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EUSIPCO 2012 20th European Signal Processing Conference

August 27-31, 2012, Bucharest

Geometric Space-Time Audio Processing

Augusto Sarti and Fabio Antonacci

Image and Sound processing Group

Dipartimento di Elettronica ed Informazione

Politecnico di Milano, Italy

A. Sarti and F. Antonacci Geometric Space-Time Audio Processing EUSIPCO 2012 , August 27th, 2012

2 Outline (1/2)

• Background and motivations – Wavefields, Rays and Beams

– Plenacoustic function

– Physical issues

– Perceptual limitations

• Building blocks – Domain: ray space

– “Sharp” primitives and descriptors

– “Graded” primitives and descriptors

• Modeling of soundfields – Ray tracing: overview

– Beam tracing: overview

– Visibility tracing

– Notes on the extension to the 3D case

– Aerial view


3 Outline (2/2)

• Measurements and images

– Acoustic measurements and corresponding constraints

– Acoustic images

– Plenacoustic images

• Applications

– Analysis • Estimation from acoustic measurements

• Estimation from acoustic and plenacoustic images

– Acoustic scene rendering/synthesis

• Conclusions and perspectives



Part 1:

Background and motivations


5 From wave equation to rays

• Generalized D’Alembert equation

This equation can be reformulated as a differential relationship btw time (L-tf variable s) and space (L-tf variables sj), which describes propagation

– Any function that complies with this relationship is a suitable candidate!

– The dimensionality of the parameter space of all possible propagation modes is far from being manageable

– We need structuring!!!

D’Alembert equation

),(),(,

, tux

btut

aji

i

j

i

ji

kk

k

k xx

ji

i

jji

k

k

k sbsa,

,



• Can we operate in a “white-box” fashion? – Including sources and environment in the modeling process

• How?

– Strong assumptions on propagation, sources, etc.

– Working with well-established geometric and algebraic tools

• Why?

– Divide and conquer approach

– If we do things well, the method will be sufficiently general and we will be able to cover a wide range of problems (calibration, structure from sound, simulation, rendering, etc.)

Issues



• Instead of implementing and modeling the PDE, we implement and model the general solution of the PDE

• In 1D this is quite straightforward

– Example: Digital WaveGuides [Smith1985,Smith1986,Smith1998] • Accommodate discontinuities in the medium and boundary conditions

(scattering)

• Boundary conditions become lego blocks for the implementation!

• The extension to 2D or 3D is all but trivial

Roadmap



• Helmholtz equation:

• General solution

• Eikonal equation:

Eikonal equation

0),(),(2

22 xx

P

cP

)(),()(),( xxx TjeASP

0)(

1)]([

2

2 x

xc

THp: • High frequencies • Non dispersive medium

Eikonal equation operates only on the direction , i.e. the direction orthogonal to the wavefront . This direction is referred to as Acoustic Ray

)(xT

)(xT



• Point-like source:

• Acoustic Impulse Response (AIR)

Eikonal equation: examples

S The wavefield emitted by a point-like

source is described through a plurality (i.e. infinite) rays departing

from the source location

S R

t

h(t)

Rays bounce according to the Snell’s law when they meet reflectors.

In the AIR each peak corresponds to an acoustic path linking source and

receiver


10 From rays to beams

Non-planar wavefronts require a plurality of rays need compact representation of a bundle of rays

acoustic beams

S

An acoustic beam is made of all rays that originate from the same pt and meet the same (planar) obstacle along their

propagation

reflectors

beams


11 Beam description

• Accommodates wave propagation in enclosures

– A source with a given radial pattern can be seen as producing a number of uniform beams (piecewise constant approximation)

– Every time a beam encounters a reflector along its propagation path, beam splitting occurs

– Inherently open-loop modeling of propagation

Pros

• General enough to accommodate spherical as well as planar wavefronts

• We can separate the geometric aspects of propagation from the modifications

that the signal undergoes during the propagation (attenuation, dispersion, etc.) o We will focus on geometry of propagation and worry about signal modifications

afterwards


12 Beam representation

– Only interactions with the environment that preserve the point-like nature of sources (both real and virtual) can be readily modeled

• In order to accommodate diffusive surfaces, we need to introduce simplifications

– The representation based on rays is not accurate at low frequency, as diffraction enables the wavefield to turn around obstacles Geometrical Theory of Diffraction (GTD)

Cons

s

s’

s

r

$ s’!

