signal reconstruction from multiscale edges
DESCRIPTION
Signal reconstruction from multiscale edges. A wavelet based algorithm. Author Yen-Ming Mark Lai ( [email protected] ) Advisor Dr. Radu Balan [email protected] CSCAMM, MATH. Motivation. Save edges. Motivation. sharp two-sided edge. sharp one-sided edge. “noisy” edges. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/1.jpg)
Signal reconstruction from multiscale edges
A wavelet based algorithm
![Page 2: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/2.jpg)
Author
Yen-Ming Mark Lai
Advisor
Dr. Radu Balan
CSCAMM, MATH
![Page 3: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/3.jpg)
Motivation
Save edges
![Page 4: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/4.jpg)
Motivation
Save edge type
sharp one-sided edge
sharp two-sided edge
“noisy” edges
![Page 5: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/5.jpg)
Motivation
edges edge type reconstruct+ =
![Page 6: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/6.jpg)
Algorithm
Decomposition + Reconstruction
![Page 7: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/7.jpg)
Decomposition
Discrete Wavelet
Transform
Save edges e.g. local extrema
Input
“edges+edge type”
![Page 8: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/8.jpg)
Decomposition
input
input
input
edge detection (scale 1)
edge detection (scale 2)
edge detection (scale 4)
=
=
=
![Page 9: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/9.jpg)
Reconstruction
Find approximation
Inverse Wavelet
Transform
Output
local extrema “edges+edge type”
![Page 10: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/10.jpg)
How to find approximation?
Find approximation
local extrema“edges+edge type”
![Page 11: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/11.jpg)
Find approximation (iterative)
Alternate projections between two spaces
![Page 12: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/12.jpg)
Find approximation (iterative)
sequences of functions
n: nf
whose H1 norm
2'2
22LnLn ff
is finite
![Page 13: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/13.jpg)
Find approximation (iterative)
0k
![Page 14: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/14.jpg)
Find approximation (iterative)
sequences of functions:
1) interpolate input signal’s wavelet extrema
2) have minimal H1 norm
![Page 15: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/15.jpg)
Q: Why minimize over H1 norm?
A: Interpolation points act like local extrema
![Page 16: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/16.jpg)
Numerical Example
algorithm interpolates between points
unclear what to do outside interpolation points
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Find approximation (iterative)
0
![Page 18: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/18.jpg)
Find approximation (iterative)
dyadic wavelet transforms of L^2 functions
![Page 19: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/19.jpg)
Find approximation (iterative)
intersection = space of solutions
![Page 20: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/20.jpg)
Find approximation (iterative)
Start at zero element to minimize solution’s norm
![Page 21: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/21.jpg)
Preliminary Results
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Step Edge (length 8)
![Page 23: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/23.jpg)
Quadratic Spline Wavelet
![Page 24: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/24.jpg)
Take DWT
![Page 25: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/25.jpg)
Take DWT
![Page 26: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/26.jpg)
[1,-1]( ,
Convolution in Matlab
*[0,0,0,0,1,1,1,1]conv )
next
1 current 1+ =next-current
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*
Convolution in Matlab
next-current =0 next-current =-1
![Page 28: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/28.jpg)
*
Convolution in Matlab
=
next-current = 0
next-current = 0
next-current = 0
next-current = 0
next-current = 1
next-current = 0
next-current = 0
next-current = 0
next-current = -1
![Page 29: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/29.jpg)
Save Local Extrema
![Page 30: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/30.jpg)
Save Local Extrema
![Page 31: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/31.jpg)
Interpolate DWT (Level 1)
interpolation to minimize H1 norm
unclear what to do outside interpolation points
![Page 32: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/32.jpg)
error
Original DWT – Level 1
Interpolated DWT – Level 1
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error
Original DWT – Level 2
Interpolated DWT – Level 2
![Page 34: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/34.jpg)
error
Original DWT – Level 3
Interpolated DWT – Level 3
![Page 35: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/35.jpg)
matrix inversion failed
Original DWT – Level 4
Interpolated DWT – Level 4
![Page 36: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/36.jpg)
Interpolated DWT
![Page 37: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/37.jpg)
Take IDWT to Recover Signal
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Recovered Signal (Red) and Original Step Edge (Blue)
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Summary
![Page 40: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/40.jpg)
Choose Input
![Page 41: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/41.jpg)
Take DWT
![Page 42: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/42.jpg)
Save Local Extrema of DWT
![Page 43: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/43.jpg)
Interpolate Local Extrema of DWT
![Page 44: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/44.jpg)
Take IDWT
![Page 45: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/45.jpg)
Issues
• Convolution detects false edges
• What to do with values outside
interpolations points?
