0:00:16we can we that then i'm here to a two percent or people title
0:00:19in compression using iteration to and and aligned action
0:00:22call for supper thinking about
0:00:24and the to jack my kids you
0:00:28yeah so that the right my talk to the three parts the first part
0:00:31for of sparse presentations
0:00:33yeah and motivate how they can be used for image compression
0:00:37and some the issues have come up in this scenario
0:00:39and i trust and the second
0:00:41we we present a present or coverage
0:00:44yeah and then
0:00:45and sorry our contributions
0:00:46and the present results
0:00:47the for
0:00:49sparse person
0:00:51yeah well
0:00:52or a signal vector Y
0:00:54are also given a dictionary matrix E
0:00:56which is a complete unique that has more calls than it has
0:01:00row support and or signal dimension
0:01:04and then
0:01:06you also have a
0:01:10so this is a signal vector Y
0:01:12that's the dictionary a matrix T and this vector X
0:01:15is a sparse representation
0:01:16and what it does this
0:01:18a set like so that's a few columns with the channel matrix D
0:01:21and a waste and to construe
0:01:23to construct an approximation of signal vector Y that's a summation is which shown to be and a vector or
0:01:29so the aim is to use as few courses possible this dictionary matrix
0:01:33and obtain nonetheless a good approximation of Y
0:01:35so the way that one can construct this vector X
0:01:38there's quite a few ways where we use in our work score
0:01:41the matching pursuit algorithm
0:01:43yeah networks like so
0:01:45we initialize the residual vector Y
0:01:47and then the first
0:01:50step of iteration
0:01:51we choose
0:01:52yeah call
0:01:54a from the dictionary one that's most correlated to are vector
0:01:57we set the coefficient
0:01:59to the projection of the rest of that
0:02:01a call and then we check the condition if we have enough of that as to me X it otherwise
0:02:05we remove the contribution of the new atom
0:02:07to give system residual
0:02:09i didn't the back
0:02:10choose another at of another coefficient
0:02:11and so so this is the matching pursuit algorithm used
0:02:16and then once we have a vector X how do we use it
0:02:18in image compression there's
0:02:21are ways in which is or don't
0:02:22in the literature
0:02:23this is way we do it
0:02:25a which is more the standard we just take
0:02:29but something and each of them use
0:02:31one block
0:02:32a a to be the signal vector Y
0:02:34yeah and this the sparse approximation X so this is sparse vector X which is that
0:02:39representation of the signal vector Y
0:02:41this is the approach we use
0:02:42and the decide which is to come up here
0:02:44the first one is
0:02:47which dictionary D we use
0:02:50and then the
0:02:51are are solution here is to use a tell which is a new their structure dictionary
0:02:55yeah i i've the duration to like dictionary
0:02:58so that's the first sign we should seconds issue issues
0:03:01hi we choose the sparsity of the blocks web image the hold the we choose how many atoms
0:03:05we used to represent each one of this block
0:03:08that gonna process for something the new approach
0:03:10just a little
0:03:11rate distortion this criterion
0:03:13distribute atoms at the image
0:03:15and the method is we should
0:03:17well as we have the spectra X
0:03:18for each block
0:03:20then how do we construct a bit stream from from that
0:03:23and the were just gonna use standard approach just to that used on you know from a decision of the
0:03:26coefficients have an encoding
0:03:28a fixed and code
0:03:29yeah for the
0:03:31so then the next
0:03:32part of my presentation one
0:03:34is going to address this to decide issues
0:03:37the the choice
0:03:38and that the distribution
0:03:40that's speak an addiction choice
0:03:42yeah so just do want to date
0:03:44the dictionary structure that we propose
0:03:46yeah we drawn here
0:03:48yeah the sparse approximation creation
0:03:50and this is a dictionary D which is vector
0:03:53a fast matrix it has more columns and or signal dimensions
0:03:58so that
0:03:59but it could be that since interesting "'cause" that's what we it's the sparsity of the vector X
0:04:03and that's what we want to one a sparse