Specular reflections Diffusive reflections


13 Additional assumptions

• 2D or 2Dx1D (separable) wavefield

– Extension to 3D is possible

• The environment is made of uniform and piecewise linear reflectors

– Generalization to curved reflectors is possible through the use of piecewise linear reflectors


14 Perceptual limitations

What comes at handy:

• Phase ignored at high frequency

• Space perception drops at low frequency (wavefield is the same for both ears)

• Limited sensitivity outside vocal bandwidth

• Space perception limited to a fairly narrow bandwidth


15 Plenacoustic function

• The Plenoptic function [Adelson1991] is a seven-dimensional function f(x,y,z,q,W,l,t), which describes the intensity of the light field seen at location (x,y,z) when looking in direction (q,W)

– Audio wavelengths are 106 times larger than light wavelengths difficult to construct a device that measures the sound pressure at any point along with its gradient

• The Plenacoustic function [Ajdler2003][Ajdler2006] is the four dimensional function p(x,y,z,t) which describes the sound pressure at location (x,y,z) and time t given an acoustic event in a room

– Knowing the plenacoustic function means knowing the soundfield at any possible location of the microphone

– Applications [Ajdler2006a]: • Head Related Transfer Function (HRTF) interpolation

• Interpolation of the Acoustic Impulse Response along a trajectory

• Modeling of the Room Impulse Response along an arbitrary trajectory

• …


16 Plenacoustic function

• We want to work with rays, therefore we need to recuperate info on direction

• We CAN extract this information (field along a direction) using arrays of mictophones (with limits due to the aperture problem)

• If we keep track of these limits, we can go back to the equivalency btw plenoptic and plenacoustic function definition

• The 2D plenacoustic function p(x,y,q,t) is the soundfield measured at location (x,y) at time t and travelling in direction q

– This representation is inherently based on rays it is convenient to adopt a representation for which rays appear as an elementary primitive

– Extension to 3D is possible (see later on)

What geometry should we use?


17 What geometry?

• Needs

– Beams (and rays) originate from points

– We need to treat planar and spherical wavefronts in a similar fashion

– We need to simplify the modeling of reflections on planar surfaces

• A natural choice: projective geometry (and beyond...)

– A “point” in P2 is a ray

– A planar wavefront originates from a source at infinity (in homogeneous coords it is a point like any other)

– A reflection on a planar surface is a homography (linear operation) Adopting projective geometry means assuming that all rays meet in some center of accumulation (projection center). Whenever this center changes (e.g. through reflection) we need to devise a projective transformation that accommodates such a change

– Extension with oriented projective geometry



Part 2:

Building blocks


19 Domain: ray space

• Acoustic ray

– In 2D a ray is characterized by three parameters. 2 possible representations:

• a point (that the ray passes through) and a direction

• the parameters of the line that the ray lies on

– We adopt the second parameterization:

– Note: rays defined in this way are also characterized by an orientation

– Projective geometry P2 does not account for the orientation

Ray characterization

],,[ 321 llll

Ray travelling from A to B Ray travelling from B to A

Oriented Projective Geometry (OPG) T2 [Stolfi87] is needed

xA

xB xB

xA

≠


20 Domain: ray space

• Given a line with parameters , all the points that lie on it must satisfy

• A ray is a point in the oriented projective space T2

• Its coordinates form a class of equivalence: – are all the same ray

– are all the same ray, with opposite orientation to the set

Definition

Tlll ],,[ 321

0,],,[ 321 kklklkl T


0k

l1

l2

l3(l1,l2,l3)

The Euclidean space spanned by such homogeneous coordinates of lines is called ray space

The acoustic ray is represented in the ray space by a half-line line with k>0


Tlll ],,[ 321

032211 lxlxl

Tlll ],,[ 321


• Acoustic source in the Euclidean space:

• Homogeneous coordinates corresponding to :

• This point lies on the line l=[l1,l2,l3]T iff xTl=0.

