Grid Tracking

Research SummaryMedical Image Analysis:Tagged Magnetic Resonance ImagesQuantifying the3D deformation of the heart is a necessity for accurate diagnosis of certain heart diseases.For example, the non-uniform contraction of different portions of the heart wall could be a sign of an unhealthy heart.A popular non-invasive method for monitoring the deformation of the heart wall is by tagged Magnetic Resonance Imaging(MRI).In a tagged MR image sequence,dark curves,forming a grid,deform with the heart wall through time.The dark curves forming the grid are”tagged”to material points,and as such follow the material points through time.My current project is splitted into two phases:(i)the tracking of the grid through time,and(ii)the subsequent recovery of the full3D deformation of the heart.Grid TrackingTracking of tagged MR image sequences is done frame-by-frame.For each frame,a quadrilateral(quad)detector is run over the image to give a set of”potential quads”.Quads are picked from the set of potential quads to form a”quilt”.The quads in the quilt are connected by a square graph structure and can be either alive or dead at each time frame.For a particular frame,live quads account for tagged materials that are visible,for example heart wall,and dead quads account for tagged materials whose structure is no longer apparent,for example blood that have been pumped out of the imaging area.A quilt is arrived at by the following Bayesian strategy.A Bayesian approach requires the specification of a likelihood function and a prior.The likelihood function models the detection of potential quads given a quilt.The prior models our information on how the quads should be joined up to form the quilt.The prior(i)encourages quads to be close to their positions predicted from the last frame,(ii)encourages neighbouring quads to be close to each other,(iii)discourages intersecting quads,(iv)avoids”tears”in the quilt,and(v)encourages connectedness of quads.With the likelihood and prior densities,a Markov Chain Monte Carlo algorithm is constructed to approximate the most probable quilt given the data—maximum of the a posterior density.The most probable quilt is our estimate for a particular frame.More details can be found in[1].Recovery of DeformationAfter the successful tracking of the grid through time,there is however a complication to the problem of deducing the 3D deformation from the correspondence of the grid points(intersections of the dark curves)through time.A tagged MR image is acquired at an imaging planefixed in3D space.However,during data acquisition,the heart moves through the imaging plane and leads to the so called through-plane problem.As a result,a labelled grid point in two successive time frames does not necessarily correspond to the same physical material point in the heart.The through-plane motion is severe enough in practice to warrant a treatment.Past techniques for tackling the through-plane problem rely on an approximation which holds if successive time frames undergo a translation and scale change,but the approximation does not hold in general.We propose a method which removes such an approximation.The deformation is recovered by minimizing an objective function which penalizes(i) the roughness of the deformation,plus(ii)the discrepancy between the observed data and thefitted deformation.The method ensures accurate reconstruction of the3D deformation when there is substantial through-plane motion during data acquisition.More details can be found in[2].Medical Imaging:Reconstruction from ProjectionsWhile at University of Pennsylvania,I was involved in an area of medical imaging research known as image reconstruction from projections.Techniques of reconstruction from projections are common in other disciplines,e.g.astronomy and geophysics.Attention was focussed on the medical imaging modality called positron emission tomography(PET).In PET,a radioactive substance is injected into the patient.The radioactive substance emits positrons.A position annihilates with an electron resulting in a pair of photons travelling in opposite directions.These pairs of photons are detected by the scanner.The measured pairs of photons for different directions are the projection data.The objective is to determine the1emission activity of the object from the measured projections,and thus allows one to quantify the uptake of the radioactive substance by the object.The reconstructed image of the emission activity is approximated by the superposition of shifted non-overlapping rectangular functions called voxels(pixels in2D).V oxel is the conventional building block for reconstruction from pro-jections.Alternative approximations using overlapping radially symmetric functions have been suggested.Different criteria exist for evaluating the choice of these basis functions.My work has focused on using the Cram´e r-Rao(CR) bound in statistical estimation theory for evaluation.The CR bound puts a fundamental lower limit on the error variances that can be achieved in image reconstruction regardless of the reconstruction algorithm.In[3],a lower bound on the mean square error(MSE)is obtained and used in the evaluation of the Kaiser-Bessel basis function with square and hexagonal packing.The two components of the lower bound on the MSE are:the square of the bias and the CR bound on the error variance.In choosing different basis functions,one is choosing a tradeoff between the bias and the error variance.The relative contribution of the two components changes for different basis functions,total number of counts and emission activities.In using the CR bound, one often asks the question of whether the bound is achievable or not.It is shown in[4]that the bound is achievable if and only if the matrix that links the measured projections and the underlying object is of a special form.In practice,the matrix is not in this special form.The question of how close algorithms used in practice are to the CR bound remains open. Seismic ImagingMy thesis concentrated on an area of seismic imaging called migration.Seismic recordings of artificial sources are made at a set of receiver positions on a surface.The objective is to produce the spatial distribution of buried scatterers as illuminated by the seismic sources.Migration involves the back-propagation of the observed wave-field on the surface. This can be regarded as running the movie of the wave-field evolution backwards in time.A method of propagation of an acoustic wave-field called the split-step Fourier method is refined in[5].The back-propagation of an acoustic wave-field requires an accurate velocity model.Errors in the velocity model give rise to fluctuation in the estimate of the scattering activity.In[6],a method called partially coherent migration is proposed to reduce suchfluctuation,albeit with some sacrifice in spatial resolution.The partially coherent migration is closely related to a technique known in medical ultrasound as spatial compounding,and is also analogous to the averaged/modified periodogram used in spectral estimation.The underlying problem of these seemingly different areas is the handling of error propagation from one domain to its Fourier domain.References[1] D.Lee,J.Kent,and K.Mardia,“Tracking of tagged MR images by Bayesian analysis of a network of quads,”in XVth InternationalConference on Information Processing in Medical Imaging,1997.[2] D.Lee,J.Kent,and K.Mardia,“The through-plane problem for tagged MR images,”in Medical Image Understanding andAnalysis,1997.Submitted.[3] D.Lee,“Lower bounds on the mean square error in emission tomography for different image approximations,”1996.Preprint.[4]R.Aharoni and D.Lee,“The Cram´e r-Rao bound for Poisson distribution,”1996.Preprint.[5] D.Lee,I.Mason,and G.Jackson,“Split-step Fourier migration with deconvolution imaging,”Geophysics,vol.56,pp.1786–1793,November1991.[6] D.Lee,G.Jackson,and I.Mason,“Partially coherent migration,”Geophysics,vol.58,pp.1301–1313,September1993.2。