Medical imaging
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Simultaneous measurements on a spatial grid. Many modalities: mainly EM radiation and sound. Medical Imaging “To invent you need a good imagination and a pile of junk.” Thomas Edison 1879 Electron rapidly decelerates at heavy metal target, giving off X-Rays. Bremsstrahlung 1896

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Medical Imaging

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Simultaneous measurements on a spatial grid.

Many modalities: mainly EM radiation and sound.

Medical Imaging


“To invent you need a good imagination and a pile of junk.”

Thomas Edison

1879


Electron rapidly decelerates at heavy metal target, giving off X-Rays.

Bremsstrahlung


1896


Projection of X-Ray silhouette onto a piece of film or detector array, with intervening fluorescent screen.

X-Ray and Fluoroscopic Images


From a series of projections, a tomographic image is reconstructed using Filtered Back Projection.

Computerized Tomography


Radioactive isotope separated by difference in inertia while bending in magnetic field.

Mass Spectrometer


Gamma camera for creating image of radioactive target. Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography).

Nuclear Medicine


Ultrasound beam formed and steered by controlling the delay between the elements of the transducer array.

Phased Array Ultrasound


Real Time 3D Ultrasound


Positron-emitting organic compounds create pairs of

high energy photons that are detected synchronously.

Positron Emission Tomography


MRI (Magnetic Resonance Imaging)

OCT (Optical Coherence Tomography)

Other Imaging Modalities


3D

Higher speed

Greater resolution

Measure function as well as structure

Combining modalities (including direct vision)

Current Trends in Imaging


Dissection:

Medical School, Day 1: Meet the Cadaver.

From Vesalius to the Visible Human

The Gold Standard


Local Operators and Global Transforms


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.

Images are n dimensional signals.


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

Global Transforms in n dimensions


Capable of geometric, similarity, or affine.

Homogeneous coordinates.

Multiply in reverse order to combine

SGI “graphics engine” 1982, now standard.

Orthographic Transform Matrix


Translation by (tx , ty)

Scale x by sx and y by sy


Rotation in 2D

  • 2 x 2 rotation portion is orthogonal (orthonormal vectors).

  • Therefore only 1 degree of freedom, .


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

  • 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.


f is usually monotonic, and shift invariant.

Inverse may not exist due to discrete values of intensity.

Brightness/contrast, “windowing”.

Thresholding.

Color Maps.

f may vary with pixel location, eg., correcting for inhomogeneity of RF field strength in MRI.

Point Operators


A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.

Histogram Equalization


Assumes bimodal distribution.

Trough represents boundary points between homogenous areas.

Adaptive Thresholding from Histogram


Assumes registration.

Averaging multiple acquisitions for noise reduction.

Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography).

Masking.

Algebraic Operators


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.

Re-Sampling on a New Lattice


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.

Convolution and Correlation


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.

Neighborhood PDE Operators


Vector

Direction of maximum change of scalar intensity I.

Normal to the boundary.

Nicely n-dimensional.

Intensity Gradient


Scalar

Maximum at the boundary

Orientation-invariant.

Intensity Gradient Magnitude


Classic Edge Detection Kernel (Sobel)


100% opaque watertight surface

Fast, 28 = 256 combinations, pre-computed

Isosurface, Marching Cubes (Lorensen)


  • Marching cubes works well with raw CT data.

  • Hounsfield units (attenuation).

  • Threshold calcium density.


Ixy = Iyx= curvature

Orientation-invariant.

What about in 3D?

Jacobian of the Intensity Gradient


Divergence of the Gradient.

Zero at the inflection point of the intensity curve.

Laplacian of the Intensity

I

Ix

Ixx


Repeated averaging of neighbors => Gaussian by Central Limit Theorem.

Binomial Kernel


Not the conventional concentric DOG

Subtracting pixels displaced along the x axis after repeated blurring with binomial kernel yields Ix

Binomial Difference of Offset Gaussian (DooG)


Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.

Texture Boundaries


2D Fourier Transform

analysis

or

synthesis


Most of the usual properties, such as linearity, etc.

Shift-invariant, rather than Time-invariant

Parsevals relation becoms Rayleigh’s Theorem

Also, Separability, Rotational Invariance, and Projection (see below)

Properties


Separability


Rotation Invariance


Projection

Combine with rotation, have arbitrary projection.


Gaussian

seperable

Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.


Hankel Transform

For radially symmetrical functions


Elliptical Fourier Series for 2D Shape

Parametric function, usually with constant velocity.

Truncate harmonics to smooth.


Fourier surface of 3D shapes (parameterized on surface).

Spherical Harmonics (parameterized in spherical coordinates).

Both require coordinate system relative to the object. How to choose? Moments?

Problem of poles: sigularities cannot be avoided

Fourier shape in 3D


Quaternions – 3D phasors

Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.


Fourier useful for image “processing”, convolution becomes multiplication.

Fourier less useful for shape.

Fourier is global, while shape is local.

Fourier requires object-specific coordinate system.

Summary


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