• If x lies in the intersection between l1 and l2, it will also lie in the

intersection between kl1 and kl2 (k>0 or k<0)

therefore kx will be the same point as x

• This means that x is homogeneous as well

21 Geometric primitives Point-like sources and receivers

T

sss xx ],[ 21x

sx T

sss xx ]1,,[ 21x

( )( )

( )( ) 0

0

0

0

2

1

2

1

lx

lx

lx

lxT

T

T

T

k

k

k

k


• A point is identified by the set of all rays that pass through it.

• In the ray space this set of lines corresponds to a plane passing through the origin, whose orthogonal is [x1,x2,1]T

• The plane [x1,x2,1]T divides the ray space in two half-spaces:

0lxT

s

0lxT

s

− lines going counterclockwise around the source

− lines going clockwise around the source

Geometric primitives Point-like sources and receivers 22

l1

l2

l3

(0,0,0)

P*

x

y

P(x,y)


Geometric primitives

• In the 2D Euclidian space the oriented (bounded) reflector is defined by its endpoints xA and xB

• The reflector divides the Euclidean space in two half-spaces: points lying at the left and at the right of the reflector.

• In the ray space the oriented reflector is defined by all the rays that intersect the reflector and travel towards one of the two half-spaces, as specified by the orientation of the reflector.

Bounded reflectors 23

An oriented reflector (AB) corresponds in the ray space to all the rays intermediate between the

planes A* and B* and with the same orientation of xA-xB



• A non-oriented reflector becomes a pair of oppositely oriented lines in T2

• An infinite non-oriented reflector is a line whose endpoints are at infinity

• All the rays in the geometric space cross an infinite reflector

Infinite Non-oriented reflector 24

an infinite non-oriented reflector is mapped onto the whole ray space



• For a compact visualization the “reduced” ray space is used

• The “reduced” ray space is obtained by intersecting the ray space with a plane, e.g. l3=1.

“Reduced” ray-space 25

Bounded reflector

Ray space Reduced ray space

Point

The point is represented by a line in the reduced ray space

The reflector is represented by two oppositely oriented beams in the reduced ray space



• An acoustic beam is the set of rays with origin xS and directions intermediate between lA and lB

• In the ray space:

– xS corresponds in the ray space to the plane with parameters xS

– lA and lB are lines both lying on the plane xS as they pass through xS .

Acoustic beams 26

xS

lA

lB

xS

lA

lB

The beam in the ray space corresponds to a portion of the plane xS limited by the directions lA and lB

l2

l1

l3

l1

l2 lA

lB

Ray space

“Reduced” ray space


Duality btw. Geometric and Ray spaces

Summary of representations in geometric and ray spaces

27

Geometric space Reduced ray space

Ray

Point

Oriented reflector

Beam

x1

x2

x1

x2

x1

x2

x1

x2

l1

l2

l1

l2

l1

l2

l1

l2


Duality btw. Geometric and Ray spaces

• A duality relationships exists between geometric and ray space:

– A point becomes a line

– A line becomes a point

– An (oriented) beam becomes an oriented segment

– A segment becomes an (oriented) beam

28

In the literature the ray space is also referred as dual space [Antonacci2008]


Visibility region

• Visibility region:

– Defines if (or what portion of) a reflector is visible from another reflector.

– In other words: which rays departing from reflector AC will hit reflector CD?

Non-oriented Reflector to reflector visibility 29

The visibility region V(RCD) of the reflector RCD as seen from the reflector RAB is the intersection of the duals of RAB and RCD

Geometric space Ray space


Visibility region

• For oriented reflectors, the visibility region is the intersection of the duals of the oriented reflectors

Oriented Reflector to Reflector visibility 30

All rays originated from a reflector RAB falling onto reflector RCD form the visibility region of RCD from RAB


Visibility diagram

• Visibility diagram (visibility from a reflector):

– Is a function that defines on which reflector (if any) a ray departing from reflector RAB falls

– In other words: given a ray l departing from RAB , which oriented reflector meets, if any, along its propagation?

