HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing
Transcript of HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing
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MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image ProcessingSpring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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HST 582 / 6.555
Image Processing II
Harvard-MIT Division of Health Sciences and TechnologyHST.582J: Biomedical Signal and Image Processing, Spring 2007Course Director: Dr. Julie Greenberg
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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2D and 3D Image Processing
• Useful linear signal processing for medical images– Interpolation– Down sampling– Hierarchical filtering– Computed Tomography (CT),
• Projection Slice Theorem
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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2D and 3D Image Processing
• Useful non-linear processing techniques– Histogram equalization– Homomorphic filtering– Edge detection
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Applications of Signal Processing in Medical Images• Linear signal processing
– Image reconstruction (tomography, MRI)– Image enhancement– Noise reduction– Artifact reduction
• Non-linear signal processing– Non-linear, adaptive filters
• Tube enhancing filters– Segmentation and beyond
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Interpolating Images
• Why– Prepare synthetic multichannel data– Prepare data sets for parallel trips down the
same processing pipeline– Comparing images during Registration
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Interpolation
• Optimal– Reconstruct the bandlimited original signal
using sinc functions– Re-sample the reconstruction
• Nearest Neighbor• Linear
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Nearest Neighbor
• Easy interpolator
– To interpolate here, assign the intensity of the closest original pixel (in this case, I11)
– leads to blocky results
…
…×
I11
I32I22I12
I21 I31
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Linear Interpolator (1D)
0 ΔT 1
f1
f2
f(ΔT) = f1 + ΔT (f2 - f1)
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Bilinear Interpolator (2D)
• 2D linear interpolator
use 1D linear interpolation for A and B, use 1D linear interpolation among A and B to get C
• Similar for 3D
B×
f22f12
f11 f21A
C
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Bilinear Interpolator...
B×
f22f12
f11 f21A
C
f(ΔT1, ΔT2) = f11 + ΔT1 (f21 - f11) + ΔT2 (f12 - f11)
+ ΔT1 ΔT2 (f22 - f21 - f12 + f11)
ΔT1
ΔT2
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Down Sampling
• Whenever a lowpass filter is applied, it may be possible to discard alternating pixels without much loss of information (down-sampling, or decimation)
• If down-sampling is desired, it may be best to do some lowpass filter to avoid aliasing– Reasonable LPF to use: [ ]1,4,6,4,1
161
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Burt (Peter) Filter
image
lowpass filter
decimate by 2
½ size image
¼ size image
1/8 size image
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Projection Slice Theorem and CT
x-rays
Focus on one ray: (discrete)
sensorsource …
photons leaving = An • photons entering
NA∏=sourceat photonssensorat photons
NAlogsourceat photonssensorat photonslog Σ=
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Continuous Analog...
∫== dttxsourceatphotonssensoratphotonsy )(
____log
x is a log-attenuation-density called: Linear Attenuation Coefficient
t2
x(t1,t2)t1
∫= 2211 ),()( dtttxtxp
we can model x-ray imaging by line integrals…
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Fourier Transform...
),( 21 FFX
)0,()( 11 FXFX p =
x(t1,t2) xp(t1)
)0,( 1FX
Transform pairs Transform pairs
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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CT Reconstruction using FT Methods
• Rotate the x-ray apparatus to many angles– Take projections
• FT the projections• Assemble the transformed projections in
frequency space:– Get frequency data on these lines at the angles the
projections were taken
• Interpolate the full frequency space• Inverse FT to get back reconstructionThere is a related method: Filtered Back-projection which isusually used
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Linear and Non-Linear Processing for Medical Images
image
better image
DataReconstruct image: usually linear
Improve image: maybe linear
Segment, find surfaces: non-linear
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Histogram Equalization
• A method for automatically setting intensity lookup table
• Apply a monotonic transformation to the data so that after the transformation, the result has a (nearly) uniform intensity histogram
• Tends to increase contrast in areas where the data is concentrated
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Histogram the Image Intensities
255N
0 255
Desired HistogramData
L A H 0 B 2550
Cumulative of Data Desired Cumulative
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Histogram: Examples
Image and its intensityhistogram Histogram equalization
Figures removed due to copyright restrictions.