0:04:05vector X
0:04:06yeah and then
0:04:08the them of a complete D is
0:04:19the map and D S
0:04:21yeah that
0:04:22the more computationally expensive it is to find the best
0:04:25you i
0:04:26the represent
0:04:27the signal vector Y well
0:04:29is still
0:04:30the second issue here
0:04:32and at that point is
0:04:33well them more absence we have a
0:04:35then the more expensive it is in terms of coding rate
0:04:39yeah yeah to to
0:04:42so that that that the fires of the atoms used that as an issue
0:04:45so for complete mess
0:04:46but is the sparsity but it also
0:04:48also increases the complexity of the decoding system
0:04:51and the coding rate
0:04:53so what we're going to do is we're going to structure
0:04:55the dictionary matrix T meaning that we're going to constrain and the way in which groups of atoms can be
0:05:02so this is the the motivation to
0:05:05the high and duration two
0:05:06and a like dictionary that this constraint
0:05:08are are going to a allow was to enjoy
0:05:10to do over complete and the sparsity of the loop
0:05:15less the constraint or without going to
0:05:17penalty in terms of
0:05:19a compact
0:05:20and coding rate
0:05:22but just a game
0:05:24i i i
0:05:26so what iteration to here
0:05:30to to illustrate that i just draw
0:05:32the matching pursuit
0:05:35block diagram for of two slides back
0:05:37is the jury matrix D
0:05:40which is constant
0:05:41for of the durations
0:05:43for the standard case
0:05:44now in our case and i three to in case
0:05:46what we do is we make this matrix D a function of the iteration
0:05:50like so
0:05:51no for with
0:05:53that that's what we call it tuition to
0:05:54because the chance of intuition
0:05:58and a
0:05:59which is the i have the same number of atoms and
0:06:05the i T king
0:06:07iteration iteration scheme
0:06:08yeah it
0:06:09more of a complete right because we have a lot more i
0:06:12i i to choose from
0:06:14but at the same time
0:06:15the complexity
0:06:17and select "'em"
0:06:18and that in this block
0:06:20the same because we have a as columns
0:06:23a when we use the would be back here
0:06:26so we have a problem compared under matching pursuit
0:06:29and also a proper coding rate we use
0:06:32code to encode
0:06:34yeah to in this just the coding rate is was going to be little to of and
0:06:37so this is structuring approach
0:06:39E allows us to enjoy over complete is
0:06:41we control
0:06:42complexity and coding rate
0:06:48i just
0:06:48drawn here
0:06:52the majors is yeah i i a we're structure so this is the iteration to structure right
0:06:57i where are is the matrix D i
0:07:00yeah yeah
0:07:01and the recording train this structure
0:07:03and the training scheme is very simple we use a top-down approach
0:07:06i so we assume we have a large set of
0:07:08training vectors Y and use all strain vectors to train
0:07:12the first layer
0:07:13the one
0:07:14and then once with trained one fixed it and we compute
0:07:17the rest use the output of the first layer so we have the rest used for the try training set
0:07:20that used to train a second there
0:07:22and so that are that to the last
0:07:27so this is
0:07:30not taken i layer
0:07:32of the i T structure at the last flight
0:07:34so that's that here
0:07:36the input that in progress you and they are dress you
0:07:39i know i
0:07:40i'm going to explore geometric we what happens when as
0:07:43you want of two atoms of this way
0:07:45so here are
0:07:46the input was that use of this the class to just use this great out here
0:07:50and then this subspace it like the screen it here
0:07:54is the i was just pose
0:07:56of the screen
0:07:57so as you can see
0:07:59in that there is a reduction of dimensionality
0:08:01between the one that was use uh i mean one
0:08:03and i rest used for i
0:08:06i rest rest of space this
0:08:08well let's dimensionality mention of that must respect
0:08:10and that was for the but i am here the but what the red
0:08:13reduces dimensionality by one
0:08:15for X
0:08:16in progress
0:08:17the problem is that
0:08:20yeah the union of this two
0:08:22rest of sub-spaces
0:08:24none of us that's entire
0:08:26original signal space
0:08:28so this is a of
0:08:29as this means that the next