Visibility from a reflector 31

The visibility diagram D(Ri) is the overlay of visibility regions V(R) of all the reflectors in the environment, according to front-to-back

order of reflectors


Visibility diagram Visibility from a reflector 32

Example 2

Example 1


Global visibility

• Global visibility:

– Defines the mutual visibility among reflectors

– The global visibility is used in the modeling of propagation, as it speeds up the evaluation of the visibility from a point (tracing of acoustic beams)…see next slides

Visibility of the whole environment 33

The global visibility is the collection of the visibility diagrams for all the reflectors present in the environment


Point to window visibility

• Given an acoustic beam that departs from a reflector Ri (i.e. it is originated from a reflection), what can we see of the real world from the position S’ behind Ri?

• How can we swiftly extract this information from the visibility diagram D(Ri) ?

34

The visibility from S’ corresponds to the lookup of the visibility diagram D(Ri) along the line S’*, representation of S’


Note: what we see from behind the reflector is what we see from the surface of the reflector Lumigraph [Levoy1996]


Point to point visibility

• Once the location of a source S’ behind a reflector and the position of a receiver R are specified, which is the ray that goes from S’ to R?

• How can we swiftly extract this information from the visibility diagram?

35

The visibility from S’ to R is the intersection between S’* and R*, representations of S’ and R on the visibility diagram D(Ri).

R*


Note: pt. to window and pt. to pt. visibility are used for modeling the sound propagation in enclosures (later on…)


“Graded” primitives

• Until now:

– Point-like sources and receivers represented in the ray space as planes

– Reflectors are represented as beams in the ray space

– Beams as intersections of the duals of sources and reflectors (i.e. segments in the reduced ray space)

• But…something is missing

– Sources, receivers and reflectors are “omnidirectional”

– Next step: assigning directive patterns to the geometric primitives

• From a different viewpoint:

– The ray space is the domain of the plenacoustic function, the directivity information of the geometric primitives constitutes the co-domain.

36


Source radial pattern

• Acoustic source in the ray space: xs ↔l*s

• The line l*s defines the domain on which the source radial pattern is

mapped to the (reduced) ray space.

37

Given an acoustic source source at xs=[x1s,x2s]T, its beampattern f(q) is

mapped in the reduced ray space to f(l1*, l2

*), with

ss

ss

xxl

xxl

21

*

2

12

*

1

)tan(

1

)tan(

)tan(

q

q

q

)1

arctan(

)tan(

*

1

*

1

*

1

*

2

s

s

xl

yl

ll

q

q

Analogous relationships hold for microphone sensitivity radial patterns


Acoustic camera

• An acoustic camera is a device that captures all acoustic rays that pass through its optical center xc.

• Example: the optical center xc in an ordinary microphone array is the central microphone

• Representation in the reduced ray space: the line lc*

corresponding to the optical center of the array

• The acoustic image is mapped on lc* as for the acoustic sources

and receivers

From geometric to ray space 38

Example in 3D


Acoustic camera

• Limited information about the acoustic scene can be extracted from the acoustic camera.

– Reason: the acoustic environment is observed along the line lc* , which

is a 1D subspace of the two-dimensional reduced ray-space

• Idea: observe a two-dimensional portion of the reduced ray space to extract more information

– Such device should be able to read the intensity of all rays passing through it, not only those passing through the optical center

Limitations 39

Plenacoustic camera


Ideal plenacoustic camera

• The ideal plenacoustic camera is a device that is able to capture all the rays that pass through it all points on the device are optical centers

• Example:

Definition and representation 40

• A plenacoustic camera CD • An acoustic source in S

• Reflector AB. • Along with the real source, also rays coming from the image source S’ are acquired • The camera acquires

oAll the rays coming from S oAll the rays departing from S and reflected by AB, which can be modeled as departing from S’

• The plenacoustic image is mapped on the ray space

Geometric space


Ideal plenacoustic camera

• The ideal plenacoustic camera is a device that is able to capture all the rays that pass through it all points on the array are optical centers

• Example:

Definition and representation 41



Ideal plenacoustic camera Definition and representation 42

Information contained in the plenacoustic image: o A segment on the line S, representation of S in the ray space

The segment is determined by the intersection btw. the region of the ray space visible from the plenacoustic camera (rays are bound to pass through the plenacoustic camera)

o A segment on the line S’, representation of S’ in the ray space

The segment is determined by the intersection btw. the region of the ray space visible from the plenacoustic camera and the visibility region of the reflector AB (rays are bound to pass through the region visible from the camera AND through the reflector AB)

Application: through the analysis of the plenacoustic imge it is possible to localize S and S’


Real plenacoustic camera

• A device with infinite optical centers is unfeasible.

