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

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

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

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