inverse problems and machine learning · 1 topics in multimedia signal processing inverse problems...
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Topics in Multimedia Signal Processing 1
Inverse Problems and Machine Learning
Julian Wörmann Research Group for Geometric Optimization and Machine Learning (GOL)
Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 02.05.2014
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 2
What are inverse problems?
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 3
Inverse Problems
cause/ excitation
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 4
Inverse Problems
cause/ excitation
System/ Process
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 5
Inverse Problems
cause/ excitation
effect/ measurement
System/ Process
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 6
Inverse Problems
cause/ excitation
effect/ measurement
System/ Process
02.05.2014
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 7
Inverse Problems
cause/ excitation
effect/ measurement
System/ Process
02.05.2014
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 8
Inverse Problems
cause/ excitation
effect/ measurement
System/ Process
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 9
Inverse Problems
cause/ excitation
effect/ measurement
System/ Process
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 10
Goal
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cause/ excitation
effect/ measurement
System/ Process
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Goal
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cause/ excitation
effect/ measurement
System/ Process
System/ -1
Process
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Goal
Model:
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cause/ excitation
effect/ measurement
System/ Process
System/ -1
Process
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Goal
noise
Model:
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cause/ excitation
effect/ measurement
System/ Process
System/ -1
Process
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 14
Inverse Problems in Image Processing
?
Denoising Deblurring Inpainting
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Topics in Multimedia Signal Processing
1. Determine the model parameters
Maschinelles Lernen und Inverse Probleme 15
Tasks
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Topics in Multimedia Signal Processing
1. Determine the model parameters
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Tasks
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Topics in Multimedia Signal Processing
1. Determine the model parameters
2. Reconstruct from
Maschinelles Lernen und Inverse Probleme 17
Tasks
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Topics in Multimedia Signal Processing
1. Determine the model parameters
2. Reconstruct from
Maschinelles Lernen und Inverse Probleme 18
Tasks
?
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 19
Approaches to solve inverse problems
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 20
Least Squares approach
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Topics in Multimedia Signal Processing
• Problems: – ill-conditioned
Example: Signal Deconvolution/Deblurring
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Least Squares approach
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Topics in Multimedia Signal Processing
• Problems: – ill-conditioned
Example: Signal Deconvolution/Deblurring – n ≠ m System under-/overdetermined
infinitely many/no solutions Example: Signal Inpainting
Maschinelles Lernen und Inverse Probleme 22
Least Squares approach
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Topics in Multimedia Signal Processing
• Problems: – ill-conditioned
Example: Signal Deconvolution/Deblurring – n ≠ m System under-/overdetermined
infinitely many/no solutions Example: Signal Inpainting
– No AWGN
Maschinelles Lernen und Inverse Probleme 23
Least Squares approach
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Topics in Multimedia Signal Processing
• Problems: – ill-conditioned
Example: Signal Deconvolution/Deblurring – n ≠ m System under-/overdetermined
infinitely many/no solutions Example: Signal Inpainting
– No AWGN • Solutions:
– Exploiting structures and properties of the data
Maschinelles Lernen und Inverse Probleme 24
Least Squares approach
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Topics in Multimedia Signal Processing
• Problems: – ill-conditioned
Example: Signal Deconvolution/Deblurring – n ≠ m System under-/overdetermined
infinitely many/no solutions Example: Signal Inpainting
– No AWGN • Solutions:
– Exploiting structures and properties of the data – Optimization under constraints
Maschinelles Lernen und Inverse Probleme 25
Least Squares approach
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 26
Optimization under constraints
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 27
Optimization under constraints
Constraint set encoded in function
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 28
Optimization under constraints
Constraint set encoded in function
assumed noise energy
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 29
What are suitable constraints?
