Medical Image Registration by Maximization of Mutual Information

Transcription

1 Medical Image Registration by Maximization of Mutual Information EE 591 Introduction to Information Theory Instructor Dr. Donald Adjeroh Submitted by Senthil.P.Ramamurthy Damodaraswamy, Umamaheswari Introduction Image registration is the process of determining correspondence between all points in two images of the same scene, and is now widely used to Medical images. In other words, image registration is finding the mapping transformation between two images both spatially and with respect to intensity. This transformation is used to match two or more pictures taken at different times, from different sensors or from different view points. Images can be viewed as deterministic or random process. 1

2 Introduction In medical imaging, segmentation, registration and interpolation play primary roles. In those, registration is the most time consuming task because we compare all voxel data and then evaluate the matching degree many times The combination of images can often lead to additional clinical information not apparent in the separate images. Registration Registration is a recently emerged task in medical image processing used to match two independently acquired images. Registration is achieved by adjustment of the relative position and Orientation until the mutual information between the images is maximized. Two basic types of medical images are made: (1) functional body images (such as SPECT or PET scans), which provide physiological information, and (2) structural images (such as CT or MRI), which provide an anatomic map of the body 2

3 Multimodal registration Registration types Integrating information taken from different sources Temporal registration Finding changes in images taken at different times or under different conditions Viewpoint registration Inferring three dimensional information from images in which either the camera or the objects in the scene have moved Template registration For model based object recognition Image Registration Methods Landmark matching methods include external fiducial landmarks or anatomic landmarks. Surface matching uses an algorithm that matches different images of the same patient surface. Intensity matching uses mutual intensity information to co-register different images. 3

4 Uses of Image Registration Image segmentation/deformable atlas Normal vs abnormal shape/variation Functional brain mapping/removing shape variation Multimodality fusion Surgical planning and evaluation Image guided surgery Template constrained reconstruction Choice of template image/model Sources of Registration Errors Feature identification errors Identification of landmarks,contours,surfaces,segmentations,etc Correspondence ambiguities Curve/surface/segmentation matching Conflicting feature information Landmarks, contours, surfaces, segmentations, intensity, modalities,etc Lack of model flexibility 4

5 Relation of MI to entropy I(A,B) = H(A) + H(B) H(A,B) = H(A) H(A/B) = H(B) H(B/A) H(A) and H(B) is the entropy of A and B, respectively and H(A B) and H(B A) are the conditional entropy of A given B and B given A respectively Relationship continued H(A) = -? a p A (a) log pa(a) H(A,B) = -? ab p AB (a,b) log p AB (a,b) H(A/B) = -? a,b p AB (a.b) log p A /B (a,b) 5

6 Mutual Information Mutual information is a measure of how much information one random variable tells about another. The basis of mutual information method is that mutual information is maximized when the images are registered. Mutual information measures the statistical dependence between the intensity of corresponding voxels of the images to be registered. Mutual Information For two images, the mutual information is computed from the joint probability distribution of the images intensity or gray-values. One of the main advantages of using mutual information is than it can be used to align images of different modalities (e.g. CT to MR-T1, MR-T1 to PET etc). In medical image processing, applications where the alignment is based on images of the same individual is known as intrapatient registration Matching datasets of different individuals is known as interpatient registration 6

7 Mutual Information The pixel values are considered as outcomes of a random variable. Pixels with common pixel values are regarded as little information sources whereas an uncommon-valued pixel is appraised at big information. While entropy for an image remains fixed, joint entropy and mutual information of two images vary as the 1-1 correspondence between the pixels from each image changes with every geometrical alignment. Maximization of Mutual Information (MMI) When mutual information is maximized, the geometric relationship, under which one image explains the other most effectively, is achieved. In other words, maximization of mutual information provides image registration. For two images, the mutual information is computed from the joint probability distribution of the images intensity or gray-values. When two images are aligned, the joint probability distribution is peaky resulting in a high mutual information value. 7

8 Maximization of Mutual Information (MMI) Mutual Information (MI), or relative entropy has been used as a similarity criterion in medical image registration. MI is a measure of the dispersive behavior of the joint histogram of geometrically related voxels ' intensities in both images. This dispersion is assumed to be smaller when the images are aligned. Mis -registration causes the distribution to disperse resulting in a low mutual information value Algorithm Let voxel of reference image be denoted by u(x), where x is the coordinates of the voxel Let voxel of the test image be denoted by v(y) Given that T is a transformation from the coordinate frame of the reference image to the test image, v (T(x)) is the test voxel associated with reference voxel u(x). 8

