Speech Transcript - SPARSITY-UNDERSAMPLING TRADEOFF OF COMPRESSED SENSING IN THE COMPLEX DOMAIN

0:00:14	okay
0:00:15	but have no i a wine
0:00:16	i today um my topic is
0:00:18	a a sparsity i'm thinking off of compressed sensing
0:00:21	in the complex domain
0:00:23	and um and a there yeah and uh this see the joint work with a for face there's session down
0:00:30	uh
0:00:32	a a a uh
0:00:33	first i will be a a a a a a brief introduction to come or scene and
0:00:37	but the face it uh transition theory
0:00:40	and uh and that we were talk about the uh
0:00:43	but sparsity a sending trade off of a complex signals
0:00:46	and uh of that would all people can
0:00:50	uh so i think everyone well be were from a uh a media with the
0:00:54	a a formulation and uh
0:00:56	take X is the uh
0:00:58	is what's we well lot no and uh
0:01:02	a sample uh a be the same how and here
0:01:05	and is to me that uh
0:01:07	but then a she the signal is kept a fine
0:01:09	and this uh sample size is three one
0:01:12	and also so uh the sparsity uh sparse as a label is K and uh
0:01:18	it is a number of a nine zero entries
0:01:20	uh so uh a base uh
0:01:22	a standard reconstruction approach
0:01:24	in uh come sensing is the basis pursuit
0:01:27	it it to minimize the error when nam uh up you to
0:01:31	uh to the uh in your system of question
0:01:35	and uh
0:01:37	so i i or uh many are standing as the rooms uh you "'cause" this
0:01:41	which uh describes how a course still be a sparse signal can be of on sample to why O
0:01:47	oh as do you present the whole underlying information
0:01:50	and uh
0:01:52	uh well you down how uh the
0:01:54	uh C around uh a based down coke earrings
0:01:58	or recent be a restricted isometry property
0:02:01	and also uh that's a repressed are
0:02:04	uh
0:02:05	uh describe the uh fits transition
0:02:08	and uh since
0:02:09	uh
0:02:11	the i P us of it the standard may start uh
0:02:15	however uh that it she the first to to unless there's uh based on earrings
0:02:20	are uh are P
0:02:21	is your already okay can
0:02:23	oh
0:02:24	um of to provide the sufficient condition and
0:02:27	uh you to the your at uh to can store uh can some T V in practice
0:02:32	and uh
0:02:35	uh a like a a it's the first two phase transition is uh
0:02:38	and necessary and sufficient conditions so it can be considered a the most the precise uh
0:02:44	so so far in sense that is
0:02:47	gives the uh this necessary and
0:02:49	speech condition
0:02:50	and uh here
0:02:52	you can uh
0:02:53	is uh it's fine is
0:02:55	the same size as R
0:02:57	uh it i is the same size and a do i is a uh second mission and uh here yeah
0:03:03	it the
0:03:03	sent me ratio and room use that small thus mask duration of is
0:03:07	uh the racial of the uh
0:03:10	uh that's quite a label oh to the us
0:03:14	simple size
0:03:15	and uh a it into be uh if uh we have uh noise is in
0:03:20	uh this area
0:03:22	uh you've we have uh less a simple why with the bus bats level is high they to use the
0:03:27	simple the signal has um a call uh nonzero entries
0:03:31	uh it is it is hard it it is your are hard to reconstruct the signal and
0:03:35	the basis pursuit to your discount to reconstruct the signal
0:03:38	and uh otherwise
0:03:40	a a a a uh the basis pursuit with sixty two
0:03:43	uh we cost to a signal if we have a uh most post
0:03:46	a a and of the signal use must uh much mouth uh simple
0:03:51	and uh based so
0:03:53	uh based on the the assumption that uh this uh this thing C matrix is costing met since and mode
0:03:58	at
0:03:59	that's uh the entries of the
0:04:01	uh of the uh matrix is uh randomly uh in the right to the for um costing distribution
0:04:07	and uh and uh the
0:04:09	uh signal time nation uh i have to an approach infinity
0:04:13	uh that uh the are where you that a shot shot boundary to divide
0:04:17	uh as the the plane L down to routine to two faces
0:04:21	it is uh
0:04:23	uh use a is a face
0:04:25	and uh the best the pursuit we
0:04:28	re uh will felt to uh reconstruct
0:04:31	see that with that overwhelming probability
0:04:33	and also in the a lower face
0:04:36	uh the base pursued a approach where O
0:04:39	sixty to reconstruct the sparse signal also with an overwhelming probability
0:04:44	and a here uh use only the uh week that's position and
0:04:47	uh we will talk about much about the definition
0:04:52	and uh
0:04:54	uh a oh
0:04:56	oh
0:04:57	so a like we uh consider a complex the signal and
0:05:01	i as two "'cause" the directions the for the first can the red it is
0:05:06	uh
0:05:07	and so far a complex that the case is seldom used are uh started it is of it especially in
0:05:12	the
0:05:13	uh phase transition us theory
0:05:15	and uh