- Pixelvalues are always positive
- Images contain homogeneous regions, i.e.
neighbouring pixels often have the same value
- Signals can be composed of „Basissignals“ (e.g. sinusoids)
Maschinelles Lernen und Inverse Probleme 29 02.05.2014
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 30
Synthesis Operator (Dictionary) idealised
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 31
Synthesis Operator (Dictionary) idealised
atoms
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Synthesis Operator (Dictionary) idealised
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 33
Synthesis Operator (Dictionary) idealised
atoms
…
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 34
Synthesis Operator (Dictionary) idealised
atoms
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
…
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 35
Synthesis Operator (Dictionary) idealised
=
atoms signal
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1
0 0
…
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 36
Synthesis Operator (Dictionary) idealised
=
atoms signal
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
1
0 0
…
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 37
Synthesis Operator (Dictionary) idealised
=
atoms signal
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
1
1 1
…
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 38
Synthesis Model
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Topics in Multimedia Signal Processing
Dictionary
Maschinelles Lernen und Inverse Probleme 39
Synthesis Model
=
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Topics in Multimedia Signal Processing
Assumption: Signal has a sparse representation
Dictionary
Maschinelles Lernen und Inverse Probleme 40
Synthesis Model
=
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Topics in Multimedia Signal Processing
Assumption: Signal has a sparse representation
Dictionary
Maschinelles Lernen und Inverse Probleme 41
Synthesis Model
=
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Topics in Multimedia Signal Processing
Assumption: Signal has a sparse representation
Dictionary • redundant • Columns are called atoms
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Synthesis Model
=
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 43
JPEG Compression
Natural images are compressible signals with a compressible representation in a DCT (JPEG) or Wavelet Basis (JPEG-2000)
Compressible Signals can be well approximated through sparse signals
Maschinelles Lernen und Inverse Probleme 43 02.05.2014
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 44
JPEG Compression
Maschinelles Lernen und Inverse Probleme 44 02.05.2014
Image from: Gregory K. Wallace, The JPEG Still Picture Compression Standard, IEEE Transactions on Consumer Electronics, vol. 38 no.1, Feb. 1992.
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 45 Maschinelles Lernen und Inverse Probleme 45 02.05.2014
Input patch Forward DCT coefficients Quantization table
Normalized quantized coefficients Reconstructed patch
Denormalized quantized coefficients
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 46 Maschinelles Lernen und Inverse Probleme 46 02.05.2014
Input patch Forward DCT coefficients Quantization table
Normalized quantized coefficients Reconstructed patch
Denormalized quantized coefficients
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 47
Synthesis Model
Goal: Find sparsest , that explains the measurements
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 48
Synthesis Model
Signal is synthesized from sparse vector Synthesis Model
Goal: Find sparsest , that explains the measurements
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 49
Synthesis Model
Signal is synthesized from sparse vector Synthesis Model
Goal: Find sparsest , that explains the measurements
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Topics in Multimedia Signal Processing 50
-1 +1
1
( ) pxxf =
x
k ppjp
j=1x = x∑
11α
22α
1p
pp
<
α0p
pp
→
α
As p → 0 we get a count of the non-zeros in the vector
0α
02.05.2014 Maschinelles Lernen und Inverse Probleme
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 51
From Synthesis Model to the Analysis Model
Synthesis Model =
Signal is synthesized from a few atoms ( = sparse vektor)
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 52
From Synthesis Model to the Analysis Model
Synthesis Model =
Analysis Model
= Assumption: Signal is mapped to sparse vector via an Analysis Operator
Signal is synthesized from a few atoms ( = sparse vektor)
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 53
From Synthesis Model to the Analysis Model
Synthesis Model =
Analysis Model
= Assumption: Signal is mapped to sparse vector via an Analysis Operator
• • Rows are called atoms
Signal is synthesized from a few atoms ( = sparse vektor)
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 54
Analysis Operator exemplarily
signal
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 55
Analysis Operator exemplarily
finite differences of adjacent pixels
signal
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 56
Analysis Operator exemplarily
=
finite differences of adjacent pixels
sparse analysed vektor signal
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Topics in Multimedia Signal Processing
Goal: Find signal , such that the analysed vector is sparse and such that explains the measurements
Maschinelles Lernen und Inverse Probleme 57
Analysis Model
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 58
Find a solution via …
OMP
FISTA TWIST
NESTA BMP
ISTA CVX L1-magic
SPARSA
… SALSA
C-SALSA
SAMP YALL1
FOCUSS
https://sites.google.com/site/igorcarron2/cs
QN CG
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Topics in Multimedia Signal Processing
1. Analytically given Advantages: Fast implementation + generalisation Drawback: Sparse representation is not optimal Examples: Wavelets, Bandlets, Curvelets, Dicrete Cosine Transform, Fourier Transform, Finite Difference Operator (Total Variation) …
Maschinelles Lernen und Inverse Probleme 59
What are appropriate Synthesis/Analysis Operators?