0:08:32the i that's one from the next layer
0:08:33it's going to a have to address the entire signal space
0:08:37so this is what to date
0:08:38yeah why of an operation we propose which works like so
0:08:42yeah so no each
0:08:46and alignment
0:08:48yeah and this
0:08:49a of takes
0:08:51for example the green at them
0:08:52and all items
0:08:53with the vertical axis
0:08:55and this score three example
0:08:57and it takes
0:08:58rested know
0:08:59space of this i
0:09:01i also
0:09:02the horizontal something
0:09:04and does the same thing that the but at the and are rest of space
0:09:07they but of is again going to file
0:09:09oh the for simple thing so able two
0:09:11of sub-spaces coincide
0:09:13and they're right on the
0:09:14or something
0:09:15meaning that i i was of space
0:09:17using this
0:09:18we rotations still
0:09:21i get get and joyce can just dimensional
0:09:25that we have about T in choosing
0:09:28it is
0:09:28rotation a she's is that i
0:09:30i was vertical axis and
0:09:33i was of is with
0:09:34the for pretty so we further change shoes
0:09:36a rotation
0:09:37majors as or are a lot of interest is
0:09:39so that they are also for
0:09:42i rest of sub-spaces
0:09:45yeah i have
0:09:46principal component
0:09:48that i was alright right
0:09:50in this for some so
0:09:51the first principal component
0:09:53of the red
0:09:54so space is going to follow along this axis
0:09:56a like was the first principal component
0:09:58of of the screen subspace is going to four
0:10:00a a lot of this
0:10:01and so one for the
0:10:06so now i'm just going to read
0:10:08are are are are are
0:10:09previous i i seen this modification
0:10:11this an interest
0:10:12but occasions
0:10:13and that's what i have a year
0:10:15so this is my and to a she two and one dictionary
0:10:18and as you can see no i have a
0:10:20well alignment i tricks per at
0:10:22yeah and because
0:10:24i i went information
0:10:28but atoms
0:10:29of the matrix with the where with this
0:10:31at this matrix here
0:10:32existing a so must also produce dimensionality
0:10:35the change my as i just what i estimator and
0:10:40so this is a are
0:10:41solution to the first sign we should what which was which to charge
0:10:45this is a each way to use because it enjoys over a complete
0:10:48we do so that C yeah in control coding rate
0:10:54now the second is issue
0:10:56well as at the distribution of process
0:10:58the image
0:10:59here we also have a
0:11:02contribution in this paper
0:11:04a a of that the standard approach used to
0:11:06so a are specified the number of atoms the number of those here is
0:11:10yeah yeah
0:11:12this is the sparse approximation
0:11:14of the input signal vector Y
0:11:16at the standard approach is us to apply a
0:11:19and or
0:11:21to this approximation to are so we choose
0:11:22this this was over at times that satisfy some at maximum or
0:11:26that's a standard approach
0:11:28you are the problem is that we have
0:11:30B blocks
0:11:31in the image
0:11:32and we want to choose the sparsity L and
0:11:34each one of this blocks Y and
0:11:36so we we
0:11:39a a a a a a global optimal
0:11:41yeah sparse functions
0:11:42approach like so
0:11:43so we want to choose a sparse sparse is of all the routes
0:11:46so that they can do
0:11:47a can look at it
0:11:49yeah block representation a
0:11:51is minimize subject to a constraint
0:11:53on the can be but the root of a box
0:11:56oh this is not very good
0:11:58so we propose
0:11:59yeah yeah an approximate
0:12:02which works like so
0:12:04yeah we first initialize a sparse is as are also said well of one to zero
0:12:08and then choose the block
0:12:11the second step that of course
0:12:12the biggest problem
0:12:13in terms of arrival
0:12:15distortion reduction it
0:12:17so this is
0:12:18a a this here is the distortion
0:12:20but we used an
0:12:22think occurred sparsity a
0:12:23and this is the
0:12:25potential distortion if we add one more at
0:12:28to twelve summation in two
0:12:30so this is the you or the reduction rather in distortion
0:12:33and that this is the
0:12:34called a penalty
0:12:37i i i and this one i here
0:12:39so this is distortion
0:12:43gain a distortion reduction distortion of bit
0:12:45and this is the power of