0:05:17	the the it's practical the ration ladies
0:05:19	are you most a a a uh in many applications
0:05:22	a a complex that signal was i are to of for example in
0:05:26	magnetic resonance you meeting and also in you right a another
0:05:30	and uh
0:05:33	uh okay are i'll we don't uh
0:05:35	will uh we won't uh use uh use per up now this your co
0:05:39	uh derive mission of the uh of the
0:05:42	that's stress to the bound
0:05:44	uh we just to give a a a a P uh give way
0:05:47	oh you pure cool out of it is a based on some
0:05:51	it is O is the as settings in you are uh estimation else phase transition
0:05:56	it is as follows we use a a moment colour um mess or two
0:06:00	uh estimate the phase transition
0:06:02	and uh than men and she's in sample in code
0:06:05	a partial for a compact lost thing that is
0:06:08	uh the uh
0:06:10	that range of the matrix i in the rated form causing and pose
0:06:14	uh are re are any men your part uh are
0:06:17	uh
0:06:18	costing
0:06:19	and also a a complex up when the only and E
0:06:23	and uh
0:06:25	oh we use uh the signal uh use simple po is also a complex austin
0:06:29	and uh we always uh oaks considers a save room at it was for symphony ratio
0:06:34	and uh
0:06:36	a a number of problems of well where be soft uh with was respect you each combination of down and
0:06:42	the room
0:06:42	and this case a reconstruction of the sparse signal
0:06:46	it's declared a if uh such and pursuing you
0:06:49	uh is met
0:06:50	and
0:06:51	a a uh a a uh after and uh the uh
0:06:54	six uh a generalized in are the uh where B
0:06:59	uh used to estimate the phase transition
0:07:01	and as things uh
0:07:04	the phase considers theory of false the real valued signal was considered
0:07:07	the cat let's uh the signal time mission approach infinity
0:07:11	this in practical uh just stick though
0:07:14	oh use
0:07:15	is uh you word fine X so
0:07:17	oh
0:07:17	that a finite and
0:07:19	fess transition uh use that is defined as the
0:07:22	uh as the value of rule
0:07:24	uh with the probability of success is uh fifty percent
0:07:30	no
0:07:32	and uh as a reason we use it in our simulation is
0:07:37	uh called a some orthonormal expansion R Y minimization
0:07:41	and uh its entries themselves that's not some basis pursuit
0:07:45	oh both real and complex mad uh value
0:07:48	uh
0:07:49	the thing that takes and a signal and
0:07:51	and also uh
0:07:53	uh
0:07:53	the i present include to uh to act in the first why use the exact what an orthonormal expansion or
0:07:59	one
0:07:59	and it can work to use the optimal solution
0:08:02	and a what is a relaxed for uh relax the version of this one
0:08:06	and and it is uh new a to optimal and i to converge to mark fast
0:08:10	it to a you ks pun sure uh you the
0:08:13	exponential right
0:08:14	and uh
0:08:15	it a it can be uh
0:08:18	sure was that it is uh you to actually is a modified version of you try to right probably
0:08:23	it is uh this the fall
0:08:25	and uh X is the uh
0:08:28	is the current solution
0:08:29	and uh S is that a soft start coding operator
0:08:33	and uh
0:08:34	uh that is the uh
0:08:37	current will have kinda view
0:08:39	uh if
0:08:40	if here a you've uh that you close to be you close
0:08:43	well be minors and a
0:08:45	a X T so uh this is uh of
0:08:48	uh this is that you N form of you try to starstruck thresholding and here the to um modifications in
0:08:54	this uh use the relax the i one
0:08:56	the first one is that
0:08:58	uh
0:09:00	a way to sum of the uh
0:09:03	all of the that's to
0:09:04	uh of the signal recovered in the last two steps
0:09:08	is uh you'd slice the used that all of the uh the current uh the current solution but also a
0:09:14	petition no a term use a it in in the temper or read you
0:09:19	and uh
0:09:20	uh uh this to change is a a pretty improve the sparsity on same pin all but you achieve uh
0:09:26	its optimal one it is
0:09:27	to achieve the uh the tradeoff um but base the pursuit
0:09:31	and also uh if
0:09:33	oh if we uh
0:09:35	a a come from out uh the complex that
0:09:38	uh that matrix and uh signal here uh the not start holding we applied choose a and P two
0:09:46	and uh this is the of the of the success rate that is
0:09:49	uh
0:09:51	a here is the uh experiments fall partial for real same
0:09:54	and uh here here can see
0:09:56	uh
0:09:57	uh we we consider a different values of all
0:10:01	uh this the uh one the sick that mission
0:10:04	and uh
0:10:06	yeah uh the meat line is the
0:10:08	oh this transition for real valued to stick that was
0:10:11	and uh
0:10:14	yeah see uh
0:10:15	and use the complex uh that's case uh basis pursuit also "'cause" this uh you "'cause" that is uh that's
0:10:21	can station that is in the white uh a area
