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### Transcript of Normalized Cuts and Image Segmentation J. Shi and J. Indian Institute of Science, Bangalore....

• Image Registration

Soma Biswas

Department of Electrical Engineering,

Indian Institute of Science, Bangalore.

• Definition

 Registration is the determination of a geometrical

transformation that aligns points in one view of an object

with corresponding points in another view of that object

Applications:

 Mosaicing: Align several images into a single

composition that represents part of a 3D scene, eg.

remote sensed images

 Build 3D Model from Images

 Medical Images: Combine the two registered

images by producing a reoriented version of one

view that can be “fused” with the other.

2 MR

SPECT registered

• Transformation

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 Relates the position of features in two images

 Rigid

 translations and rotations

 preserve all distances

 Preserve the straightness of lines and all nonzero angles between straight lines

 Affine

 Also allows scaling and shearing

 Preserves the straightness of lines

 Allows angles between lines to change

 curved

 Allows the mapping of straight lines to curves

 perspective

 The parallelism of lines need not be preserved

• Registration algorithms

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 Method used to find the transformation

 Landmark based

 Intensity based

 Information theory based

 Registration using basis functions

 Registration using splines

 Physics based

 Elastic, Fluid, Optical flow, etc.

• Point/Landmark based

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 Identifying corresponding points in the images and inferring the image transformation

 Types of landmarks

 Extrinsic

 artificial objects attached to the patient

 Intrinsic

 internal anatomical structures: fiducial points,

 To be reliable, they must lie in clearly discernible features, (fiducial features)

 Displacement in the determination of the fiducial point - fiducial localization error (FLE).

The transformation that aligns the corresponding fiducial points will then interpolate the

mapping from these points to other points in the views

 Computing the average or “centroid” of each set of points  translation

 Rotated this point set about the new centroid until the sum of the squared distances between each corresponding point pair is minimized

• Point-based Method

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fiducial registration error

• Point-based, rigid registration

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• Point-based, registration – with scaling

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• Point-based registration: Nonisotropic scaling

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• Surface-based

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 Method

 Extracting corresponding surfaces

 Computing the transformation by minimizing some measure of

distance between the two surfaces

 Algorithms used

 The “Head and Hat” Algorithm

 The Iterative Closest Point Algorithm

 Registration using crest lines

• Disparity Functions

 Search for the transformation that minimizes some disparity function or metric

between the two surfaces X and Y

 Distance between two feature sets A and B is normally defined as the minimum

distance between a point in A and a point in B

d(A,B) is small if one pair of points in these two sets are close

 Hausdorff distance from A to B

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• ICP

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• Insights

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• Details

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assume closest points correspond

• Algorithm

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Init the error to ∞

Calculate correspondence

Calculate alignment

Apply alignment

Update error

If error > threshold

• Intensity Based Method

 Calculates a transformation between two images using the pixel values alone.

 Registration transformation is determined by iteratively optimizing some “similarity

measure” calculated from all pixel values.

 Reference Image: A

 Target Image: B is iteratively transformed to

 Similarity measures are invariably calculated for the set of pixels in the overlapping

region which is a function of T and so changes with iteration

 Similarity Measures: Image Subtraction

 If images being registered are identical, except for misalignment

 SSD = 0, for correct registration, and will increase with registration error

 If A and B differ only by Gaussian noise, SSD is the optimal measure

 Eg. Serial registration of MR images, images identical except for small changes due to

disease progression or response to treatment

 Problems: non-gaussian noise, if small number of pixel intensity change by large

amount

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• Similarity Measure

 Intensities in images A and B are linearly related, CC can be shown to the ideal

similarity measure

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• Similarity Measure

 Ratio-Image Uniformity (RIU): initially devised for registration of multiple PET

images of the same subject

 For each estimate of the registration transformation, a ratio image R s calculated

 R: divide each pixel in A by each pixel in B’.

 Uniformity determined by calculating normalized std of R

 Algorithm iteratively determines the transformation T that minimizes the normalized

std, i.e. maximizes uniformity

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• Joint Histogram

 n-dim, n is the number of images used to generate it

 Axis is intensity in each image

 Value at each point: no. of pixels with that combination of intensities

 Joint histogram normalized: estimate of joint probability distribution function (pdf) of

intensities in the n images

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• Joint Entropy

 Shannon entropy widely used as a measure of information

 Describes the average information supplied by a set of symbols {s} whose

probabilities are given by {p(s)}

 If all symbols s have equal probability, then entropy will be at maximum.

 If one symbol has a probability of 1 and all others have a probability of zero, then

entropy will have a minimum value.

 Images correctly aligned -> joint histograms have tight clusters, surrounded by large

dark regions -> clusters disperse as the images become less well registered.

 Tight clusters in the histograms represent a small number of symbols s having high

probabilities P(s) & Surrounding dark regions in the joint histogram represent

large numbers of symbols with probability zero.

- As the clusters disperse, the high intensity regions of the joint histogram become less

intense, and previously dark regions in the histograms become brighter

 Misregistration results in an increase in histogram entropy.

 Entropy of the PDF calculated from images A and B’ should be iteratively minimized

to register these images

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