michael elad the computer science department the technion – israel institute of technology

Sparse & Redundant Representations and Their Use in

Signal and Image Processing CS Course 236862 – Winter 2012

Michael EladThe Computer Science Department

The Technion – Israel Institute of technologyHaifa 32000, Israel

October 28, 2012

Michael EladThe Computer-Science DepartmentThe Technion

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What This Field is all About ?Depends whom you ask, as the researchers in this field come from the following disciplines: • Mathematics• Applied Mathematics• Statistics• Signal & Image Processing: CS, EE, Bio-medical, …• Computer-Science Theory• Machine-Learning • Physics (optics) • Geo-Physics• Astronomy• Psychology (neuroscience)• …


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My Answer (For Now)

A New Transform for

Signals We are all well-aware of the idea of transforming a signal and changing its representation.

We apply a transform to gain something – efficiency, simplicity of the subsequent processing, speed, …

There is a new transform in town, based on sparse and redundant representations.


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Transforms – The General Picture

Invertible Transforms

Linear

Unitary

SeparableStructured

n

n

x

nD


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Redundancy? In a redundant transform,

the representation vector is longer (m>n).

This can still be done while preserving the linearity of the transform:

m

n

x

nD

m x

n†D

†x

x

xI

DDD


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Sparse & Redundant Representation m

n

x

nD We shall keep the linearity

of the inverse-transform. As for the forward (computing

from x), there are infinitely many possible solutions.

We shall seek the sparsest of all solutions – the one with the fewest non-zeros.

This makes the forward transform a highly non-linear operation.

The field of sparse and redundant representations is all about defining clearly this transform, solving various theoretical and numerical issues related to it, and showing how to use it in practice.

Sounds … Boring !!!! Who cares about a new transform?

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Lets Take a Wider Perspective

Voice Signal Radar Imaging

Still Image

Stock Market

Heart Signal

CT

Traffic Information We are surrounded by various

sources of massive information of different nature.

All these sources have some internal structure, which can be exploited.


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Model?

Effective removal of noise (and many other applications) relies on an proper modeling of the signal


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Which Model to Choose? There are many different

ways to mathematically model signals and images with varying degrees of success.

The following is a partial list of such models (for images):

Good models should be simple while matching the signals:

Principal-Component-Analysis Anisotropic diffusionMarkov Random Field Wienner FilteringDCT and JPEG Wavelet & JPEG-2000Piece-Wise-Smooth C2-smoothnessBesov-Spaces Total-VariationBeltrami-Flow

Simplicity

Reliability


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An Example: JPEG and DCT178KB – Raw data

4KB

8KB

12KB20KB24KB

How & why does it works?

Discrete Cosine Trans.

The model assumption: after DCT, the top left coefficients to be dominant and the rest zeros.



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Research in Signal/Image Processing

Model Problem (Application) Signal

Numerical Scheme

A New Research Work (and Paper) is Born

The fields of signal & image processing are essentially built of an evolution of models and ways to use them for various tasks

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Again: What This Field is all About?

A Data Model and

Its Use Almost any task in data processing requires a model – true for denoising, deblurring, super-resolution, inpainting, compression, anomaly-detection, sampling, and more.

There is a new model in town – sparse and redundant representation – we will call it Sparseland.

We will be interested in a flexible model that can adjust to the signal.


Machine Learning

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MathematicsSignal

Processing

A New Emerging Model

Sparseland and

Example-Based Models

Wavelet Theory

Signal Transforms

Multi-Scale Analysis

Approximation Theory

Linear Algebra

Optimization Theory

Denoising

Compression InpaintingBlind Source Separation Demosaicing

Super-Resolution


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The Sparseland Model

Task: model image patches of size 10×10 pixels.

We assume that a dictionary of such image patches is given, containing 256 atom images.

The Sparseland model assumption: every image patch can be described as a linear combination of few atoms.

α1 α2 α3

Σ


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The Sparseland Model

We start with a 10-by-10 pixels patch and represent it using 256 numbers – This is a redundant representation.

However, out of those 256 elements in the representation, only 3 are non-zeros – This is a sparse representation.

Bottom line in this case: 100 numbers representing the patch are replaced by 6 (3 for the indices of the non-zeros, and 3 for their entries).

Properties of this model: Sparsity and Redundancy.

α1 α2 α3

Σ


Chemistry of Data


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Model vs. Transform ? m

n

x

nD The relation between the

signal x and its representation is the following linear system, just as described earlier.

We shall be interested in seeking sparse solutions to this system when deploying the sparse and redundant representation model.

This is EXACTLY the transform we discussed earlier.Bottom Line: The transform and the

model we described above are the same thing, and their impact on

signal/image processing is profound and worth studying.

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Difficulties With Sparseland Problem 1: Given an image patch, how

can we find its atom decomposition ? A simple example:

There are 2000 atoms in the dictionary The signal is known to be built of 15 atoms

possibilities

If each of these takes 1nano-sec to test, this will take ~7.5e20 years to finish !!!!!!