• What to do when matrix inversion fails?
![Page 46: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/46.jpg)
Timeline
Dec – write up mid-year report
Jan– code local extrema search
Oct/Nov – code Alternate Projections(90%)
(85%)
(100%)
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Timeline
• February/March – test and debug entire
system (8 weeks)
• April – run code against database (4
weeks)
• May – write up final report (2 weeks)
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Questions?
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Supplemental Slides
![Page 50: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/50.jpg)
Input Signal (256 points)
Which points to save?
![Page 51: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/51.jpg)
Compressed Signal (37 points)
What else for reconstruction?
![Page 52: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/52.jpg)
Compressed Signal (37 points)
sharp one-sided edge
![Page 53: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/53.jpg)
Compressed Signal (37 points)
sharp two-sided edge
![Page 54: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/54.jpg)
Compressed Signal (37 points)
“noisy” edges
![Page 55: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/55.jpg)
Calculation
Reconstruction:
• edges
• edge type information
Original: (256 points)
(37 points)
(x points)
![Page 56: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/56.jpg)
37
Compression
edges edge type
+ x < 256
![Page 57: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/57.jpg)
Summary
Save edges
![Page 58: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/58.jpg)
Summary
Save edge type
sharp one-sided edge
sharp two-sided edge
“noisy” edges
![Page 59: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/59.jpg)
Summary
edges edge type reconstruct+ =
![Page 60: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/60.jpg)
Algorithm
Decomposition + Reconstruction
![Page 61: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/61.jpg)
Decomposition
Discrete Wavelet
Transform
Save edges e.g. local extrema
Input
“edges+edge type”
![Page 62: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/62.jpg)
Reconstruction
Find approximation
Inverse Wavelet
Transform
Output
local extrema “edges+edge type”
![Page 63: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/63.jpg)
What is Discrete Wavelet Transform?
Discrete Wavelet
Transform
Input
![Page 64: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/64.jpg)
What is DWT?
1) Choose mother wavelet
2) Dilate mother wavelet
3) Convolve family with input
DWT
![Page 65: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/65.jpg)
1) Choose mother wavelet
![Page 66: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/66.jpg)
2) Dilate mother waveletmother wavelet
dilate
![Page 67: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/67.jpg)
2) Dilate mother wavelet
![Page 68: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/68.jpg)
Convolve family with input
input
input
input
wavelet scale 1
wavelet scale 2
wavelet scale 4
=
=
=
![Page 69: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/69.jpg)
Convolve “family”
input
input
input
wavelet scale 1
wavelet scale 2
wavelet scale 4
=
=
=
DWT
multiscale
![Page 70: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/70.jpg)
What is DWT?
(mathematically)
![Page 71: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/71.jpg)
How to dilate?
)2(
2
1)(
2 jj
xxj
mother wavelet
![Page 72: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/72.jpg)
How to dilate?
)2(
2
1)(
2 jj
xxj
dyadic (powers of two)
![Page 73: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/73.jpg)
How to dilate?
)2(
2
1)(
2 jj
xxj
scale
![Page 74: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/74.jpg)
How to dilate?
