medical image analysis

Medical Image AnalysisMedical Image AnalysisImage Segmentation

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

Edge-Based Image Edge-Based Image SegmentationSegmentationUseful links

◦The Berkeley Segmentation Dataset and Benchmark

◦TurtleSeg

Edge-Based Image Edge-Based Image SegmentationSegmentation


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 Edge Detection OperationsOperationsGradient magnitude and

directional information from the Sobel horizontal and vertical direction masks

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Edge Detection Edge Detection OperationsOperationsThe second-order gradient

operator Laplacian can be computed by convolving one pf the following masks

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Edge Detection Edge Detection OperationsOperationsA smoothing filter first before

taking a Laplacian of the imageCombined into a single Laplacian

of Gaussian function as

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Edge Detection Edge Detection OperationsOperations

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Edge Detection Edge Detection OperationsOperationsA Laplacian of Gaussian (LOG)

mask of pixels, :

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Boundary TrackingBoundary Tracking


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 TrackingBoundary TrackingThe neighborhood search method

◦ : edge magnitude◦ : edge orientation◦ : a boundary pixel◦ : a successor boundary pixel◦ , , :pre-determined

thresholds

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Boundary TrackingBoundary Tracking

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Boundary TrackingBoundary TrackingA 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 TrackingBoundary TrackingA* 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

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◦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

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◦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

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Hough TransformHough TransformHough 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

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Hough TransformHough Transform

Figure comes from the Wikipedia, www.wikipedia.org.


Gradient e

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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 Pixel-Based Direct Classification MethodsClassification Methods


Example the histogram for bimodal distribution

Find the deepest valley point between the two consecutive major peaks

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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 Optimal Global ThresholdingThresholdingAssume

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

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Optimal Global Optimal Global ThresholdingThresholding

The error probabilities of misclassifying a pixel

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Optimal Global Optimal Global ThresholdingThresholding

Assume the Gaussian probability density functions

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Pixel Classification Through Pixel Classification Through ClusteringClusteringFeature 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 distancePost-processing

◦Region growing, pixel connectivity

K-Means ClusteringK-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

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K-Means ClusteringK-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.

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K-Means ClusteringK-Means ClusteringObjective function

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Fuzzy c-Means ClusteringFuzzy c-Means ClusteringThe objective function

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Region-Based Region-Based SegmentationSegmentationRegion-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-GrowingRegion-GrowingTwo 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

Center Pixel

Pixels satisfying the similarity criterionPixels not satisfying the similarity criterion3x3 neighborhood

5x5 neighborhood

7x7 neighborhood

Segmented 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-SplittingRegion-SplittingThe 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

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Figure 7.7. An image with quad region-splitting process (top) and the corresponding quad-tree structure (bottom).

Recent Advances in Recent Advances in SegmentationSegmentationModel-based estimation methodsRule-based systemsAutomatic segmentation

Estimation-Model Based Estimation-Model Based Adaptive SegmentationAdaptive Segmentation


A multi-level adaptive segmentation (MAS) method

Define Classes

Determination of Model Parameters

(From a set ofmanually segmented and labeled images)

Classification of ImagePixels Using Model

Signatures

Identification of TissueClasses and Pathology

Formation ofa New ClassNecessary?

All pixelsclassified?

ParameterRelaxation

Yes

Yes

No

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 7.10. 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 Image Segmentation Using Neural NetworksNeural Networks


Backpropagation neural network for classification

Radial basis function (RBF) network

Segmentation of arterial structure in digital subtraction angiograms

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Non-Linear Activation Function F

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Figure 7.11. A basic computational neural element or Perceptron for classification.

Hidden Layer Neurons

Output Layer Neurons

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Figure 7.12. A feedforward Backpropagation neural network with one hidden layer.

RBF Unit 1 RBF Unit 2 RBF Unit n

Input ImageSliding

Image Window

Output

Linear Combiner

RBF Layer

Figure 7.13. An RBF network classifier for image segmentation.

Figure 7.14. 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.

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