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<div id="slides" style="display: none;">
<div class="pz_slide" id="slide_toc" section="introduction">
===Compressive Sensing===
<div class="pz_slide_toc">
Table of contents:
* <a href="#introduction">Introduction</a>
* <a href="#overview">Overview</a>
* <a href="#solving">Solving the Riddle</a>
* <a href="#significance">Significance of the Riddle</a>
* <a href="#two_important_bases">Two Important Bases</a>
* <a href="#sensing_coordinates">Sensing Coordinates</a>
* <a href="#sparsity_basis">Sparsity Basis Projections</a>
* <a href="#aliases">K-Sparse Aliases</a>
* <a href="#key_equation">Key Equation</a>
* <a href="#reconstruction">Practical Reconstruction</a>
* <a href="#extensions">Summary and Extensions</a>
<span class="pz_text" ref="slide_tips">(Keyboard shortcuts and tips)</span>
</div>
<div class="pz_note" id="slide_tips">
<!--div class="pz_slide_tips"-->
<div style="font-size: 10pt;">
* *Esc*: Close/open slides window.
* *Left/right*: Navigate slides.
* *Up/down*: Scroll main page.
* *Return*: Go to corresponding section in main page.
* Links navigate between slides and main page.
* Drag window anywhere on page.
</div>
</div>
</div>
<div class="pz_slide" id="slide_introduction" section="introduction">
<div class="pz_slide_heading">
===Polynomial Riddle===
</div>
<div class="pz_slide_body">
<img src="riddle.svg">
* Three polynomial signals. I pick one.
* Which samples do you need to correctly determine the signal?
* Just <span style="color: red;">red</span> and
<span style="color: blue;">blue</span> samples? All three samples?
</div>
</div>
<div class="pz_slide" id="slide_overview" section="overview">
<div class="pz_slide_heading">
===Overview===
</div>
<style>.overview td {background:#ddd;}</style>
<div class="pz_slide_body">
* Promise of CS: transmit or store signals using fewer bits.
* Represent one signal according to two *bases*.
<br><br>
<table class="overview" width="400" cellpadding="4">
<th>Basis<th>Coordinates
<tr><td>1. Sparsity<td>At least one zero coefficient
<tr><td>2. Sensing<td>At least one redundant coefficient
</table><br>
* Polynomial example, $f(t)=x_0+x_1 t+ x_2 t^2$:
<table class="overview" width="480" cellpadding="4">
<th>Representation<th>Basis<th>Coordinates
<tr>
<td>1. Sparsity
<td>$\{1,t,t^2\}$
<td>$[x_0~x_1~x_2]^T$ (one non-zero)
<tr>
<td>2. Sensing
<td>Lagrange
<td>Sample values
</table>
</div>
</div>
<div class="pz_slide" id="slide_solving" section="solving">
<div class="pz_slide_heading">
=== Solving the Riddle ===
</div>
<div class="pz_slide_body" style="margin-top: -10px;">
<img src="riddle.svg">
* 3 samples unambiguously distinguish __any__ quadratic polynomial.
* 2 samples?
** <span style="color: blue;">blue</span> and
<span style="color: green;">green</span>:
line and parabola agree for $x_1=2x_2$.
** <span style="color: red;">red</span> and
<span style="color: blue;">blue</span>:
line and parabola agree for $x_1=-x_2$.
** <span style="color: red;">red</span> and
<span style="color: green;">green</span>: <b>no two signals ever agree</b>
<br/>(for any $x_0$, $x_1$, $x_2$).
* *No 1-sparse aliases using the <span style="color: red;">red</span> and
<span style="color: green;">green</span> samples*.
</div>
</div>
<div class="pz_slide" id="slide_significance" section="significance">
<div class="pz_slide_heading">
=== Significance of the Riddle ===
</div>
<div class="pz_slide_body">
* Two samples is __compression__.
** Sparsity *enables* compression.
** Compression requires absence of K-Sparse aliases
* Periodic functions are (trigonometric) polynomials.
** Useful in applications.
** E.g., signals with sparse frequency spectrum.
* Two samples is __sub-Nyquist__
** But, sparsity is *severe* constraint.
* The key is a *pair of bases*: sparse and compressible.