)2(
2
1)(
2 jj
xxj z
halve amplitude
double support
![Page 75: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/75.jpg)
Mother Wavelet (Haar)scale 1, j=0
![Page 76: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/76.jpg)
Mother Wavelet (Haar)
scale 2, j=1
![Page 77: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/77.jpg)
Mother Wavelet (Haar)
scale 4, j=2
![Page 78: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/78.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
![Page 79: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/79.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
convolution
![Page 80: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/80.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
dilates
![Page 81: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/81.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
original signal
![Page 82: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/82.jpg)
What is convolution?(best match operation)
Discrete Wavelet
Transform
Input1)mother wavelet
2)dilation
3)convolution
![Page 83: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/83.jpg)
Convolution (best match operator)
dtgftgf )()()(
dummy variable
![Page 84: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/84.jpg)
Convolution (best match operator)
flip g around y axis
dtgftgf )()()(
![Page 85: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/85.jpg)
dtgftgf )()()(
Convolution (best match operator)
shifts g by t
![Page 86: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/86.jpg)
dtgftgf )()()(
do nothing to f
Convolution (best match operator)
![Page 87: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/87.jpg)
dtgftgf )()()(
Convolution (best match operator)
pointwise multiplication
![Page 88: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/88.jpg)
dtgftgf )()()(
Convolution (best match operator)
integrate over R
![Page 89: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/89.jpg)
)7.7(gf
flip g and shift by 7.7
dgf )7.7()(
Convolution (one point)
![Page 90: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/90.jpg)
)7.7(gf
do nothing to f
dgf )7.7()(
Convolution (one point)
![Page 91: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/91.jpg)
)7.7(gf
multiply f and g pointwise
dgf )7.7()(
Convolution (one point)
![Page 92: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/92.jpg)
)7.7(gf
integrate over R
dgf )7.7()(
Convolution (one point)
![Page 93: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/93.jpg)
)7.7(gf
Convolution (one point)
scalar
![Page 94: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/94.jpg)
Convolution of two boxes
f
![Page 95: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/95.jpg)
Convolution of two boxes
45.1g
![Page 96: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/96.jpg)
Convolution of two boxes
45.1t
![Page 97: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/97.jpg)
Convolution of two boxes
dgfgf )45.1()()45.1( 0
![Page 98: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/98.jpg)
Convolution of two boxes
f
![Page 99: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/99.jpg)
Convolution of two boxes
5.0g
![Page 100: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/100.jpg)
Convolution of two boxes
5.0t
![Page 101: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/101.jpg)
Convolution of two boxes
dgfgf )5.0()()5.0( 5.0
![Page 102: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/102.jpg)
Convolution of two boxes
![Page 103: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/103.jpg)
Convolution of two boxes
gf
![Page 104: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/104.jpg)
Why convolution?
Location of maximum best fit
![Page 105: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/105.jpg)
Where does red box most look like blue box?
![Page 106: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/106.jpg)
Why convolution?
Location of maximum best fit
maximum
![Page 107: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/107.jpg)
Why convolution?
Location of maximum best fit
maxima best fit location
![Page 108: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/108.jpg)
Where does exponential most look like box?
![Page 109: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/109.jpg)
Where does exponential most look like box?
![Page 110: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/110.jpg)
Where does exponential most look like box?
maximum
![Page 111: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/111.jpg)
Where does exponential most look like box?
maximum best fit location
![Page 112: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/112.jpg)
So what?
If wavelet is an edge, convolution detects location of edges
![Page 113: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/113.jpg)
Mother Wavelet (Haar)
![Page 114: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/114.jpg)
Mother Wavelet (Haar)
![Page 115: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/115.jpg)
Mother Wavelet (Haar)
![Page 116: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/116.jpg)
What is edge?
Local extrema of wavelet transform
![Page 117: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/117.jpg)
Summary of Decomposition
Discrete Wavelet
Transform
Save “edges” e.g. local extrema
Input
“edges+edge type”
![Page 118: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/118.jpg)
Summary of Decomposition
input
input
input
edge detection (scale 1)
edge detection (scale 2)
edge detection (scale 4)
=
=
=
![Page 119: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/119.jpg)
How to find approximation?
Find approximation
local extrema“edges+edge type”
![Page 120: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/120.jpg)
Find approximation (iterative)
Alternate projections between two spaces
![Page 121: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/121.jpg)
Find approximation (iterative)
2'2
22:LnLnn ffnf
![Page 122: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/122.jpg)
Find approximation (iterative)
2'2
22:LnLnn ffnf
H_1 Sobolev Norm
![Page 123: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/123.jpg)
Find approximation (iterative)
functions that interpolate given local maxima points
![Page 124: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/124.jpg)
Find approximation (iterative)
dyadic wavelet transforms of L^2 functions
![Page 125: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/125.jpg)
Find approximation (iterative)
intersection = space of solutions
![Page 126: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/126.jpg)
Find approximation (iterative)
Start at zero element to minimize solution’s norm
![Page 127: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/127.jpg)
Q: Why minimize over K?
