![Page 1: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/1.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization and Image Processing
Erin E. Tripp
Syracuse University
November 2018
![Page 2: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/2.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Table of Contents
1 Overview
2 Image Denoising
3 Foreground Detection
![Page 3: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/3.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Image Denoising
We receive a blurry, noisy image:
and we want to recover the original.
![Page 4: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/4.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Image Denoising
For example, even a visually imperceptible amount of noise cancause a neural net to misclassify the image with highprobability. 1
1From Explaining and Harnessing Adversarial Examples by Goodfellow, Shlens, and Szegedy.
![Page 5: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/5.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Foreground Detection
We want to separate moving objects from the nonmovingbackground in video data:
![Page 6: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/6.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization
How are these optimization problems?
First, we develop a mathematical model for our problem.
This consists of:
a function C (x) that measures the quantity of interest(variations in pixel values, similarity to input, etc)a constraint set S which restricts the acceptable answers(matrices satisfying certain inequalities, etc.)
So the problem becomes
arg minx∈S
C (x)
We can then use a variety of algorithms for findingminimizers of functions.
![Page 7: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/7.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization
How are these optimization problems?
First, we develop a mathematical model for our problem.
This consists of:
a function C (x) that measures the quantity of interest(variations in pixel values, similarity to input, etc)a constraint set S which restricts the acceptable answers(matrices satisfying certain inequalities, etc.)
So the problem becomes
arg minx∈S
C (x)
We can then use a variety of algorithms for findingminimizers of functions.
![Page 8: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/8.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization
How are these optimization problems?
First, we develop a mathematical model for our problem.
This consists of:
a function C (x) that measures the quantity of interest(variations in pixel values, similarity to input, etc)
a constraint set S which restricts the acceptable answers(matrices satisfying certain inequalities, etc.)
So the problem becomes
arg minx∈S
C (x)
We can then use a variety of algorithms for findingminimizers of functions.
![Page 9: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/9.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization
How are these optimization problems?
First, we develop a mathematical model for our problem.
This consists of:
a function C (x) that measures the quantity of interest(variations in pixel values, similarity to input, etc)a constraint set S which restricts the acceptable answers(matrices satisfying certain inequalities, etc.)
So the problem becomes
arg minx∈S
C (x)
We can then use a variety of algorithms for findingminimizers of functions.
![Page 10: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/10.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization
How are these optimization problems?
First, we develop a mathematical model for our problem.
This consists of:
a function C (x) that measures the quantity of interest(variations in pixel values, similarity to input, etc)a constraint set S which restricts the acceptable answers(matrices satisfying certain inequalities, etc.)
So the problem becomes
arg minx∈S
C (x)
We can then use a variety of algorithms for findingminimizers of functions.
![Page 11: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/11.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Optimization
How are these optimization problems?
First, we develop a mathematical model for our problem.
This consists of:
a function C (x) that measures the quantity of interest(variations in pixel values, similarity to input, etc)a constraint set S which restricts the acceptable answers(matrices satisfying certain inequalities, etc.)
So the problem becomes
arg minx∈S
C (x)
We can then use a variety of algorithms for findingminimizers of functions.
![Page 12: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/12.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing image data
How do we turn an image into a mathematical object?
The image becomes a matrix, and the entries of the matrixare the pixel values, ranging from 0 (black) to 255 (white).For example, 0 10 83
25 103 15011 60 98
represents a 3× 3 pixel grayscale image.
The cameraman example is 256× 256 (65536) pixels.
![Page 13: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/13.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing image data
How do we turn an image into a mathematical object?
The image becomes a matrix, and the entries of the matrixare the pixel values, ranging from 0 (black) to 255 (white).For example, 0 10 83
25 103 15011 60 98
represents a 3× 3 pixel grayscale image.
The cameraman example is 256× 256 (65536) pixels.
![Page 14: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/14.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing image data
How do we turn an image into a mathematical object?
The image becomes a matrix, and the entries of the matrixare the pixel values, ranging from 0 (black) to 255 (white).For example, 0 10 83
25 103 15011 60 98
represents a 3× 3 pixel grayscale image.
The cameraman example is 256× 256 (65536) pixels.