• Real plenacoustic camera [Markovic2012]: a linear microphone array on which a finite number of optical centers are defined

• Implementation:

43

• The array is organized in sub-arrays.

•Each sub-array is composed by N>2 microphones •A sliding window selects microphones in each sub-array

• An acoustic image is acquired for each sub-array.

• The optical center of each acoustic image lies in the center of the sub-array.



• Example:

• Each sub-array is composed by a limited number of microphones and its extension is limited acoustic images are blurred and exhibit a limited angular resolution

• The microphone array has a limited size only a portion of the whole ray space is observed.

Example and Limitations 44



• Minimum angular distance between two acoustic sources to be observed as separate primitives in the acoustic image (and therefore in the plenacoustic image)

Finite resolution 45

))(cos(||||2

||||||min

'2'1

2

'1'2

2

'2

2

'1 q

mpmp

ppmpmp

ss

ssss

According to the specific technique adopted for the computation of the acoustic image (MVDR, Music, …) a different resolution min can be achieved. Two sources ps1’ and ps2’ must honor in order to appear as distinct in the plenacoustic image



Part 3:

Modeling of soundfields


Modeling propagation

• Goal: given

– the geometry of the environment

– the location of the source and of the receiver

provide an estimate of the Acoustic Impulse Response btw. source and receiver

• Complex task using FEM or BEM approaches.

• Approximate solutions come from geometrical acoustics.

• Some techniques:

– Ray tracing [Krokstad1968], [Rindel1995], [Naylor1992]

– Image source [Allen1979], [Kirzenstein1984], [Huopaniemi 1997]

– Beam Tracing [Funkhouser1998], [Laine2009], [Dadoun1985]

– Visibility-based beam tracing [Antonacci2008, Markovic2010]

47


Ray tracing 48

Steps:

1. Cast a finite number of acoustic rays from the source 2. Propagate through the environment 3. Interact with objects in the environment

Determine the first intersection of the ray with the scene

surface (ray shooting ) Reflect rays Some sound is absorbed (absorption coefficient depends on material)


Ray tracing Implementation 49 49

Steps:

1. Cast a finite number of acoustic rays from the source 2. Propagate through the environment 3. Interact with objects in the environment

Delay:

Attenuation:

Room impulse response

Anechoic recording or synthetized signal

Output sound

d

c1

d

*


Ray tracing Pros and cons 50

Disadvantages:

Inefficient Suffers from aliasing New tracing whenever either the source or the receiver

moves

Advantages:

Simple to implement; can model specular reflections, diffusion and diffraction; models all types of surfaces (e.g. curved surfaces)


Image source

• The acoustic source is interactively mirrored to the other side of all reflectors that constitute the environment (virtual sources)

• All specular paths up to the desired reflection order are found

• Extremely efficient for simple, rectangular rooms

Implementation 51

Image sources


Image source

• Generates approximately n(n-1)r-1 virtual sources, where n is the number of reflectors and r the reflection order

• A big number of computed virtual sources are actually invisible, either from a reflected source (i.e. do not generate any acoustic field – physically invalid specular reflection) or the receiver

Cons 52


Image source

• All virtual sources should be tested for occlusion by all potential occludes – O(nr) visibility checks in arbitrary environments

Cons 53

Disadvantages: Only specular reflections Computationally demanding in

complex environments Dependence on source and

receiver positions

Advantages: Efficient in rectangular rooms Finds all specular reflections


Beam tracing

• Takes advantage of spatial coherence – groups rays in beams

• The interactive mirroring of the source is done only with respect to reflectors that beam encounters during the propagation

• Beams are organized in trees according to the splitting hierarchy

• Path tracing (finding paths between source and receiver) becomes an efficient process (lookup of the beam tree data structure)

54


Beam tracing

• Advantages:

