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Edge Detection Operations Gradient magnitude and directional information from the Sobel horizontal and vertical direction masks
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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 SegmentationSegmentation
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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
1)( Tbe j
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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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Boundary TrackingBoundary TrackingA* 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
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Boundary TrackingBoundary TrackingA* 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
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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.
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Gradient e
r
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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
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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 7.3. 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
◦The histogram)()()( 2211 zpPzpPzp
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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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Optimal Global Optimal Global ThresholdingThresholdingThe optimal global threshold T
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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
kiv ki ,...,2,1
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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
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
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
Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.
Backpropagation neural network for classification
Radial basis function (RBF) network
Segmentation of arterial structure in digital subtraction angiograms
x1
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Non-Linear Activation Function F
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w2
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Figure 7.11. A basic computational neural element or Perceptron for classification.
Hidden Layer Neurons
Output Layer Neurons
x1 x2 x3xn 1
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.
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.