![Page 15: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/15.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing video data
For video footage, we want to deal with several frames(individual images) at once.
Each frame is entered as a matrix then transformed into acolumn:
0 10 8325 103 15011 60 98
−→
0251110
1036083
15088
![Page 16: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/16.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing video data
For video footage, we want to deal with several frames(individual images) at once.
Each frame is entered as a matrix then transformed into acolumn:
0 10 8325 103 15011 60 98
−→
0251110
1036083
15088
![Page 17: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/17.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing video data
0 10 8325 103 15011 60 98
20 100 01 0 270 12 201
17 0 55234 60 0
0 106 111
frame 1 frame 2 frame 3
Multiple frames are then concatenated:
0 20 1725 1 23411 0 010 100 0
103 0 6060 12 10683 0 55
150 27 098 201 111
![Page 18: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/18.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing video data
In our video example, each frame is 120× 160 (19200)pixels. This becomes a 19200× 1 column.
If we use 200 frames, our video matrix will be a19200× 200 matrix!
Note: More frames = more accurate foregroundseparation, but also more computations.
![Page 19: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/19.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing video data
In our video example, each frame is 120× 160 (19200)pixels. This becomes a 19200× 1 column.
If we use 200 frames, our video matrix will be a19200× 200 matrix!
Note: More frames = more accurate foregroundseparation, but also more computations.
![Page 20: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/20.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Processing video data
In our video example, each frame is 120× 160 (19200)pixels. This becomes a 19200× 1 column.
If we use 200 frames, our video matrix will be a19200× 200 matrix!
Note: More frames = more accurate foregroundseparation, but also more computations.
![Page 21: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/21.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Table of Contents
1 Overview
2 Image Denoising
3 Foreground Detection
![Page 22: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/22.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How do we represent image denoising as an optimizationproblem?
arg min f (Y ) +1
2λ‖X − Z‖2F
subject to Y = DX
Z is the noisy input imageD computes the difference between adjacent pixelsf penalizes small differences in the data
![Page 23: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/23.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How do we represent image denoising as an optimizationproblem?
arg min f (Y ) +1
2λ‖X − Z‖2F
subject to Y = DX
Z is the noisy input imageD computes the difference between adjacent pixelsf penalizes small differences in the data
![Page 24: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/24.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How do we represent image denoising as an optimizationproblem?
arg min f (Y ) +1
2λ‖X − Z‖2F
subject to Y = DX
Z is the noisy input image
D computes the difference between adjacent pixelsf penalizes small differences in the data
![Page 25: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/25.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How do we represent image denoising as an optimizationproblem?
arg min f (Y ) +1
2λ‖X − Z‖2F
subject to Y = DX
Z is the noisy input imageD computes the difference between adjacent pixels
f penalizes small differences in the data
![Page 26: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/26.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How do we represent image denoising as an optimizationproblem?
arg min f (Y ) +1
2λ‖X − Z‖2F
subject to Y = DX
Z is the noisy input imageD computes the difference between adjacent pixelsf penalizes small differences in the data
![Page 27: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/27.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Algorithm
We use the Alternating Direction Method of Multipliers.
![Page 28: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/28.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Algorithm
We use the Alternating Direction Method of Multipliers.
First write the augmented Lagrangian Lη(X ,Y ,P) for thisproblem:
f (Y ) +1
2λ‖X − Z‖2F − 〈P,DX − Y 〉+
η
2‖DX − Y ‖2F
![Page 29: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/29.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Algorithm
We use the Alternating Direction Method of Multipliers.
First write the augmented Lagrangian Lη(X ,Y ,P) for thisproblem:
f (Y ) +1
2λ‖X − Z‖2F − 〈P,DX − Y 〉+
η
2‖DX − Y ‖2F
ADMM:
Input X0,Y0
For k = 1, . . . ,N
Yk+1 = arg minY
Lη(Xk ,Y ,Pk)
Xk+1 = arg minX
Lη(X ,Yk+1,Pk)
Pk+1 = Pk − η(DXk+1 − Yk+1)
![Page 30: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/30.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Algorithm
We use the Alternating Direction Method of Multipliers.