Solution: Approximation algorithms

α1 α2 α3

Σ

2000 2.4e 3715


α1 α2 α3

Σ

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Difficulties With Sparseland

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2

-1

0

1

2

Iteration 0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2

-1

0

1

2

Iteration 1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

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0

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Iteration 2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

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Iteration 3

0 200 400 600 800 1000 1200 1400 1600 1800 2000

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Iteration 4

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2

-1

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1

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Iteration 5

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-2

-1

0

1

2

Iteration 6

Various algorithms exist. Their theoretical analysis guarantees their success if the solution is sparse enough

Here is an example – the Iterative Reweighted LS:


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α1 α2 α3

Σ Problem 2: Given a family of signals, how do

we find the dictionary to represent it well? Solution: Learn! Gather a large set of

signals (many thousands), and find the dictionary that sparsifies them.

Such algorithms were developed in the past 5 years (e.g., K-SVD), and their performance is surprisingly good.

This is only the beginning of a new era in signal processing …


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α1 α2 α3

Σ Problem 3: Is this model flexible enough to

describe various sources? e.g., Is it good for images? Audio? Stocks? …

General answer: Yes, this model is extremely effective in representing various sources. Theoretical answer: yet to be given. Empirical answer: we will see in this

course, several image processing applications, where this model leads to the best known results (benchmark tests).


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Difficulties With Sparseland ?

Problem 1: Given an image patch, how can we find its atom decomposition ?

Problem 2: Given a family of signals, how do we find the dictionary to represent it well?

Problem 3: Is this model flexible enough to describe various sources? E.g., Is it good for images? audio? …

ALL ANSW

ERED

POSITIVE

LY AND

CONSTRUC

TIVELY

α1 α2 α3

Σ


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This Course

Sparse and Redundant Representations

Will review a decade of tremendous progress in the field of

Theory Numerical Problems

Applications (image processing)


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Who is Working on This? Donoho, Candes – Stanford

Tropp – CalTech

Baraniuk, W. Yin – Rice Texas

Gilbert, Strauss – U-Michigan

Gribonval, Fuchs – INRIA France

Starck – CEA – France

Vandergheynst, Cehver– EPFL Swiss

Rao, Delgado – UC San-Diego

Do, Ma – U-Illinois

Tanner, Davies – Edinbourgh UK

Elad, Zibulevsky, Bruckstein, Eldar – Technion

Goyal – MIT

Mallat – Ecole-Polytec. Paris

Daubechies – Princeton

Coifman – Yale

Romberg – GaTech

Lustig, Wainwright – Berkeley

Sapiro – UMN

Friedlander – UBC Canada

Tarokh – Harvard

Cohen, Combettes – Paris VI


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This Field is rapidly Growing …


Searching ISI-Web-of-Science: Topic=((spars* and (represent* or approx* or solution) and (dictionary or pursuit)) or (compres* and sens* and spars*))

led to 1368 papers

Here is how they spread over time:

25Michael EladThe Computer-Science DepartmentThe Technion

Which Countries?

26Michael EladThe Computer-Science DepartmentThe Technion

Who is Publishing in This Area?


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Here Are Few Examples for the Things That We Did

With This Model So Far …


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Image Separation [Starck, Elad, & Donoho (`04)]

The original image - Galaxy SBS 0335-052

as photographed

by Gemini

The texture part spanned

by global DCT

The residual being additive noise

The Cartoon part spanned by wavelets


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Inpainting [Starck, Elad, and Donoho (‘05)]

Outcome

Source


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Initial dictionary (overcomplete DCT)

64×256

Image Denoising (Gray) [Elad & Aharon (`06)]

Source

Result 30.829dB

The obtained dictionary after 10 iterations

Noisy image 20


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Original Noisy (12.77dB) Result (29.87dB)

Denoising (Color) [Mairal, Elad & Sapiro, (‘06)]

Original Noisy (20.43dB) Result (30.75dB)


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Deblurring [Elad, Zibulevsky and Matalon, (‘07)]

original (left), Measured (middle), and Restored (right): Iteration: 0 ISNR=-16.7728 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 1 ISNR=0.069583 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 2 ISNR=2.46924 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 3 ISNR=4.1824 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 4 ISNR=4.9726 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 5 ISNR=5.5875 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 6 ISNR=6.2188 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 7 ISNR=6.6479 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 8 ISNR=6.6789 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 12 ISNR=6.9416 dBoriginal (left), Measured (middle), and Restored (right): Iteration: 19 ISNR=7.0322 dB


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Result Original 80% missing

Inpainting (Again!) [Mairal, Elad & Sapiro, (‘06)]

Original 80% missing Result


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Original Noisy (σ=25) Denoised

Original Noisy (σ=50) Denoised

Video Denoising [Protter & Elad (‘06)]


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Results for 550

Bytes per

each file

15.81

14.67

15.30

13.89

12.41

12.57

6.60

5.49

6.36

Facial Image Compression [Brytt and Elad (`07)]


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Results for 400

Bytes per

each file

18.62

16.12

16.81

7.61

6.31

7.20

?