A: Interpolation points act like local extrema
2'2
22:LnLnn ffnf
![Page 128: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/128.jpg)
Reconstruction
Find approximation(minimization
problem)
Inverse Wavelet
Transform
Output
![Page 129: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/129.jpg)
Example
Input of 256 points
![Page 130: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/130.jpg)
Input Signal (256 points)
![Page 131: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/131.jpg)
Input Signal (256 points)
major edges
![Page 132: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/132.jpg)
Input Signal (256 points)
minor edges (many)
![Page 133: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/133.jpg)
Discrete Wavelet Transform
fW dj2
Dyadic (powers of 2)
= DWT of “f” at scale 2^j
![Page 134: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/134.jpg)
DWT (9 scales, 256 points each)
![Page 135: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/135.jpg)
DWT (9 scales, 256 points each)
major edges
![Page 136: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/136.jpg)
Input Signal (256 points)
major edges
![Page 137: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/137.jpg)
DWT (9 scales, 256 points each)
minor edges (many)
![Page 138: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/138.jpg)
Input Signal (256 points)
minor edges (many)
![Page 139: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/139.jpg)
Decomposition
Discrete Wavelet
Transform
Save “edges” e.g. local extrema
Input
![Page 140: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/140.jpg)
DWT (9 scales, 256 points each)
![Page 141: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/141.jpg)
Save Local Maxima
![Page 142: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/142.jpg)
Local Maxima of Transform
![Page 143: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/143.jpg)
Local Maxima of Transform
low scale most sensitive
![Page 144: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/144.jpg)
Mother Wavelet (Haar)
![Page 145: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/145.jpg)
Local Maxima of Transform
high scale least sensitive
![Page 146: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/146.jpg)
Mother Wavelet (Haar)
![Page 147: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/147.jpg)
Decomposition
Discrete Wavelet
Transform
Save “edges” e.g. local extrema
Input
![Page 148: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/148.jpg)
Local Maxima of Transform
![Page 149: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/149.jpg)
Find approximation (iterative)
Alternate projections between two spaces
![Page 150: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/150.jpg)
Reconstruction
Find approximation(minimization
problem)
Inverse Wavelet
Transform
Output
![Page 151: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/151.jpg)
Mallat’s Reconstruction (20 iterations)
![Page 152: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/152.jpg)
original
reconstruction (20 iterations)
![Page 153: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/153.jpg)
Implementation
Language: MATLAB – Matlab wavelet toolbox
Complexity: convergence criteria
![Page 154: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/154.jpg)
Databases
• Baseline signals
– sinusoids, Gaussians, step edges, Diracs
• Audio signals
![Page 155: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/155.jpg)
Validation
• Unit testing of components
– DWT/IDWT
– Local extrema search
– Projection onto interpolation space (\Gamma)
![Page 156: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/156.jpg)
Testing
• L2 norm of the error (sum of squares)
versus iterations
• Saturation point in iteration (knee)
![Page 157: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/157.jpg)
Schedule (Coding)
• October/November – code Alternate
Projections (8 weeks)
• December – write up mid-year report (2
weeks)
• January – code local extrema search (1
week)
![Page 158: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/158.jpg)
Schedule (Testing)
• February/March – test and debug entire
system (8 weeks)
• April – run code against database (4
weeks)
• May – write up final report (2 weeks)
![Page 159: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/159.jpg)
Milestones
• December 1, 2010 – Alternate Projections
code passes unit test
• February 1, 2011 – local extrema search
code passes unit test
• April 1, 2011 - codes passes system test
![Page 160: Signal reconstruction from multiscale edges](https://reader036.vdocuments.us/reader036/viewer/2022062409/56814d19550346895dba52fb/html5/thumbnails/160.jpg)
Deliverables
• Documented MATLAB code
• Testing results (reproducible)
• Mid-year report/Final report