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Medical Imaging

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Simultaneous measurements on a spatial grid.

Many modalities: mainly EM radiation and sound.

“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.

1896

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

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

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

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

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

Positron-emitting organic compounds create pairs of

high energy photons that are detected synchronously.

MRI (Magnetic Resonance Imaging)

OCT (Optical Coherence Tomography)

3D

Higher speed

Greater resolution

Measure function as well as structure

Combining modalities (including direct vision)

Dissection:

Medical School, Day 1: Meet the Cadaver.

From Vesalius to the Visible Human

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.

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

Capable of geometric, similarity, or affine.

Homogeneous coordinates.

Multiply in reverse order to combine

SGI “graphics engine” 1982, now standard.

Scale x by sx and y by sy

- 2 x 2 rotation portion is orthogonal (orthonormal vectors).
- Therefore only 1 degree of freedom, .

- 3 x 3 rotation portion is orthogonal (orthonormal vectors).
- 3 degree of freedom (dotted circled), , as expected.

- 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.

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

Assumes bimodal distribution.

Trough represents boundary points between homogenous areas.

Assumes registration.

Averaging multiple acquisitions for noise reduction.

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

Masking.

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.

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.

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.

Vector

Direction of maximum change of scalar intensity I.

Normal to the boundary.

Nicely n-dimensional.

Scalar

Maximum at the boundary

Orientation-invariant.

100% opaque watertight surface

Fast, 28 = 256 combinations, pre-computed

- Marching cubes works well with raw CT data.
- Hounsfield units (attenuation).
- Threshold calcium density.

Ixy = Iyx= curvature

Orientation-invariant.

What about in 3D?

Divergence of the Gradient.

Zero at the inflection point of the intensity curve.

I

Ix

Ixx

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

Not the conventional concentric DOG

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

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

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)

Combine with rotation, have arbitrary projection.

seperable

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

For radially symmetrical functions

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

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.