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Methods in Image Analysis – Lecture 2 Local Operators and Global Transforms. George Stetten, M.D., Ph.D. CMU Robotics Institute 16-725 U. Pitt Bioengineering 2630 Spring Term, 2006. Preface. Some things work in n dimensions, some don’t. It is often easier to present a concept in 2D.

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methods in image analysis lecture 2 local operators and global transforms

Methods in Image Analysis – Lecture 2Local Operators and Global Transforms

George Stetten, M.D., Ph.D.

CMU Robotics Institute 16-725

U. Pitt Bioengineering 2630

Spring Term, 2006

preface
Preface
  • Some things work in n dimensions, some don’t.
  • It is often easier to present a concept in 2D.
  • I will use the word “pixel” for n dimensions.
point operators
Point Operators
  • f is usually monotonic, and shift invariant.
  • Inverse may not exist.
  • Brightness/contrast, “windowing”.
  • Thresholding.
  • Color Maps.
  • f may vary with pixel location, eg., correcting for inhomogeneity of RF field strength in MRI.
histogram equalization
Histogram Equalization
  • A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.
adaptive thresholding from histogram
Adaptive Thresholding from Histogram
  • Assumes bimodal distribution.
  • Trough represents boundary points between homogenous areas.
algebraic operators
Algebraic Operators
  • Assumes registration.
  • Averaging multiple acquisitions for noise reduction.
  • Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography).
  • Masking.
re sampling on a new lattice
Re-Sampling on a New Lattice
  • Can result in denser or sparser pixels.
  • Two general approaches:
    • Forward Mapping (Splatting)
    • Backward Mapping (Interpolation)
      • Nearest Neighbor
      • Bilinear
      • Cubic
  • 2D and 3D texture mapping hardware acceleration.
neighborhood operators
Neighborhood Operators
  • Kernels
  • Cliques
  • Markov Random Fields
  • Must limit the relationships to be practical.
convolution and correlation
Convolution and Correlation
  • Template matching uses correlation, the primordial form of image analysis.
  • Kernels are mostly used for “convolution” although with symmetrical kernels equivalent to correlation.
  • Convolution flips the kernel and does not normalize.
  • Correlation subtracts the mean and generally does normalize.
neighborhood pde operators
Neighborhood PDE Operators
  • For discrete images, always requires a specific scale.
  • “Inner scale” is the original pixel grid.
  • Size of the kernel determines scale.
  • Concept of Scale Space, Course-to-Fine.
intensity gradient
Intensity Gradient
  • Vector
  • Direction of maximum change of scalar intensity I.
  • Normal to the boundary.
  • Nicely n-dimensional.
intensity gradient magnitude
Intensity Gradient Magnitude
  • Scalar
  • Maximum at the boundary
  • Orientation-invariant.
isosurface marching cubes lorensen
Isosurface, Marching Cubes (Lorensen)
  • 100% opaque watertight surface
  • Fast, 28 = 256 combinations, pre-computed
slide19

Marching cubes works well with raw CT data.

  • Hounsfield units (attenuation).
  • Threshold calcium density.
other useful 3d rendering methods
Other Useful 3D Rendering Methods

Direct shading from gradient (Levoy, Drebin)

  • Voxels are blended (translucent).
  • Opacity proportional to gradient magnitude.
  • Rendering uses gradient direction as “surface” normal.

Maximum Intensity Projection (MIP)

  • Maximum Intensity Pixel along each projected line of sight.
  • Good for vasculature
slide21

Drebin, R. A., Carpenter, L., AND Hanrahan, P. Volume rendering. Computer Graphics 22, 4 (Aug. 1988), 65--74.

For a nice review of graphics rendering techniques see

http://accad.osu.edu/~waynec/history/lesson18.html

jacobian of the intensity gradient
Jacobian of the Intensity Gradient
  • Ixy = Iyx= curvature
  • Orientation-invariant.
  • What about in 3D?
viewing the intensity as height
Viewing the intensity as “height”
  • Differential geometry: the surface and tangent plane
  • Example: cylindrical surface, curvature = 0.
  • Move in the y direction, no roll: Ixy= 0
  • Move in the x direction, no pitch: Iyx = 0

= Iy

= Ix

laplacian of the intensity
Laplacian of the Intensity
  • Divergence of the Gradient.
  • Zero at the inflection point of the intensity curve.