</div>
</div>
<div class="pz_slide" id="slide_twobases_1" section="two_important_bases">
<div class="pz_slide_heading">
=== Two Important Bases ===
<p class="subslide">(Slide 1 of 2)</p>
</div>
<div class="pz_slide_body">
==== Sparsity Basis ====
* E.g., $\mathcal{B}_N=\{1,t,t^2\}$
* $K$-sparse signal has $K$ non-zero coefficients
* E.g., 1-sparse $f(t)=x_0+x_1 t+x_2t^2$
** One of $x_0$, $x_1$ or $x_2$ is non-zero
** I.e., constant, line or parabola
</div>
</div>
<div class="pz_slide" id="slide_twobases_2" section="two_important_bases">
<div class="pz_slide_heading">
=== Two Important Bases ===
<p class="subslide">(Slide 2 of 2)</p>
</div>
<div class="pz_slide_body">
==== Sensing Basis ====
* Coefficients (sensed) from measurements
* Prototype: *Lagrange* basis
** Coefficients are sample values.
** Basis polynomials unity at one instant, zero at others.
** Plot is for sampling instants $\{-1,0,2\}$
<img src="lagrange_basis.svg">
</div>
</div>
<div class="pz_slide" id="slide_sensing_coordinates_1"
section="sensing_coordinates" codenodes="270,209,235">
<div class="pz_slide_heading">
=== Sensing Coordinates ===
<p class="subslide">(Slide 1 of 3)</p>
</div>
<div class="pz_slide_left_panel_one_third" style="padding-top: 0px">
==== Constant ====
* Shown in black:
** $f(t)=1$
* Each sample value has an axis (see inset).
</div>
<div class="pz_slide_right_panel_two_thirds"
style="padding-top: 0px;">
<div style="margin-left: -100px;">
<img src="geom_1.svg" height="420px">
</div>
</div>
</div>
<div class="pz_slide" id="slide_sensing_coordinates_2"
section="sensing_coordinates" codenodes="270,209,210,241">
<div class="pz_slide_heading">
=== Sensing Coordinates ===
<p class="subslide">(Slide 2 of 3)</p>
</div>
<div class="pz_slide_left_panel_one_third" style="padding-top: 0px">
==== Constant & Line ====
* Shown in black:
** $f(t)=1$
** $f(t)=t$
</div>
<div class="pz_slide_right_panel_two_thirds" style="padding-top: 0px;">
<div style="margin-left: -143px;">
<img src="geom_2.svg" height="420px">
</div>
</div>
</div>
<div class="pz_slide" id="slide_sensing_coordinates_3"
section="sensing_coordinates" codenodes="270,209,210,211,242">
<div class="pz_slide_heading">
=== Sensing Coordinates ===
<p class="subslide">(Slide 3 of 3)</p>
</div>
<div class="pz_slide_left_panel_one_third" style="padding-top: 0px">
==== Constant & Line & Parabola ====
* Shown in black:
** $f(t)=1$
** $f(t)=t$
** $f(t)=t^2$
</div>
<div class="pz_slide_right_panel_two_thirds" style="padding-top: 0px;">
<div style="margin-top: -9px; margin-left: -143px;">
<img src="geom_3.svg" height="429px">
</div>
</div>
</div>
<div class="pz_slide" id="slide_sparsity_basis"
section="sparsity_basis" codenodes="212,243">
<div class="pz_slide_heading">
=== Sparsity Basis Projections ===
</div>
<div class="pz_slide_left_panel_one_third" style="padding-top: 0px">
* Project sparsity basis elements onto $b_0$,$b_2$ plane.
* Each projection has only two coordinates.
</div>
<div class="pz_slide_right_panel_two_thirds" style="padding-top: 28px;">
<div style="margin-left: -100px;">
<img src="geom_4.svg" height="420px">
</div>
</div>
</div>
<div class="pz_slide" id="slide_aliases_1"
section="aliases" codenodes="215,236">
<div class="pz_slide_heading">
=== Aliases ===
</div>
<div class="pz_slide_left_panel_one_third" style="padding-top: 60px">
* Project sparsity basis onto
$\color{red}{b_0}$,$\color{green}{b_2}$ plane.
* $\begin{bmatrix}1\\1\end{bmatrix}$, $\begin{bmatrix}-1\\2\end{bmatrix}$,
$\begin{bmatrix}1\\4\end{bmatrix}$
* Each <font color="#C0C000">projection</font> is distinct.
