Transcript Document

Medical Imaging
• 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
Bremsstrahlung
Electron rapidly decelerates at heavy metal target,
giving off X-Rays.
1896
X-Ray and Fluoroscopic Images
Projection of X-Ray silhouette onto a piece of film or
detector array, with intervening fluorescent screen.
Computerized Tomography
From a series of projections, a tomographic image is
reconstructed using Filtered Back Projection.
Mass Spectrometer
Radioactive isotope separated by difference in
inertia while bending in magnetic field.
Nuclear Medicine
Gamma camera for creating image of radioactive
target. Camera is rotated around patient in SPECT
(Single Photon Emission Computed Tomography).
Phased Array Ultrasound
Ultrasound beam formed and steered by controlling the
delay between the elements of the transducer array.
Real Time 3D Ultrasound
Positron Emission Tomography
Positron-emitting organic compounds create pairs of
high energy photons that are detected synchronously.
Other Imaging Modalities
• MRI (Magnetic Resonance Imaging)
• OCT (Optical Coherence Tomography)
Current Trends in Imaging
•
•
•
•
•
3D
Higher speed
Greater resolution
Measure function as well as structure
Combining modalities (including direct
vision)
The Gold Standard
• Dissection:
– Medical School, Day 1: Meet the Cadaver.
– From Vesalius to the Visible Human
Local Operators and Global Transforms
Images are n dimensional signals.
• 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.
Global Transforms in n
dimensions
• Geometric (rigid body)
n
– n translations and   rotations.
2
 
• Similarity
– Add 1 scale (isometric).
• Affine
– Add n scales (combined with rotation => skew).
– Parallel lines remain parallel.
• Projection
Orthographic Transform Matrix
•
•
•
•
Capable of geometric, similarity, or affine.
Homogeneous coordinates.
Multiply in reverse order to combine
SGI “graphics engine” 1982, now standard.
 x  a1,1 a1, 2
 y   a
a2 , 2
2
,
1
  
0
 1   0
a1,3   x 
a2 , 3   y 
 
1   1 
Translation by (tx , ty)
 x 1 0 t x   x 
 y   0 1 t   y 
y  
  
 1  0 0 1   1 
Scale x by sx and y by sy
 x   s x
 y    0
  
 1   0
0
sy
0
0  x 
0  y 
 
1  1 
Rotation in 2D
 x cos
 y    sin 
  
 1   0
 sin 
cos
0
0  x 



0 y
 
1  1 
• 2 x 2 rotation portion is orthogonal (orthonormal vectors).
• Therefore only 1 degree of freedom,  .
Rotation in 3D
 x  a1,1 a1, 2
 y  a
a2 , 2
2
,
1
 
 z   a3,1 a3, 2
  
0
1  0
a1, 2
a2 , 3
a3,3
0
0  x 
0  y 
 
0  z 
 
1  1 
• 3 x 3 rotation portion is orthogonal (orthonormal vectors).
• 3 degree of freedom (dotted circled),  n  , as expected.
2
 
Non-Orthographic Projection in 3D
 x 1
 y 0
 
 z  0
  
 1  0
0
1
0
0
0
0
1
k
0  x 



0 y
 
0  z 
 
1  1 
• 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
I ' x, y   f I x, y 
• 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.
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
• Assumes bimodal distribution.
• Trough represents boundary points between
homogenous areas.
Algebraic Operators
I ' x, y   gI1 x, y , I 2 x, y 
• 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
• 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.
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
• 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
dI
dI
dI
I  xˆ  yˆ  zˆ  I x xˆ  I y yˆ  I z zˆ
dx
dy
dz
• Vector
• Direction of maximum change of scalar
intensity I.
• Normal to the boundary.
• Nicely n-dimensional.
Intensity Gradient Magnitude

I  I x  I y  I z
2
2
• Scalar
• Maximum at the boundary
• Orientation-invariant.