I

Ix

Ixx

binomial kernel
Binomial Kernel
  • Repeated averaging of neighbors => Gaussian by Central Limit Theorem.
slide26

From

Insight into

Images, Yoo

slide27

From

Insight into

Images, Yoo

Blurring the derivative of an image =

Convolving with the derivative of a Gaussian

binomial difference of offset gaussian doog
Binomial Difference of Offset Gaussian (DooG)
  • Not the conventional concentric DOG
  • Subtracting pixels displaced along the x axis after repeated blurring with binomial kernel yields Ix
difference of gaussian operators dog not offset
Difference of Gaussian Operators (DOG)(not offset)
  • Conventionally, 2 concentric Gaussians of different scale.
  • Acts like a Laplacian, zero at boundary.

From

Insight into

Images, Yoo

slide30

From

Insight into

Images, Yoo

DOG used for Edge Enhancement

slide31

Edge

Enhancement

(area under

kernel = 0)

Contrast

Enhancement

(area under

kernel = 1)

boundary profiles tamburo
Boundary Profiles (Tamburo)
  • Splatting in an ellipsoid along the gradient.
boundary profiles
Boundary Profiles
  • Fitting the cumulative Gaussian
variable conductance diffusion vcd
Variable Conductance Diffusion (VCD)
  • Attempt to get around the global nature of Fourier.
  • Smoothing with a Gaussian in the spatial domain lowers noise, but also blurs boundaries.
  • Gaussian smoothing simulates uniform heat diffusion.
  • VCD makes conductance an inverse function of gradient, so that “heat” does not flow well across boundaries. This homogenizes already homogenious regions while preserving boundaries.
other basis sets for boundary kernel
Other basis sets for boundary kernel.
  • Wavelets
  • Statistical texture analysis
  • Conjacent spheres
texture boundaries
Texture Boundaries
  • Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.
ridges
Ridges
  • Attempt to tie the image structure together in a locally continual manner.
  • Along some manifold in less than n dimensions.
  • Local maximum along the normal to the ridge.
  • Canny edge detector is a boundariness ridge.
  • A “core” is a medialness ridge.
  • Medialness within an object is the property of being equidistant from two boundaries.
global transforms in n dimensions
Global Transforms in n dimensions
  • Geometric (rigid body)
    • n translations and rotations.
  • Similarity
    • Add 1 scale (isometric).
  • Affine
    • Add n scales (combined with rotation => skew).
    • Parallel lines remain parallel.
  • Projection
orthographic transform matrix
Orthographic Transform Matrix
  • Homogeneous coordinates (divide vector by scalar to keep

last component = 1.

  • Capable of geometric, similarity, or affine.
  • Homogeneous coordinates.
  • Multiply in reverse order to combine
  • SGI “graphics engine” 1982, now standard.
translation by t x t y
Translation by (tx , ty)

Scale x by sx and y by sy

rotation in 2d
Rotation in 2D
  • 2 x 2 rotation portion is orthogonal (orthonormal vectors).
  • Therefore only 1 degree of freedom, .
rotation in 3d
Rotation in 3D
  • 3 x 3 rotation portion is orthogonal (orthonormal vectors).
  • 3 degree of freedom (dotted circled), , as expected.
non orthographic projection in 3d
Non-Orthographic Projection in 3D
  • When vector is made homogeneous (dividing to make last component 1), x and y are made smaller in the distance.
  • For X-ray or direct vision, projects onto the (x,y) plane.
  • Rescales x and y for “perspective” by changing the “1” in the homogeneous coordinates, as a function of z.
slide45

Anisotropic Scaling of Vectors(ITK Software Guide 4.25, 4.26)

itkVector

dy

dx1

dx2

itkVector objects are used to represent distances between locations, velocities, etc.

itkCovarientVector

dy

dx1

dx2

itkCovarientVector objects are used to represent properties such as the gradient,

since stretching the image in the x dimension lowers the x gradient component (slope).

dI

dI

dx1

dx2

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