?

?

Facial Image Compression [Brytt and Elad (`07)]


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Super-Resolution [Zeyde, Protter & Elad (‘09)]

Ideal Image

Given Image

SR ResultPSNR=16.95dB

Bicubic interpolation

PSNR=14.68dB


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Super-Resolution [Zeyde, Protter & Elad (‘09)]

The Original Bicubic Interpolation SR result


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Are they working well?

To Summarize

Sparse and redundant representations and other example-based modeling methods are drawing a considerable

attention in recent years

Which model to

choose?

Yes, these methods have been deployed to a series of

applications, leading to state-of-the-art results. In parallel, theoretical results provide the backbone for these algorithms’ stability and good-performance

An effective (yet simple) model for

signals/images is key in getting better

algorithms for various applications


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And now some Administrative issues …

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This Course – General

ויתירים דלילים ותמונות ייצוגים אותות בעיבוד ושימושיהם : הקורס 236862מספר

מרצה: מיכאל אלעד

זיכוי אקדמי: נקודות2

שעות הרצאה ומקום: 4, טאוב 12:30 – 10:30יום א',

דרישות קדם: (תלמידי מוסמכים אינם נדרשים לקדם)046200 או 236860

ספרות נדרשת: מאמרים שיוזכרו במהלך הסמסטר וספר (ראו בהמשך)

אתר הקורס: (כתובת האתר) http://www.cs.technion.ac.il/~elad/teaching (ומשם יש לעקוב אחר הקישורים לקורס זה)

מועדי הבחינה: – יום ו'5.4.2012 – יום ב' ו- 4.2.2012


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Course Material

We shall follow this book.

No need to buy the book. The lectures will be self-contained.

The material we will cover has appeared in 40-60 research papers that were published mostly (not all) in the past 6-7 years.


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This Course Sitehttp://www.cs.technion.ac.il/~elad/teaching/courses/Sparse_Representations_Winter_2012/index.htm

Go to my home page, click the “teaching” tab, then “courses”, and choose the top on the list


http://www.cs.technion.ac.il/~elad/teaching/courses/Sparse_Representations_Winter_2012/index.htm



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This Course – Lectures and HW הרצאה פרק הנושא

1 1 מבוא כללי2 2 יחידות פתרונות דלילים3 3 [Batch-OMP – 1]תרגיל בית # pursuitאלגוריתמי 4 4 – משפטי שקילות pursuitביצועי אלגוריתמי 5 5 התייחסות לרעש – יחידות ואלגוריתמים6 5,6 [FISTA - #2]תרגיל בית iterated shrinkageהתייחסות לרעש –יציבות, 7 8 ניתוח ביצועים ממוצעים – יסודות וניתוח שיטת הסף8 9 Danzig-selector 9 8 , מגוון יישומים אפשרייםSparse-Landעיבוד אותות בעזרת מודל

10 9 ,10 - אופציונלי[Gibbs Sampler – #3]תרגיל בית – MMSE ו – MAPמשערכים 11 11 – שיטותMMSE ו- MAPמשערכים 12 11 ), דחיסת תמונות פניםK-SVD ו- MODלימוד מילונים (13 12 ,13 [THR עם K-SVD –#4]תרגיל בית ניקוי רעש – שיטות שונות וקשריהן 14 14 סיכום


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This Course - Gradesהקורס דרישות

.) " האחראי ) המרצה י ע תינתנה ההרצאות כל רגיל בפורמט קורס זהו יינתנו הקורס תכנות 4במהלך על דגש עם בזוגות) (הגשה בית תרגילי

. MATLABב- על המבוסס פרויקט יבצעו סטודנטים צמד האחרונה. 1-3כל מהעת מאמרים

ובו תיאור של המאמרים ודו"ח מסכם יידרשו הסטודנטים להכין מצגת בפרויקט עמודים). 20-30הללו, תרומתם, והשאלות הפתוחות שהותירו (היקף של כ-

.בסיום הקורס יאורגן יום עיון ובו משתתפי הקורס יציגו את הפרויקטים בת בחינה תיערך הקורס כללית 20-30בסיום התמצאות תבדוק אשר שאלות

בחומר .

מבנה הציוןבית, 30% תרגילי - הפרויקט, 20% על סמינר - הפרויקט, 20% על דו"ח - 30% -

בחינת התמצאות.

למעוניינים.שומעים חופשיים המעוניינים להצטרף יתקבלו בברכה -ל אימייל שילחו לרשימת [email protected]אנא להיכנס מנת על

Michael Eladהתפוצה.The Computer-Science DepartmentThe Technion

mailto:[email protected]

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This Course - Projects


Read the instruction in the course’s site

michael elad the computer science department the technion – israel institute of technology

Documents

redundant transform

field of sparse

sparse representations

nowa new transform

model signals

model assumption

jpeg wavelet jpeg

image processing cs