1
2 2
 I  nˆ
I
Ix
Iy
I
Classic Edge Detection Kernel
(Sobel)
  1 0 1
  2 0 2  I
x


  1 0 1
2
1
1
0

0
0  Iy


  1  2  1
Isosurface, Marching Cubes
(Lorensen)
• 100% opaque watertight surface
• Fast, 28 = 256 combinations, pre-computed
• Marching cubes works well
with raw CT data.
• Hounsfield units (attenuation).
• Threshold calcium density.
Jacobian of the Intensity Gradient
 d I
 dx2
 2
 d I
 dy dx
2
d I 

 I xx
dx dy

2
d I   I yx
2 
dy 
2
• Ixy = Iyx = curvature
• Orientation-invariant.
• What about in 3D?
I xy 

I yy 
Laplacian of the Intensity
2
2
2
d I  d I  d I 
2
2
2
 I   2    2    2   I xx  I yy  I zz
 dx   dy   dz 
2
2
2
2
I
Ix
Ixx
• Divergence of the Gradient.
• Zero at the inflection point of the
intensity curve.
  1  1  1
  1 8  1


  1  1  1
Binomial Kernel
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 1
2 4 2
3 9 9 3
4 16 24 16 4
1 2 1
3 9 9 3
6 24 36 24 6
1 3 3 1
4 16 24 16 4
1 4 6 4 1
• Repeated averaging of neighbors =>
Gaussian by Central Limit Theorem.
Binomial Difference of Offset
Gaussian (DooG)
-1
0
1
-1 -2
0
2
1
-1 -4 -6 -4
0
4
6
4
1
-2 -4
0
4
2
-4 -16 -24 -16 0 16 24 16 4
-1 -2
0
2
1
-6 -24 -36 -24 0 24 36 24 6
-4 -16 -24 -16 0 16 24 16 4
-1 -4 -6 -4
0
4
6
4
• Not the conventional concentric DOG
• Subtracting pixels displaced along the x axis after
repeated blurring with binomial kernel yields Ix
1
Texture Boundaries
• Two regions with the same intensity but
differentiated by texture are easily discriminated
by the human visual system.
2D Fourier Transform
F  u, v  
 
analysis
  f  x, x  e
 
 j 2  ux  vy 
dx dy
or
 
 j 2 ux
F  u, v      f  x , y  e
dx
  

  j 2 vy
dy
e

synthesis
f  x, y  
 
  F  u, v  e
 
j 2  ux  vy 
du dv
Properties
• 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)
Separability
if
f  x , y   f1  x  f 2  y 
f  x, y   F u, v 
then
F
f1  x  f 2  y   F1 u F2 v   F u, v 
F
f1  x   F1 u 
F
f 2  y   F2 v 
F
Rotation Invariance
 x  cos
 y   sin 
  
sin    x 



cos   y 
F
f x cos  y sin  ,  x sin   y cos 
F u cos  v sin  ,  u sin   v cos 
Projection
px    f x, y  dy


Pu   F u, 0
Combine with rotation, have arbitrary projection.
Gaussian
F
g  x  
 G u 
g1  x   g2  x   g3  u 
G1  u  G2  u   G3  u 
seperable

e
 x2  y2
2 2

e
 x2
2 2
e
 y2
2 2
Since the Fourier Transform is also
separable, the spectra of the 1D
Gaussians are, themselves, separable.
Hankel Transform
For radially symmetrical functions
f  x, y   f r r , r  x  y
2
2
F u, v   Fr q , q  u  v
2
F u, v  
  
  f  x, y  e
2
2
2
 j 2 ux  vy 
dx dy 
  


0
2  j 2qr cos 
f r r   e
d  r dr  Fr q 
0

Elliptical Fourier Series for 2D
ShapeParametric function, usually
with constant velocity.
xt  ak
Fs
yt bk
Fs
center  a0, b0 
Truncate harmonics to smooth.
Fourier shape in 3D
• 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
Quaternions – 3D phasors
a  a1  ia2  ja3  ka4
i  j  k  ijk  1
2
2
2
a  a1  ia2  ja3  ka4
*

a  a  a2  a  a4
2
1
2
2
3
2

1
2
a  b  a1  b1   ia2  b2   ja3  b3   k a4  b4 
Product is defined such that rotation by arbitrary angles from
arbitrary starting points become simple multiplication.
Summary
• 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.