0:10:24	uh
0:10:25	a the basis pursuit we or uh reconstruct the smell thing though actually
0:10:29	and in the uh at uh in the black area
0:10:32	oh the uh best pursued uh will found to reconstruct
0:10:36	but this my signal
0:10:37	and
0:10:38	also
0:10:39	oh the phase transition occurs as about of the same
0:10:42	the same position
0:10:44	as as and a as a uh we can larger and larger uh the uh
0:10:49	the transition with a P can less and less
0:10:53	and uh
0:10:54	here uh uh uh use the uh of they have uh estimate the phase transition is is uh we also
0:10:59	can see
0:11:00	uh
0:11:01	but different values uh of the sick that i mission
0:11:04	and uh i was a phase transition occur
0:11:07	coincide which each other and also the uh the both uh are all of them uh superior to the real
0:11:14	uh if its transition
0:11:16	it it a to the office its shape for the real value the signal
0:11:20	and uh
0:11:25	and uh here uh used the not use power meant a consider an a different um if you same both
0:11:30	uh of the uh
0:11:33	a a sensing matrix uh encoding oh E a
0:11:36	oh complex gaussian but now only and turn the re
0:11:39	a if can see uh the
0:11:40	also this transition also occur it's it
0:11:43	a coincide inside which you as the which the each other ladies
0:11:47	uh the universe of of that's and see also do
0:11:50	across different um at since same bows
0:11:53	uh
0:11:56	as so a
0:11:57	a here is this some discussion it is
0:11:59	oh i
0:12:00	a from here of from uh
0:12:02	about uh
0:12:03	a as a nation K C
0:12:05	uh
0:12:06	the complex that sparse signal i use your to
0:12:08	we construct then the real that signal
0:12:11	oh you is that uh a less same are required to
0:12:14	exactly reconstruct signal
0:12:16	so oh
0:12:18	and you need to achieve uh explanation is that
0:12:21	it was supposed that step aside time
0:12:23	as sparsity level it's K uh it is they are K nonzero entries in the
0:12:28	a complex about the signal
0:12:30	so a U T C pavement to recover to care bear a boats from two and same pose
0:12:35	if we
0:12:36	oh consider a complex than the number it to real number
0:12:40	a uh are ever in a complex in case the are
0:12:44	but additional constraints and that is
0:12:46	uh that's uk variabilities effect at K pairs
0:12:50	so that is a
0:12:52	oh okay
0:12:54	uh so uh
0:12:57	uh this is the difference between uh uh from the uh
0:13:01	uh the real real white if the way that just consider a complex that a signal as a real wine
0:13:06	yeah why recombine the real and other major part
0:13:10	and so uh the
0:13:11	uh
0:13:12	it is that supports where be different of from the case uh we can see that the fear
0:13:19	and uh
0:13:21	uh so i here uh is uh what to find a
0:13:24	is
0:13:25	of of a complex that the uh sent P matrix and just stick though with a time mission
0:13:30	and a find that uh the best the pursuit to reconstruction approach
0:13:34	well i like "'cause" is uh
0:13:36	that's can see shane in play of the two and the room
0:13:39	and also the complex uh the phase transition thought complex men sick though
0:13:43	is a the rich use a real why you in says that's
0:13:47	which is uh
0:13:49	uh it requires a less po to re construct the complex signal
0:13:54	and also uh the universe not see of of that's transitions
0:13:58	also hold uh of course men different battery same
0:14:02	okay that's so thank you
0:14:07	right
0:14:15	when considering the complex case do you need to worry about
0:14:19	how are you
0:14:20	windowing doing a google phase transition how you drawing
0:14:23	uh you nonzero coefficients
0:14:26	but yeah uh that the the could uh
0:14:28	the
0:14:29	as the stick the of the a think to use uh
0:14:32	is from a costing in distribution
0:14:34	it is the real and the you menu part of the random
0:14:37	what i'm am six does it does does it affect the how you how you tool that's in the in
0:14:42	the uh we'll case nine that
0:14:44	uh basically is just design happens that affect the
0:14:47	uh the phase transition
0:14:49	uh not the not the magnitude of the coefficient
0:14:51	but do we know in the in the complex case what what is the constraints
0:14:57	that the can
0:14:57	right
0:14:58	if you by if like to my uh my coefficients make a complex gaussian of but different distribution
0:15:04	we like it a different stations
0:15:06	oh are you you are expand uh experiments i one it where with then but you don't know
0:15:12	be on the improved level
0:15:14	oh yeah or we can to give it here
0:15:25	okay

SPARSITY-UNDERSAMPLING TRADEOFF OF COMPRESSED SENSING IN THE COMPLEX DOMAIN

Compressed Sensing: Theory and Methods

Presented by: Zai Yang, Author(s): Zai Yang, Cishen Zhang, Nanyang Technological University, Singapore