

Mixture of 6 Gaussian distributions:

- Pure Voxels : CSF, GM , WM- Mixed Voxels : CSF+GM, GM+WM- Outliers

Use EM to learn the distributions

Matter Distribution Estimation

The Template

The Template obtained with a translation and HoH as a landmark

CSF

GM WM

Recovering the Transformation

Information Map : Information contained at each voxel with a translation, left: with SCC, right: with HoH.

HoH SCC

Results

Landmark Error on training set Error on testing set

SCC 1.81 mm (1.42 mm) 2.46 mm (1.92 mm)

HoH 2.75 mm (1.97 mm) 3.70 mm (1.48 mm)

HT 0.26 mm (0.51 mm) 2.19 mm (1.11mm)

translation, 38 training images, 9 images for testing

Using more complex transformationsIf has more parameters ,

Gradient descent on the transformation parameters:

Current extensions

• Affine Transformations– Able to deal with several landmarks simultaneously– Estimation by gradient descent in the parameter space– Uniqueness issues – C. Izard, B. Jedynak, Bayesian Registration for

Landmark detection, ISBI, april 2006

• Splines transformations– Able to deal with several landmarks at the same time,– Flexibility of the model to various number of

landmarks,– Unicity of the transformation– Estimation by gradient descent in the parameter space