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

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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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