First write the augmented Lagrangian Lη(X ,Y ,P) for thisproblem:
f (Y ) +1
2λ‖X − Z‖2F − 〈P,DX − Y 〉+
η
2‖DX − Y ‖2F
ADMM:
Input X0,Y0
For k = 1, . . . ,N
Yk+1 = arg minY
Lη(Xk ,Y ,Pk)
Xk+1 = arg minX
Lη(X ,Yk+1,Pk)
Pk+1 = Pk − η(DXk+1 − Yk+1)
![Page 31: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/31.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Results
![Page 32: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/32.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Table of Contents
1 Overview
2 Image Denoising
3 Foreground Detection
![Page 33: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/33.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How can we model foreground detection as an optimizationproblem?
![Page 34: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/34.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How can we model foreground detection as an optimizationproblem?Let Y be the video matrix. We assume Y can be decomposedas Y = M + S , where
M is a low rank matrix (the stable background)
S is a sparse matrix (the moving foreground)
![Page 35: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/35.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How can we model foreground detection as an optimizationproblem?Let Y be the video matrix. We assume Y can be decomposedas Y = M + S , where
M is a low rank matrix (the stable background)
S is a sparse matrix (the moving foreground)
![Page 36: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/36.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
How can we model foreground detection as an optimizationproblem?Let Y be the video matrix. We assume Y can be decomposedas Y = M + S , where
M is a low rank matrix (the stable background)
S is a sparse matrix (the moving foreground)
![Page 37: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/37.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
Since we don’t know M and S we want to find approximationsM̃ and S̃ .
![Page 38: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/38.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
Since we don’t know M and S we want to find approximationsM̃ and S̃ .
Then the problem can be rephrased as, “How close is ourapproximation to the actual solution?”
![Page 39: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/39.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Modeling the Problem
Since we don’t know M and S we want to find approximationsM̃ and S̃ .
Then the problem can be rephrased as, “How close is ourapproximation to the actual solution?”
Mathematically, we want to minimize
‖M̃ + S̃ − Y ‖2F
![Page 40: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/40.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Sketch of Algorithm
Projected gradient descent:
Create an initial estimate S0 of S and set M0 = Y − S0.
Factor M0 = U0V>0
Perform gradient descent on the function
‖UV> + S − Y ‖2F
with respect to U and V respectively, projecting each stepback onto the constraint sets. Update S accordingly.
![Page 41: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/41.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Sketch of Algorithm
Projected gradient descent:
Create an initial estimate S0 of S and set M0 = Y − S0.
Factor M0 = U0V>0
Perform gradient descent on the function
‖UV> + S − Y ‖2F
with respect to U and V respectively, projecting each stepback onto the constraint sets. Update S accordingly.
![Page 42: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/42.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Sketch of Algorithm
Projected gradient descent:
Create an initial estimate S0 of S and set M0 = Y − S0.
Factor M0 = U0V>0
Perform gradient descent on the function
‖UV> + S − Y ‖2F
with respect to U and V respectively, projecting each stepback onto the constraint sets. Update S accordingly.
![Page 43: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/43.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Sketch of Algorithm
Projected gradient descent:
Create an initial estimate S0 of S and set M0 = Y − S0.
Factor M0 = U0V>0
Perform gradient descent on the function
‖UV> + S − Y ‖2F
with respect to U and V respectively, projecting each stepback onto the constraint sets. Update S accordingly.
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Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
Results
The original frame; the foreground; and the background.
![Page 45: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/45.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection
References
S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein,Distributed Optimization and Statistical Learning via theAlternating Direction Method of Multipliers, Foundations andTrends in Machine Learning, Vol. 3, No. 1 (2010) 1–122.
X. Yi, D. Park, Y. Chen, and C. Caramanis, Fast Algorithms forRobust PCA via Gradient Descent, Preprint, arXiv:1605.07784v1 [cs.IT] (2016).
![Page 46: Optimization and Image Processing · Optimization and Image Processing Erin E. Tripp Overview Image Denoising Foreground Detection Optimization How are these optimization problems?](https://reader030.vdocument.in/reader030/viewer/2022040922/5e9bf6dcae2c410d2866fbdc/html5/thumbnails/46.jpg)
Optimizationand ImageProcessing
Erin E. Tripp
Overview
ImageDenoising
ForegroundDetection Thank you!