Maschinelles Lernen und Inverse Probleme 59 02.05.2014
Topics in Multimedia Signal Processing
1. Analytically given Advantages: Fast implementation + generalisation Drawback: Sparse representation is not optimal Examples: Wavelets, Bandlets, Curvelets, Dicrete Cosine Transform, Fourier Transform, Finite Difference Operator (Total Variation) …
2. Learned from trainingdata
Advantages: Optimal sparse representation, performance Drawback: Slow implementation Examples : CURRENT RESEARCH!
Maschinelles Lernen und Inverse Probleme 60
What are appropriate Synthesis/Analysis Operators?
Maschinelles Lernen und Inverse Probleme 60 02.05.2014
Topics in Multimedia Signal Processing
1. Analytically given Advantages: Fast implementation + generalisation Drawback: Sparse representation is not optimal Examples: Wavelets, Bandlets, Curvelets, Dicrete Cosine Transform, Fourier Transform, Finite Difference Operator (Total Variation) …
2. Learned from trainingdata
Advantages: Optimal sparse representation, performance Drawback: Slow implementation Examples : CURRENT RESEARCH!
Example: Dictionary Learning
Maschinelles Lernen und Inverse Probleme 61
What are appropriate Synthesis/Analysis Operators?
Maschinelles Lernen und Inverse Probleme 61 02.05.2014
Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 62
Analysis Operator Learning
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Operator Learning example for Image Processing
vectorised patches
Operator Learning
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 64
Analysis Operator Learning Basics
Required: N representative training signals
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 65
Analysis Operator Learning Basics
Required: N representative training signals
Sought: Analysis Operator, such that N analysed vectors are sparse
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 66
Analysis Operator Learning Basics
Required: N representative training signals
Sought: Analysis Operator, such that N analysed vectors are sparse
Analysis Operator Atoms
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 67
Analysis Operator Learning Basics
Required: N representative training signals
Sought: Analysis Operator, such that N analysed vectors are sparse
Analysis Operator Atoms
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 68
Analysis Operator Learning Basics
Required: N representative training signals
Sought: Analysis Operator, such that N analysed vectors are sparse
Constraint set to avoid trivial solution
Analysis Operator Atoms
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 69
Geometric Analysis Operator Learning (GOAL)
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Topics in Multimedia Signal Processing
• Constraints
Maschinelles Lernen und Inverse Probleme 70
Geometric Analysis Operator Learning (GOAL)
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Topics in Multimedia Signal Processing
• Constraints 1. Atoms/rows of are normalised, i.e.
Maschinelles Lernen und Inverse Probleme 71
Geometric Analysis Operator Learning (GOAL)
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Topics in Multimedia Signal Processing
• Constraints 1. Atoms/rows of are normalised, i.e. 2. has full rank, i.e.
Maschinelles Lernen und Inverse Probleme 72
Geometric Analysis Operator Learning (GOAL)
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Topics in Multimedia Signal Processing
• Constraints 1. Atoms/rows of are normalised, i.e. 2. has full rank, i.e. 3. Rows are not trivially linear dependent,
Maschinelles Lernen und Inverse Probleme 73
Geometric Analysis Operator Learning (GOAL)
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Topics in Multimedia Signal Processing
• Constraints 1. Atoms/rows of are normalised, i.e. 2. has full rank, i.e. 3. Rows are not trivially linear dependent,
• From constraints 1+2 element of a special manifold efficient method to find a solution (e.g. Conjugate Gradient, Quasi-Newton)
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Geometric Analysis Operator Learning (GOAL)
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 75
Example: Manifold Learning
• Normalised rows lie on the surface of a sphere (with radius = 1)
• Step along geodesics
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 76
Applied to solve inverse problems
?