0:12:47the because problems that true
0:12:49and those it turns
0:12:50so we just a a one more at to this choice and
0:12:54i i its sparsity P and the P
0:12:56scheme for the second step just a block
0:12:58yeah i i didn't and you add to the choice weapon and so one all still
0:13:03the but but just for the image is one
0:13:06so that's
0:13:08that was a second
0:13:09the site issue
0:13:10and now have some to percent
0:13:12yeah yeah for of but that's that the using is as follows
0:13:15yeah we use the product that the set which is is that a set of non homogeneous face images
0:13:19so the right conditions of the poses not controlled
0:13:23in as we take a training set of the
0:13:25a four hundred
0:13:27i that
0:13:27training and just and that i a test set a hundred images of this or the showing just right
0:13:32so we use this training set to train
0:13:34i i type structure
0:13:35iteration to align dictionary structure
0:13:38yeah and then test
0:13:40use this
0:13:41yeah this test
0:13:44so just examples of for it
0:13:51so here are we distortion or cells
0:13:53i have a
0:13:54i a first of all this curves
0:13:57E i was or a one hundred test image
0:14:00so is it's a two thousand but here that the sort
0:14:03right now i
0:14:04this is true in last
0:14:07i then i have a three
0:14:08curves for i
0:14:09yeah this is
0:14:11this but curve as with lots of size times can
0:14:13the green of about size twelve and twelve
0:14:16and this one
0:14:16for a of size sixteen ten sixteen
0:14:18so as you can see i to of is quite claim
0:14:22is not and all rates
0:14:25even greater than for nice
0:14:27and than at highest rates
0:14:29this is
0:14:29still at of point nine
0:14:34the just one out
0:14:35that the coding scheme used to encode
0:14:37the sparse vector X is present
0:14:39so was
0:14:41yeah in in there rate distortion
0:14:43yeah are work at that
0:14:45transform that we use
0:14:46yeah and the are
0:14:47oh of the of the proposed to
0:14:51at the location scheme
0:14:53okay so now i also have some
0:14:56i only to the results
0:14:58slight have a a two images here
0:15:00that i code using the a two thousand
0:15:03and i
0:15:04as you can see
0:15:05because for use and i are better than of that can you
0:15:11concluding remarks i started by
0:15:14summarizing a really let's possible since the presentation are and how we can use the
0:15:18yeah in image compression
0:15:21and then
0:15:22in doing so we ran to three decide issues
0:15:24the first one was what transformation
0:15:26we applied to the signal
0:15:28or to the image blocks well
0:15:30what dictionary use
0:15:31there we propose using
0:15:33and new dictionary structure
0:15:34the i that's true
0:15:36yeah and then there was a question of how do we
0:15:38i i atoms across the image
0:15:41and there are proposed new
0:15:43gram with distortion this approach
0:15:45and then
0:15:46in terms of
0:15:47and and the are X just use very standard approaches
0:15:51so there was nothing there
0:15:52yeah yeah but the best results
0:15:54yeah we're we're good
0:15:56a from a given to of house
0:15:58yes is was only for the cost features
0:16:01i thank you very much tension
0:16:03you have any questions or that that's
0:16:10i i question
0:16:21do you can put but it exactly a scheme
0:16:24and so i'm how have compared the
0:16:28so there's
0:16:29there's a few things that come to play here
0:16:31so the vector X
0:16:32we have to specify in terms of a i in this is
0:16:35and one does coefficients
0:16:37so for that the since we use the fixed month code
0:16:39it's just going to be a a range up to
0:16:41of the number of that
0:16:43and and for the coefficients week while custom assuming the quantizer
0:16:46then we use a huffman code from that
0:16:51by a special property of the gain of the coefficients because many you think can be most likely that the
0:16:56value of the complete and you give the exponent at no is and that's
0:17:00but does that something of multiple red
0:17:06one more question right you
0:17:12so recorded sessions so we need a microphone
0:17:18a close to encode addiction on yeah right
0:17:21no we we make the assumption that lectures available
0:17:24at the decoder
0:17:28okay let's think a speaker