9 Algorithm The estimation of the transformation that registers the reference image u and test image v by maximizing their mutual information, ˆT = arg max T I (u(x), v (T(x)) Here we treat x as a random variable over coordinate locations in the reference image. We draw samples from x in order to approximate I and its derivative. Transformation 9

10 Transformation Each of the images is associated an image coordinate frame with its origin positioned in a corner of the image, with the x axis along the row direction, the y axis along the column direction, and the z axis along the plane direction. In order to align two images, we need to know the transformation that relates the position of features in one image or coordinate space with the position of the corresponding feature in another image or coordinate space. Algorithm Mutual information is defined in terms of entropy in the following way: I (u(x), v (T(x)) = H (u(x)) + H (v (T(x)) H (u(x), v (T(x)) H (.) is the entropy of a random variable, and is defined as H(x) = -? p(x) ln p(x) dx While joint entropy of two random variables x and y is H(x,y) = -? p(x,y) ln p(x,y) dx dy Entropy can be interpreted as a measure of uncertainty, variability or complexity. 10

11 Algorithm The exact transformation that maps the first image u (the model) onto the second image v (the image) should give rise to the largest mutual information. Mutual information then becomes an optimization criterion, optimized w.r.t. T: MI(T) = H( u(x) ) + H( v(t(x)) ) - H( u(x), v(t(x)) ) Estimating Entropy and its derivatives The entropy of a random variable z may be expressed as an expectation of the negative logarithm of the probability density H (z) = E Z (-ln p (z)) We believe that mutual information provides some advantage over joint entropy by providing larger capture range 11

12 Stochastic Maximization of Mutual Information An approximation to the derivative of maximum of mutual information: d/ dt (I (T)) = d/dt H (v(t(x)) d/dt H(u(x), v(t(x))) Assuming that the covariance matrices of the component densities used in the approximation scheme for the joint density are block diagonal:?uv-1 = DIAG (?uu-1,?vv-1 ) We can obtain an estimate for the derivative of the mutual information as follows: ^di / dt = 1 / NB? xi e B? xj e A (vi - vj)t [Wu (vi, vj)?v-1 - Wuv (wi, wj))?vv-1] d/dt(vi- vj) Stochastic maximization of mutual information The weighing factors are defined as Wu(vi, vj) = G?v(vi- vj) /? xk e A G?v(vi - vk), and Wuv(wi, wj) = G?uv (wi-wj) /? xk e A G?uv (wi - wk), Similarly for indices j and k, Ui = u(xi), vi = v(t(xi)), and wi = [ui, vi]t Stochastic maximization algorithm A? { sample of size NA drawn from x} B? { sample of size NB drawn from x} T? T +? ^di / dt 12

13 Stochastic maximization of mutual information The parameter? is called the learning rate. The above procedure is repeated a fixed number of times or until convergence is detected. A good estimate of the derivative of the mutual information could be obtained by exhaustively sampling the data. Stochastic approximation is a scheme that uses noisy derivative estimate instead of the true derivative for optimizing a function. source target warped source 13

14 The whole story in images The floating Image 14

15 The Reference Image Result for Translational movement 15

16 Applications Image registration has been used in many clinical situations: Radiotherapy tumor Stroke blood flow plastic and cranio-planar surgery.. diagnosis of breast cancer, colon cancer, cardiac studies, wrist and other injuries, inflammatory diseases and different neurological disorders including brain tumors, Alzheimer's disease and schizophrenia. radiotherapy, mostly for brain tumors Conclusion Mostly, image registration is applied in merging the functional SPECT or PET data with anatomical CT and MRI images, providing additional useful clinical information. With further development of the computer technology and physical methods for registration and visualization, medical image fusion will definitely find an even wider clinical application 16

17 References Dr.Diego Martin, Medical Director MRI, Dept of Radiology, HSC, WVU Multimodality Image Registration by Maximization of Mutual Information, Fredrik,Andre,Dirk,Guy and Paul Suetens Multi-Modal Volume Registration by Maximization of Mutual Information, William, Paul and Ron Kikinis GaryCmiccai2002.pdf 17