Denoising Deblurring Inpainting
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 77
Applied to solve inverse problems
!
Denoising Deblurring Inpainting
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Topics in Multimedia Signal Processing
Demo: GOAL + Lena
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 79
Bimodal signal reconstruction
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 80
Application: 3D Reconstruction in HD
3D scene analysis with high-resolution camera
and depth sensor
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 81
Application: 3D Reconstruction in HD
3D scene analysis with high-resolution camera
and depth sensor
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 82
Application: 3D Reconstruction in HD
3D scene analysis with high-resolution camera
and depth sensor
Bimodal Analysis Operator
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 83
Learning from bimodal signals
Intensity Depth
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 84
Learning from bimodal signals
Intensity Depth bright
dark
signal pair
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 85
Learning from bimodal signals
Intensity Depth bright
dark
Intensity operator 𝛀𝐼 Depth operator 𝛀𝐷
minimize𝛀𝐼,𝛀𝐷
𝐺 𝛀𝐼𝑺𝐼 ,𝛀𝐷𝑺𝐷
learn 𝛀𝑰 and 𝛀𝐷 such that both analyzed vectors 𝛀𝐼𝒔𝑖 and 𝛀D𝒔d are maximally sparse
signal pair
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 86
Bimodal reconstruction Unimodal
𝐬∗ ∈ arg min𝐬 ∈ ℝ𝑵
𝑔 𝛀𝐬 subject to 𝒜𝐬 − 𝒚 22 ≤ 𝜀
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 87
Bimodal reconstruction Unimodal
𝐬∗ ∈ arg min𝐬 ∈ ℝ𝑵
𝑔 𝛀𝐬 subject to 𝒜𝐬 − 𝒚 22 ≤ 𝜀
Bimodal
(𝒔𝐼∗, 𝒔𝐷∗ ) ∈ arg min𝒔𝐼,𝒔𝐷∈ ℝ𝑵
𝐺 𝛀𝐼𝒔𝐼 ,𝛀𝐷𝒔𝐷 subj. to 𝒜𝐼𝒔𝐼 − 𝒚𝐼 22 + 𝒜𝐷𝒔𝐷 − 𝒚𝐷 2
2 ≤ 𝜀
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 88
Bimodal reconstruction Unimodal
𝐬∗ ∈ arg min𝐬 ∈ ℝ𝑵
𝑔 𝛀𝐬 subject to 𝒜𝐬 − 𝒚 22 ≤ 𝜀
Bimodal
𝒔𝐷∗ ∈ arg min𝒔𝐷∈ ℝ𝑵
𝜆𝐺 𝒄,𝛀𝐷𝒔𝐷 + 𝒜𝐷𝒔𝐷 − 𝒚𝐷 22
𝒄 0 Intensity image fixed
(𝒔𝐼∗, 𝒔𝐷∗ ) ∈ arg min𝒔𝐼,𝒔𝐷∈ ℝ𝑵
𝐺 𝛀𝐼𝒔𝐼 ,𝛀𝐷𝒔𝐷 subj. to 𝒜𝐼𝒔𝐼 − 𝒚𝐼 22 + 𝒜𝐷𝒔𝐷 − 𝒚𝐷 2
2 ≤ 𝜀
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 89
Results of the bimodal reconstruction (JID)
bicubic interpolation NN interpolation JID
8x
Depth Map Super-Resolution
3D Scene Reconstruction
original bicubic interpolation JID
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Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 90
Analysis Based Blind Compressive Sensing
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Topics in Multimedia Signal Processing
• Only a few linear and non-adaptive measurements are sufficient to reconstruct the signal with high accuracy.
Maschinelles Lernen und Inverse Probleme 91
Concept of Compressive Sensing
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Topics in Multimedia Signal Processing
• Only a few linear and non-adaptive measurements are sufficient to reconstruct the signal with high accuracy.
• Exploitation of the sparse representation with a (analytically) given Dictionary or Operator.
