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## PowerPoint Slideshow about 'Medical Imaging Simultaneous measurements on a spatial grid.' - adamdaniel

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

Simultaneous measurements on a spatial grid.

Many modalities: mainly EM radiation and sound.

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

Bremsstrahlung1896 off X-Rays.

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

X-Ray and Fluoroscopic ImagesFrom a series of projections, a tomographic image is reconstructed using Filtered Back Projection.

Computerized TomographyRadioactive isotope separated by difference in inertia while bending in magnetic field.

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

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

Phased Array UltrasoundReal Time 3D Ultrasound between the elements of the transducer array.

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

high energy photons that are detected synchronously.

Positron Emission TomographyMRI (Magnetic Resonance Imaging) between the elements of the transducer array.

OCT (Optical Coherence Tomography)

Other Imaging Modalities3D 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 ImagingDissection: between the elements of the transducer array.

Medical School, Day 1: Meet the Cadaver.

From Vesalius to the Visible Human

The Gold StandardSome 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.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 dimensionsCapable 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 MatrixTranslation by ( between the elements of the transducer array.tx , ty)

Scale x by sx and y by sy

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 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 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.

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

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

Trough represents boundary points between homogenous areas.

Adaptive Thresholding from HistogramAssumes 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 OperatorsCan 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 LatticeTemplate 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 CorrelationDiscrete 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 OperatorsVector image analysis.

Direction of maximum change of scalar intensity I.

Normal to the boundary.

Nicely n-dimensional.

Intensity GradientClassic Edge Detection Kernel (Sobel) image analysis.

100% opaque watertight surface image analysis.

Fast, 28 = 256 combinations, pre-computed

Isosurface, Marching Cubes (Lorensen)- Marching cubes works well with raw CT data. image analysis.
- Hounsfield units (attenuation).
- Threshold calcium density.

I image analysis.xy = Iyx= curvature

Orientation-invariant.

What about in 3D?

Jacobian of the Intensity GradientDivergence of the Gradient. image analysis.

Zero at the inflection point of the intensity curve.

Laplacian of the IntensityI

Ix

Ixx

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

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

Texture Boundaries2D Fourier Transform texture are easily discriminated by the human visual system.

analysis

or

synthesis

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)

PropertiesSeparability texture are easily discriminated by the human visual system.

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

Projection texture are easily discriminated by the human visual system.

Combine with rotation, have arbitrary projection.

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 texture are easily discriminated by the human visual system.

For radially symmetrical functions

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 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 3DQuaternions – 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.

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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