okay
this is gonna be
a short presentation
i problems
and the title of the talk is the general framework for a a choice D based indexing and retrieval with
a order data
it this is done with my
i a student should lee and formers to
and channel shown she who is now it to nokia research
and the
the goal is to do retrieval
and indexing indexing and retrieval
based on motion trajectory data
this is an an all problem people first begin to look at this
issue in the late nineties
and
a done a quite a bit of work
a most of the work centres round
yeah
doing it and using different modalities to reduce the dimensionality
we introduced in two thousand and three a method of doing it using pca
but then we wanted to extend it to working with
yeah have multiple trajectory simultaneously
and so we had to work in tensor space
and so we try to do something like pca intense or space
and we to the number of different techniques
a yeah
for
for tensor decomposition
based on
yeah
using higher order S D
and paris type model and another technique which we develop their cells
and the word various off
what we don't gonna focus on to they use the issue of how to deal
this
problem when you're dealing with tensor
but now of the query dimensionality does not mention
does not match the tense a dimensionality
meaning
you may have a different number of objects
a um
different like
or
in particular the case that we have here is different number of camp
so we actually are dealing with a different number of objects and different number of cameras
and the query we for example may have a single camera
and the database has multiple camera
and so the question is how can you do this the without having to we compute
a separate indexing for each scenario
and some than the talk a a a a little bit about
the invariance properties of the H of as U D
and how to apply to the indexing retrieval problem and present some experimental result
and so the basic scenario of using a a high order svd for indexing and retrieval
consist of
looking at multiple motion trajectory
and of from
yeah of multiple targets simultaneously
and then
i
having a compact representation in the form of a tensor as
and finally reducing the dimensionality and in this particular case but going to focus on a order is but
a more more properly people refer to it as tucker decomposition that would be a more accurate
in know processing that term
a choice of D of becoming brain
even though it's not use the terminology
uh they
origin of this is the following so if you look at a single
trajectory we can model it to say to L usually X and Y coordinates
of the trajectory over time
if we have two trajectories
we model it as a matrix
if
and we look at then i at the space of all of these
pair of trajectories
we get a tensor
a three dimensional array
and this is from what a single camera
if we now want to extend that for looking at multiple cameras in particular and this case two cameras
we have to three dimensional or rate or a four dimensional array
and so it forms a higher-order tensor
and you can continue this using multi modality you could like this
same trick for doing indexing and retrieval
yeah
for having different modalities
you can go higher dimension and higher dimension
no the reason we wanna work with a choice is with is because of the following theorem and what i
done the here is i've actually just the
loosely paraphrase
the the and words
that precise mathematical description of the theory
a paper
and about uh um
a page and a half of the paper
devoted to the proof of the theorem
but basically with the cr says is something which is quite into it
a we are all familiar with the for a transform
and if you have a multi the dimension of for a transform any now wanna take yeah the three dimensional
for transform that thing
and you now to take that two dimensional fourier transform only
it's sufficient to just simply look at the corresponding to the mentioned
they will have
the the you can just take the inverse with respect to the third one and will have the right
two dimensional fourier transform
and the reason for that is because of the orthogonality property
of the four yeah base
and the same thing is true a here
that is if i take a age of is he D and i decompose at it's decomposed into a tensor
and
in
unitary matrices
and so because of the a or orthogonality with the unitary property of those matrix
if find out think the scene it's sub tensor
so to get portion of the original tensor
and
i you can apply to a H of P D
i will get the same corresponding unitary matrices
for the dimensions of a in which i chosen for the subtensor tensor
and i do not need to calculate them again from scratch
which means of the corresponding indexing of the sub tensor
would be identical
a a of the same mold
or the same unit are a major
so if you want to precise mathematical description of what i just said and what's written here
it's in the paper and a proof of it is in the paper
and i should say one more thing this is uh a a result that was first
oh for three the mention tensor
yeah three order tensor
by that change how how as part of is a P D as at university of london
and what have done in this paper is
extended to our bit-rate dimension
the result
it
always true no matter what dimension
but it is a critically important thing for us because if we were to work with a different type of
decomposition
like paris
or parallel factor analysis
or can a call or any of the other one
a property fail
and we would be unable to do anything that we're doing in this paper
because you would have
to we compute everything from scratch for each such that
and so that that we have this property we can proceed along the lines of the original work that we
did for tensor decomposition
X this time we do it a lot a sub tensor is only
so the indexing part
and proceeds along the very same lines we have a H of ways P the we compute for the tensor
in this case the four dimensional tensor
and
a with take the mode
of the query
and do it
similar decomposition but this time we do it only along
the M
a modes if we choose
and then yeah are we
slice
and a T have
in index set tensor as
and with the number of index tensor is is computed
a the following
for
and for the retrieval procedure we simply
and a compare the query index
and yeah
to the to the query tensor that we that we have obtained before
and then a just simply do a frobenius norm between the two
so the algorithm to be compute
is essentially the same
as we presented a uh several years back
on
tensor
base
a a comparison for indexing and retrieval of motion trajectory
the main difference between this work and uh uh and our previous
is in our previous work it was generic didn't care what
tensor decomposition channel
and it applied it on the same
a dimensionality of then sir for the query
and for the data
and that's a
a strong assumption
yeah because we we have no control over the query size
and this is especially true when you're dealing with multiple cameras
and multiple camera tensor
a queries
because
not all cameras have access to the same trajectory simultaneously
and so of the main difference here is that we are only looking at the substance or
for which they gave available
and then
comparing compare and then
obtaining the corresponding a uh
query representation from our original in
which is index over all possible modality
and so here the uh experimental results for work
and uh these are collection of
tensor is of a from the caviar datasets from in
and
these are from two cameras sets
and this is the uh precision-recall recall curve
corresponding and this is for complete queries
and
the
resulting yeah uh these that the in matrix sizes
and here are are are the indexing time and retrieval time
and i should say that the uh
indexing time is
for a choice of be D are traditionally very good
and where they suffer is a which remote time
we do not
remedy this
and yeah
we
the of five well perform the retrieval times here
and what we have to say is that we have to pay this price
if we want to have the flexibility
of dealing with different yeah size subtensor as
in the query and a database
and
here we do the same thing but for partial queries
so they query and the data size are not same size
and these are the corresponding precision recall curve
so
short
our our am am main
messages
a shows with D or type decomposition
because of its sort the orthogonality
is particularly useful in applications where
you
do not know in advance
what are the dimensionality is and you need to make
a mix and match it during query time
and so we have applied this general principle in our case to motion trajectories
but it can be applied to any higher order data
an analysis with the retrieval or not
and show that it actually yeah
the
very well
thank you very much