Maschinelles Lernen und Inverse Probleme 92
Concept of Compressive Sensing
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Topics in Multimedia Signal Processing
• Reconstruction of the signals under the assumption that there exists a sparse representation
Maschinelles Lernen und Inverse Probleme 93
Analysis Based Compressive Sensing
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Topics in Multimedia Signal Processing
• Exploiting the property that learned operators admit a sparser representation
• Adaptive, signal dependent regularisation of the inverse problem under
consideration of the error model
Maschinelles Lernen und Inverse Probleme 94
Analysis Based Blind Compressive Sensing
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Topics in Multimedia Signal Processing
• Exploiting the property that learned operators admit a sparser representation
• Adaptive, signal dependent regularisation of the inverse problem under
consideration of the error model
Maschinelles Lernen und Inverse Probleme 95
Analysis Based Blind Compressive Sensing
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Topics in Multimedia Signal Processing
Image reconstruction
Maschinelles Lernen und Inverse Probleme 96
Analysis Based Blind Compressive Sensing
(1) ABCS (2) TV Operator
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Topics in Multimedia Signal Processing
Learned Analysis Operators
Maschinelles Lernen und Inverse Probleme 97
Analysis Based Blind Compressive Sensing
(1) Random Input (2) Barbara (3) Piecewise constant
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Topics in Multimedia Signal Processing
• Simultaneous reconstructing and learning allows one to find an operator that adaptively fits the underlying image structure
Maschinelles Lernen und Inverse Probleme 98
Analysis Based Blind Compressive Sensing
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Topics in Multimedia Signal Processing
• Simultaneous reconstructing and learning allows one to find an operator that adaptively fits the underlying image structure
• The Analysis Operator does not need to be learned before the
reconstruction
Maschinelles Lernen und Inverse Probleme 99
Analysis Based Blind Compressive Sensing
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Topics in Multimedia Signal Processing
• Simultaneous reconstructing and learning allows one to find an operator that adaptively fits the underlying image structure
• The Analysis Operator does not need to be learned before the
reconstruction
• Ability to reconstruct different signal/image classes by simply exchanging the error model
Maschinelles Lernen und Inverse Probleme 100
Analysis Based Blind Compressive Sensing
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Topics in Multimedia Signal Processing
Maschinelles Lernen und Inverse Probleme 101
Take Home Messages
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Topics in Multimedia Signal Processing
• Structure in data is extremely important and can be utilized to
regularize inverse problems
Maschinelles Lernen und Inverse Probleme 102
Take Home Messages
02.05.2014
Topics in Multimedia Signal Processing
• Structure in data is extremely important and can be utilized to
regularize inverse problems
• Sparsity is a valuable property of many signals
Maschinelles Lernen und Inverse Probleme 103
Take Home Messages
02.05.2014
Topics in Multimedia Signal Processing
• Structure in data is extremely important and can be utilized to
regularize inverse problems
• Sparsity is a valuable property of many signals
• Machine Learning can help to find such structures
Maschinelles Lernen und Inverse Probleme 104
Take Home Messages
02.05.2014
Topics in Multimedia Signal Processing
• Structure in data is extremely important and can be utilized to
regularize inverse problems
• Sparsity is a valuable property of many signals
• Machine Learning can help to find such structures
• Geometric aspects of a problem can be exploited in the optimization
Maschinelles Lernen und Inverse Probleme 105
Take Home Messages
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Topics in Multimedia Signal Processing
Weiterführende Literatur
02.05.2014 Maschinelles Lernen und Inverse Probleme 106
S. Hawe, M. Kleinsteuber, and K. Diepold. Analysis Operator Learning and its Application to Image Reconstruction. IEEE Transactions on Image Processing, vol.22, no.6, pp.2138-2150, June 2013. J. Wörmann, S. Hawe, and M. Kleinsteuber. Analysis Based Blind Compressive Sensing. IEEE Signal Processing Letters, 20(5) 491-494, 2013. M. Kiechle, S. Hawe, and M. Kleinsteuber. A Joint Intensity and Depth Co-Sparse Analysis Model for Depth Map Super-Resolution. IEEE International Conference on Computer Vision 2013. M. Aharon, M. Elad, and A. Bruckstein. K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation. IEEE Transactions on Signal Processing, vol. 54, no.11, 2006.