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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

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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?

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 3

Inverse Problems

cause/ excitation

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 4

Inverse Problems

cause/ excitation

System/ Process

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 5

Inverse Problems

cause/ excitation

effect/ measurement

System/ Process

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 9

Inverse Problems

cause/ excitation

effect/ measurement

System/ Process

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 10

Goal

02.05.2014

cause/ excitation

effect/ measurement

System/ Process

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 11

Goal

02.05.2014

cause/ excitation

effect/ measurement

System/ Process

System/ -1

Process

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 12

Goal

Model:

02.05.2014

cause/ excitation

effect/ measurement

System/ Process

System/ -1

Process

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 13

Goal

noise

Model:

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing

1. Determine the model parameters

Maschinelles Lernen und Inverse Probleme 15

Tasks

02.05.2014

Topics in Multimedia Signal Processing

1. Determine the model parameters

Maschinelles Lernen und Inverse Probleme 16

Tasks

02.05.2014

Topics in Multimedia Signal Processing

1. Determine the model parameters

2. Reconstruct from

Maschinelles Lernen und Inverse Probleme 17

Tasks

02.05.2014

Topics in Multimedia Signal Processing

1. Determine the model parameters

2. Reconstruct from

Maschinelles Lernen und Inverse Probleme 18

Tasks

?

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 19

Approaches to solve inverse problems

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 20

Least Squares approach

02.05.2014

Topics in Multimedia Signal Processing

• Problems: – ill-conditioned

Example: Signal Deconvolution/Deblurring

Maschinelles Lernen und Inverse Probleme 21

Least Squares approach

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 26

Optimization under constraints

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 27

Optimization under constraints

Constraint set encoded in function

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 28

Optimization under constraints

Constraint set encoded in function

assumed noise energy

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 31

Synthesis Operator (Dictionary) idealised

atoms

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 32

Synthesis Operator (Dictionary) idealised

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 33

Synthesis Operator (Dictionary) idealised

atoms

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 38

Synthesis Model

02.05.2014

Topics in Multimedia Signal Processing

Dictionary

Maschinelles Lernen und Inverse Probleme 39

Synthesis Model

=

02.05.2014

Topics in Multimedia Signal Processing

Assumption: Signal has a sparse representation

Dictionary

Maschinelles Lernen und Inverse Probleme 40

Synthesis Model

=

02.05.2014

Topics in Multimedia Signal Processing

Assumption: Signal has a sparse representation

Dictionary

Maschinelles Lernen und Inverse Probleme 41

Synthesis Model

=

02.05.2014

Topics in Multimedia Signal Processing

Assumption: Signal has a sparse representation

Dictionary • redundant • Columns are called atoms

Maschinelles Lernen und Inverse Probleme 42

Synthesis Model

=

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

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)

02.05.2014

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)

02.05.2014

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)

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 54

Analysis Operator exemplarily

signal

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 55

Analysis Operator exemplarily

finite differences of adjacent pixels

signal

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 56

Analysis Operator exemplarily

=

finite differences of adjacent pixels

sparse analysed vektor signal

02.05.2014

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

02.05.2014

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

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) …

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

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 63

Operator Learning example for Image Processing

vectorised patches

Operator Learning

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 64

Analysis Operator Learning Basics

Required: N representative training signals

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 69

Geometric Analysis Operator Learning (GOAL)

02.05.2014

Topics in Multimedia Signal Processing

• Constraints

Maschinelles Lernen und Inverse Probleme 70

Geometric Analysis Operator Learning (GOAL)

02.05.2014

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)

02.05.2014

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)

02.05.2014

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)

02.05.2014

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)

Maschinelles Lernen und Inverse Probleme 74

Geometric Analysis Operator Learning (GOAL)

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 76

Applied to solve inverse problems

?

Denoising Deblurring Inpainting

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 77

Applied to solve inverse problems

!

Denoising Deblurring Inpainting

02.05.2014

Topics in Multimedia Signal Processing

Demo: GOAL + Lena

02.05.2014 Maschinelles Lernen und Inverse Probleme 78

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 79

Bimodal signal reconstruction

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 83

Learning from bimodal signals

Intensity Depth

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 84

Learning from bimodal signals

Intensity Depth bright

dark

signal pair

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing Maschinelles Lernen und Inverse Probleme 90

Analysis Based Blind Compressive Sensing

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing

Image reconstruction

Maschinelles Lernen und Inverse Probleme 96

Analysis Based Blind Compressive Sensing

(1) ABCS (2) TV Operator

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

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

02.05.2014

Topics in Multimedia Signal Processing

Maschinelles Lernen und Inverse Probleme 101

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

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

02.05.2014

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.