* Need *two* projections to alias another.
* Impossible since *1-sparse*.
</div>
<div class="pz_slide_right_panel_two_thirds" style="padding-top: 40px;">
<div style="margin-left: -25px;">
<img src="spark_1.svg" height="350px">
</div>
</div>
</div>
<div class="pz_slide" id="slide_key_equation" section="key_equation">
<div class="pz_slide_heading">
===Key equation: $Ax=b$===
</div>
<div class="pz_slide_body">
* Columns of $A$ are *sparsity basis projections*
($\mathcal{SB}$-projections)
** E.g., $A=\begin{bmatrix}1&-1&1\\1&2&4\end{bmatrix}$ from last slide.
* $x$ = *sparsity coordinates*, $[x_0,...,x_N]^T$ (only $K$ non-zero)
* $Ax$ = combination of $\mathcal{SB}$-projections = *signal's projection* = $b$
* $b$ = *compressed sampling coordinates* ($b=[b_0,...,b_M]^T$)
* $Ax=b$ *under determined* (fewer rows than columns).
** Unique solution if no fewer than $2K+1$ columns of $A$ combine to zero
($2K+1=3$ above).
** i.e., $\mathrm{spark}(A) \geq 2K+1$
</div>
</div>
<div class="pz_slide" id="slide_reconstruction_1"
section="reconstruction" codenodes="196,207,238">
<div class="pz_slide_heading">
=== Reconstruction ===
<p class="subslide">(Slide 1 of 4)</p>
</div>
<div class="pz_slide_body" style="margin-top: -30px;">
<img src="recon1_1.svg" height="460px">
</div>
</div>
<div class="pz_slide" id="slide_reconstruction_2"
section="reconstruction" codenodes="196,207,203,250">
<div class="pz_slide_heading">
=== Reconstruction ===
<p class="subslide">(Slide 2 of 4)</p>
</div>
<div class="pz_slide_body" style="margin-top: -30px;">
<img src="recon1_2.svg" height="460px">
</div>
</div>
<div class="pz_slide" id="slide_reconstruction_3"
section="reconstruction" codenodes="196,207,203,204,249">
<div class="pz_slide_heading">
=== Reconstruction ===
<p class="subslide">(Slide 3 of 4)</p>
</div>
<div class="pz_slide_body" style="margin-top: -32px;">
<img src="recon1_3.svg" height="460px">
</div>
</div>
<div class="pz_slide" id="slide_reconstruction_4"
section="reconstruction" codenodes="197,208,254">
<div class="pz_slide_heading">
=== Reconstruction ===
<p class="subslide">(Slide 4 of 4)</p>
</div>
<div class="pz_slide_left_panel_one_third" style="padding-top: 40px">
* Traverse solutions of $Ax=b$
* $\color{blue}{\ell_0}$-norm difficult
* $\color{red}{\ell_2}$-norm inaccurate
* $\color{green}{\ell_1}$-norm just right
</div>
<div class="pz_slide_right_panel_two_thirds" style="padding-top: 10px;">
<div style="margin-left: -50px;">
<img src="norms.svg" height="350px">
</div>
</div>
</div>
<div class="pz_slide" id="slide_extensions" section="extensions">
<div class="pz_slide_heading">
=== Extensions ===
</div>
<div class="pz_slide_body">
Bridging the gap between intuition and applications:
* the relationship between bases
* uniform sampling (i.e., if you can't choose sampling instants)
* the signal class (e.g., periodic signals)
* the effect of noise
* choosing bases for large-scale applications
* computational tools
Companion module <a href="/?id=b0080" target="_blank">Compressive
Sensing Primer</a> addresses these areas.
</div>
</div>
<div class="pz_slide" id="slide_references">
<div class="pz_slide_heading">
===References===
</div>
<ol class="bib">
1. Bruckstein, A.M.; Donoho, D.L.; Elad, M.,
"[http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.102.4697,
From Sparse Solutions of Systems of Equations to Sparse Modelling of
Signals and Images]," SIAM Review, Vol.51, No.1, pp.34,81.
1. Engelberg, S.,
"[http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6145261&isnumber=6145243,
"Compressive sensing (Instrumentation Notes)],"
Instrumentation & Measurement Magazine, IEEE ,
vol.15, no.1, pp.42,46, February 2012.
</ol>
</div>
</div>