Medical Image Analysis Image Segmentation Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Edge-Based Image Segmentation Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, Edge-based approach ◦ Spatial filtering to compute the first-order or second-order gradient information of the image: Sobel, Laplacian masks ◦ Edges need to be linked to form closed regions ◦ Uncertainties in the gradient information due to noise and artifacts in the image

Edge Detection Operations Gradient magnitude and directional information from the Sobel horizontal and vertical direction masks

Edge Detection Operations The second-order gradient operator Laplacian can be computed by convolving one pf the following masks

Edge Detection Operations A smoothing filter first before taking a Laplacian of the image Combined into a single Laplacian of Gaussian function as

Edge Detection Operations

A Laplacian of Gaussian (LOG) mask of pixels, :

Boundary Tracking Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, Edge-linking ◦ Pixel-by-pixel search to find connectivity among the edge segments ◦ Connectivity can be defined using a similarity criterion among edge pixels ◦ Geometrical proximity or topographical properties

Boundary Tracking The neighborhood search method ◦ : edge magnitude ◦ : edge orientation ◦ : a boundary pixel ◦ : a successor boundary pixel ◦,, :pre-determined thresholds

Boundary Tracking

A graph-based search method ◦ Find paths between the start and end nodes minimizing a cost function that may be established based on the distance and transition probabilities ◦ The start and end nodes are determined from scanning the edge pixels based on some heuristic criterion

Start Node End Node Figure 7.1. Top: An edge map with magnitude and direction information; Bottom: A graph derived from the edge map with a minimum cost path (darker arrows) between the start and end nodes.

Boundary Tracking A* search algorithm ◦ 1. Select an edge pixel as the start node of the boundary and put all of the successor boundary pixels in a list, OPEN ◦ 2. If there is no node in the OPEN list, stop; otherwise continue ◦ 3. For all nodes in the OPEN list, compute the cost function and select the node with the smallest cost. Remove the node from the OPEN list and label it as CLOSED. The cost function may be computed as

Boundary Tracking A* search algorithm ◦ 4. If is the end node, exit with the solution path by backtracking the pointers; otherwise continue ◦ 5. Expand the node by finding all successors of. If there is no successor, go to Step 2; otherwise continue

Boundary Tracking A* search algorithm ◦ 6. If a successor is not labeled yet in any list, put it in the list OPEN with updated cost as and a pointer to its predecessor ◦ 7. If a successor is already labeled as CLOSED or OPEN, update its value by ◦. Put those CLOSED successors whose cost functions were lowered, in the OPEN list and redirect to the pointers from all nodes whose costs were lowered. Go to Step 2

Hough Transform Hough transform ◦ Similar to the Radon transform ◦ Detect straight lines and other parametric curves such as circles, ellipses ◦ A line in the image space forms a point in the parameter space

Hough Transform Figure comes from the Wikipedia,

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, Gradient e r p O Figure 7.2. A model of the object shape to be detected in the image using Hough transform. The vector r connects the Centroid and a tangent point p. The magnitude and angle of the vector r are stored in the R-table at a location indexed by the gradient of the tangent point p.

Pixel-Based Direct Classification Methods Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, Example the histogram for bimodal distribution Find the deepest valley point between the two consecutive major peaks

Figure 7.3. The original MR brain image (top), its gray-level histogram (middle) and the segmented image (bottom) using a gray value threshold T=12 at the first major valley point in the histogram.

Figure 7.4. Two segmented MR brain images using a gray value threshold T=166 (top) and T=225 (bottom).