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Homomorphic Filtering
• An automatic method for spatially varying contrast adjustment
lowpass
highpass
explog
β>1
α<1
α
β H(ω)
√(ω12+ω2
2)
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Homomorphic Processing: Example
Original image Image after homomorphicprocessing
related: unsharp masking From Lim
Image removed due to copyright restrictions.Figure 8.11 in: Lim, J. S. Two-Dimensional Signal and Image Processing.
Upper Saddle River, NJ: Prentice Hall, 1989. ISBN: 9780139353222
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Duality
• Focus on regions Segmentation• Focus on boundaries Edges
• Sometimes best to do both.
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Edge Detection• From computer vision
– Roberts 1965 Lincoln Lab– Horn 1972 MIT AI– Marr and Hildreth 1980 MIT AI– Canny 1983 MIT AI
Picture HigherProcessor
a compact descriptionCite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Basic Idea: 1D Edge
• Motivation: ideal step edge + noise
find peak
space
space
intensitythis is the important
information
1st derivative
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Edges in 2D
• Strategy:– Find direction of
gradient
– Analyze derivative of signal in direction of gradients
profile
gradient
contour lines
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Zero Crossing Style Edges: 1D
signal
1st derivative
2nd derivative
peak
zero crossing
Mark edges where 2nd
derivative = 0(vigorously)
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Noise
• Noise is an issue with derivative operators– How do we fight the noise
• Assume white noise• Assume good stuff has important low frequency
content
• recall Wiener filtering...
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Noise
• Noise is an issue with derivative operators– How do we fight the noise
• Assume white noise• Assume good stuff has important low frequency
content
• USE LOWPASS FILTER (Gaussian ?)
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Difference of Gaussians
GaussianBlur
2nd
Derivativesignalmark zerocrossings
+≈
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Works in 2D, 3D, …
• Gaussian separates• There exists a rotationally-symmetric 2nd
derivative operator:Laplacian
similar for 3D
Discrete analog:-1-1
-1
-1
-1-1
-1 -182
2
2
22 ),(
yf
xfyxf
∂∂
+∂∂
≡∇
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2D ∇2G Operator
• The zero crossing contours of the result of an operator like this form closed contours.
“Mexican hat”
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Seminal Edge References...
• “Theory of Edge Detection”, David Marr, Ellen Hildreth, Proc. Royal Statistical Society of London, B, vol 207, pp 187 --217, 1980– good paper, often cited
• VISION, David Marr– good book!
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• Threshold Bug…
• Excess zero crossings from noise, etc.– Usual fix: threshold edges based on “strength.”
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Edge Detection
signal
1st derivative
2nd derivative
big 1st derivativemagnitude
“strong” zerocrossing
In 2D: thresholdbased on gradientstrength
22
⎥⎦
⎤⎢⎣
⎡∂∂
+⎥⎦⎤
⎢⎣⎡∂∂ f
yf
x
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Ultimate Edge Operator: Canny
• For improved noise behavior go back to 1D directional derivatives
• For less fragmentation– Use Hysteresis thresholding
• The problem:edge
fragmentationCite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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In 1D Along 2D Contour
edge
edge strength
threshold
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Schmidt Trigger uses Hysteresis
edge
edge strength
High thresholdLow Threshold
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John Canny MIT MS Thesis
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Edge Detection: Example
Original image
Source: Canny, J. F. “The complexity of robot motion planning.” MIT Ph.D. thesis, 1987.
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Edge Detection: Example
Detected edges
Source: Canny, J. F. “The complexity of robot motion planning.” MIT Ph.D. thesis, 1987.
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Edges in 3D are Surfaces
• Somewhat useful for finding organ boundaries.– Simple.– May leave the problem of figuring out which
boundary is what.
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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3D Medical Edge Finding...
• Recursive Filtering and Edge Tracking: Two Primary Tools for 3D Edge Detection– Olivier Monga, Rachid Deriche, Gergoire
Malandain, Jean Pierre Cocquerez– Image and Vision Computing Vol 9, Nr. 4,
1991
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Source: Monga, O., et al. "Recursive filtering and edge tracking: two primary tools for 3D edge detection." Image and Vision Computing 9 no. 4 (1991): 203-214. doi:10.1016/0262-8856(91)90025-K. Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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the end.Cite as: William (Sandy) Wells. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].