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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 l.jpg

Simultaneous measurements on a spatial grid.

Many modalities: mainly EM radiation and sound.

Medical Imaging




Slide4 l.jpg

1896 off X-Rays.


X ray and fluoroscopic images l.jpg

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

X-Ray and Fluoroscopic Images


Computerized tomography l.jpg

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

Computerized Tomography


Mass spectrometer l.jpg

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

Mass Spectrometer


Nuclear medicine l.jpg

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

Nuclear Medicine


Phased array ultrasound l.jpg

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

Phased Array Ultrasound


Real time 3d ultrasound l.jpg
Real Time 3D Ultrasound between the elements of the transducer array.


Positron emission tomography l.jpg

Positron-emitting organic compounds create pairs of between the elements of the transducer array.

high energy photons that are detected synchronously.

Positron Emission Tomography


Other imaging modalities l.jpg

MRI (Magnetic Resonance Imaging) between the elements of the transducer array.

OCT (Optical Coherence Tomography)

Other Imaging Modalities


Current trends in imaging l.jpg

3D between the elements of the transducer array.

Higher speed

Greater resolution

Measure function as well as structure

Combining modalities (including direct vision)

Current Trends in Imaging


The gold standard l.jpg

Dissection: between the elements of the transducer array.

Medical School, Day 1: Meet the Cadaver.

From Vesalius to the Visible Human

The Gold Standard


Local operators and global transforms l.jpg

Local Operators and Global Transforms between the elements of the transducer array.


Images are n dimensional signals l.jpg

Some things work in between the elements of the transducer array.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.


Global transforms in n dimensions l.jpg

Geometric (rigid body) between the elements of the transducer array.

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


Orthographic transform matrix l.jpg

Capable of geometric, similarity, or affine. between the elements of the transducer array.

Homogeneous coordinates.

Multiply in reverse order to combine

SGI “graphics engine” 1982, now standard.

Orthographic Transform Matrix


Translation by t x t y l.jpg
Translation by ( between the elements of the transducer array.tx , ty)

Scale x by sx and y by sy


Rotation in 2d l.jpg
Rotation in 2D between the elements of the transducer array.

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

  • Therefore only 1 degree of freedom, .


Rotation in 3d l.jpg
Rotation in 3D between the elements of the transducer array.

  • 3 x 3 rotation portion is orthogonal (orthonormal vectors).

  • 3 degree of freedom (dotted circled), , as expected.


Non orthographic projection in 3d l.jpg
Non-Orthographic Projection in 3D between the elements of the transducer array.

  • 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.


Point operators l.jpg

f between the elements of the transducer array. 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


Histogram equalization l.jpg

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

Histogram Equalization


Adaptive thresholding from histogram l.jpg

Assumes bimodal distribution. uniform density of pixel intensity across the dynamic range.

Trough represents boundary points between homogenous areas.

Adaptive Thresholding from Histogram


Algebraic operators l.jpg

Assumes registration. uniform density of pixel intensity across the dynamic range.

Averaging multiple acquisitions for noise reduction.

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

Masking.

Algebraic Operators


Re sampling on a new lattice l.jpg

Can result in denser or sparser pixels. uniform density of pixel intensity across the dynamic range.

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


Convolution and correlation l.jpg

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


Neighborhood pde operators l.jpg

Discrete images always requires a specific scale. image analysis.

“Inner scale” is the original pixel grid.

Size of the kernel determines scale.

Concept of Scale Space, Course-to-Fine.

Neighborhood PDE Operators


Intensity gradient l.jpg

Vector image analysis.

Direction of maximum change of scalar intensity I.

Normal to the boundary.

Nicely n-dimensional.

Intensity Gradient


Intensity gradient magnitude l.jpg

Scalar image analysis.

Maximum at the boundary

Orientation-invariant.

Intensity Gradient Magnitude



Isosurface marching cubes lorensen l.jpg

100% opaque watertight surface image analysis.

Fast, 28 = 256 combinations, pre-computed

Isosurface, Marching Cubes (Lorensen)


Slide35 l.jpg


Jacobian of the intensity gradient l.jpg

I image analysis.xy = Iyx= curvature

Orientation-invariant.

What about in 3D?

Jacobian of the Intensity Gradient


Laplacian of the intensity l.jpg

Divergence of the Gradient. image analysis.

Zero at the inflection point of the intensity curve.

Laplacian of the Intensity

I

Ix

Ixx



Binomial difference of offset gaussian doog l.jpg

Not the conventional concentric DOG Theorem.

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

Binomial Difference of Offset Gaussian (DooG)


Texture boundaries l.jpg

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

Texture Boundaries


2d fourier transform l.jpg
2D Fourier Transform texture are easily discriminated by the human visual system.

analysis

or

synthesis


Properties l.jpg

Most of the usual properties, such as linearity, etc. texture are easily discriminated by the human visual system.

Shift-invariant, rather than Time-invariant

Parsevals relation becoms Rayleigh’s Theorem

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

Properties


Separability l.jpg
Separability texture are easily discriminated by the human visual system.


Rotation invariance l.jpg
Rotation Invariance texture are easily discriminated by the human visual system.


Projection l.jpg
Projection texture are easily discriminated by the human visual system.

Combine with rotation, have arbitrary projection.


Gaussian l.jpg
Gaussian texture are easily discriminated by the human visual system.

seperable

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


Hankel transform l.jpg
Hankel Transform texture are easily discriminated by the human visual system.

For radially symmetrical functions


Elliptical fourier series for 2d shape l.jpg
Elliptical Fourier Series for 2D Shape texture are easily discriminated by the human visual system.

Parametric function, usually with constant velocity.

Truncate harmonics to smooth.


Fourier shape in 3d l.jpg

Fourier surface of 3D shapes (parameterized on surface). texture are easily discriminated by the human visual system.

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 l.jpg
Quaternions – 3D phasors texture are easily discriminated by the human visual system.

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


Summary l.jpg

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