Optimal Global Thresholding Assume ◦ The histogram of an image to be segmented has two Gaussian distributions belonging to two respective classes such as background and object ◦ The histogram

Optimal Global Thresholding The error probabilities of misclassifying a pixel

Optimal Global Thresholding Assume the Gaussian probability density functions

Optimal Global Thresholding The optimal global threshold

Pixel Classification Through Clustering Feature vector of pixels ◦ Gray value, contrast, local texture measure, red, green, or blue components Clusters in the multi-dimensional feature space ◦ Group data points with similar feature vectors together in a single cluster ◦ Distance measure: Euclidean or Mahalanobis distance Post-processing ◦ Region growing, pixel connectivity

K-Means Clustering ◦ 1. Select the number of clusters with initial cluster centroids ; ◦ 2. Partition the input data points into clusters by assigning each data point to the closest cluster centroid using the selected distance measure ◦ 3. Compute a cluster assignment matrix representing the partition of the data points with the binary membership value of the th data point to the th cluster such that

K-Means Clustering ◦ 4. Re-compute the centroids using the membership values as ◦ 5. If cluster centroids or the assignment matrix does not change from the previous iteration, stop; otherwise go to Step 2.

K-Means Clustering Objective function

Fuzzy c-Means Clustering The objective function

Region-Based Segmentation Region-growing based segmentation ◦ Examine pixels in the neighborhood based on a pre-defined similarity criterion ◦ The neighborhood pixels with similar properties are merged to form closed regions Region splitting ◦ The entire image or large regions are split into two or more regions based on a heterogeneity or dissimilarity criterion

Region-Growing Two criteria ◦ A similarity criterion that defines the basis for inclusion of pixels in the growth of the region ◦ A stopping criterion that stops the growth of the region

Figure 7.5. A pixel map of an image (top) with the region-growing process (middle) and the segmented region (bottom).

Figure 7.6. A T-2 weighted MR brain image (top) and the segmented ventricles (bottom) using the region-growing method.

Region-Splitting The following conditions are met: ◦ 1. Each region, ; is connected ◦ 2. ◦ 3. for all, ; ◦ 4. = TRUE for ◦ 5. = FALSE for, where is a logical predicate for the homogeneity criterion on the region

R R4R4 R3R3 R2R2 R1R1 R24R24 R23R23 R22R22 R21R21 R1R1 R 21 R 22 R 23 R 41 R 43 R 24 R 42 R 44 R3R3 R44R44 R43R43 R42R42 R41R41 Figure 7.7. An image with quad region-splitting process (top) and the corresponding quad-tree structure (bottom).

Recent Advances in Segmentation Model-based estimation methods Rule-based systems Automatic segmentation

Estimation-Model Based Adaptive Segmentation Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, A multi-level adaptive segmentation (MAS) method

Define Classes Determination of Model Parameters (From a set of manually segmented and labeled images) Classification of Image Pixels Using Model Signatures Identification of Tissue Classes and Pathology Formation of a New Class Necessary? All pixels classified? Parameter Relaxation Yes No Figure 7.8: The overall approach of the MAS method.

Figure 7.9: (a) Proton Density MR and (b) perfusion image of a patient 48 hours after stroke.

Figure Results of MAS method with 4x4 pixel probability cell size and 4 pixel wide averaging. (a) pixel classification as obtained on the basis of maximum probability, (b) as obtained with p>0.9.

Image Segmentation Using Neural Networks Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, Backpropagation neural network for classification Radial basis function (RBF) network Segmentation of arterial structure in digital subtraction angiograms

x1x1 xnxn x2x2 1 Non-Linear Activation Function F w n+1 w1w1 w2w2 wnwn Figure A basic computational neural element or Perceptron for classification.

Hidden Layer Neurons Output Layer Neurons x1x1 x2x2 x3x3 xnxn 1 Figure A feedforward Backpropagation neural network with one hidden layer.

RBF Unit 1 RBF Unit 2 RBF Unit n Input Image Sliding Image Window Output Linear Combiner RBF Layer Figure An RBF network classifier for image segmentation.

Figure RBF Segmentation of Angiogram Data of Pig-cast Phantom image (top left) with using a set of 10 clusters (top right) and 12 clusters (bottom) respectively.