– Good accuracy

– No need to recalculate the beam tree if receiver moves -> enables real-time rendering when receiver is moving (with static source)

– All paths between source and receiver are found (unlike ray tracing)

– Finds a minimum number of virtual sources (unlike image source method) – beam tracing automatically eliminates occluded paths (virtual sources that are never visible) during the construction of the beam tree

• Disadvantages:

– Moving the source requires the construction of a new beam tree which is generally a computationally demanding task

– Complex to implement

– In its initial implementation only specular reflections are modeled

Pros and cons 55


Visibility-based beam tracing (2D)

Goal: overcome the limitations of beam tracing, speeding up the computation of the beams

56

Iterative procedure For each beam that falls onto a reflector: 1.compute the reflected bundle of rays:

2.split the reflected bundle of rays into sub-beams (intersect its ray space representation with reflector’s visibility diagram); 3. add new beams to the beam tree data structure; 4.repeat the procedure for each beam that falls onto a reflector (the recursive procedure stops when the preassigned order of reflection is reached or when the beams die out).

3 33 1 3 1' ABR

S M S

( ) ( )1

3 1 3 3 3 3 3 1' det AB AB

T

R R

l M M l


2D versus 3D

2D:

• In 2D there are two degrees of

freedom in the description of a line;

• A line in 2D is a hyper-plane;

57

3D:

• In 3D there are four degrees of freedom in the description of a line (increase is two, not one);

• A 3D line does not separate two half-spaces as in 2D;

• Visual events are complex in 3D.


Discontinuity surfaces 58

Discontinuity surfaces:

2D [Teller1992]

3D [Drettakis1994]

2D versus 3D


Extension to 3D Representation

Extension to 3D

• a line in 2D has 2 DOF; a line in 3D has 4 DOF ; • unlike a line in 2D, a line in 3D does not separate two half-spaces; • visual events are complex in 3D.

five-dimensional oriented projective space

Plucker coordinates of a oriented line l passing through points A and B:

6 1

B - Al

A×B

Not every point in corresponds to ray in 3D - only those points that lay on a four-dimensional ruled surface known as Plucker quadric:

0 T l Ql

3

3

0 IQ

I 0

5P

5P


Beam tracing in 3D

Visibility regions: Only events on the surface of the Plucker quadric have to be taken into account while evaluating the mutual visibility between reflectors.

Visibility evaluation is no more a linear program. A new, dedicated, algorithm has to be developed.

Beam Tracing (visibility from point evaluation): • similar procedure to 2D case; • increase in dimension is exactly one (2 dof); • no intersection with the quadric is needed.

The beam tracing procedure has a reasonable computational burden that allows efficient tracing of acoustic beams.


Beam tracing in 3D

DEMO

61



Part 4:

Measurements and images


Acoustic measurements and images

• An acoustic system can extract:

– Acoustic measurements: Time Of Arrival, Time Difference Of Arrival, Direction Of Arrival,…

– Acoustic images, images of acoustic events extracted from traditional acoustic cameras

– Plenacoustic images, extracted from plenacoustic cameras

• We aim at

– finding a convenient representation in the ray space for measurements and images;

– estimate primitives in the acoustic scene (source localization, reflector localization, self-calibration of microphone arrays,…).

63


Acoustic measurements

• Examples of measurements that concern the propagation are – Time of Arrival (TOA)

– Time Difference of Arrival (TDOA)

depending on whether we have control over the source (sync)

• Such measurements may refer to – a direct acoustic path

– an indirect (specularly reflected) acoustic path

• A measurement carries informaton that can be turned into a constraint – This constraint will be useful for characterizing primitives only if it is expressed

in terms of the primitives themselves

Categorization 64


Time Of Arrival on direct path

• HP – Visibility between source and receiver

– First arrival corresponds to direct path from source to receiver

(source beam directly illuminates receiver)

• Given the TOA t and the speed of propagation c the source is bound to lie on a circle (or a sphere) of radius tc centered on the receiver

In the geometric domain 65

M

S

t c


Time Of Arrival on direct path

• The point xP2 corresponds to the location of the source iff

where C is the matrix representation of a circle (a=c, b=0) centered in the receiver [Contini2012]

• The (scalable) parameters (a, d, e, f) can be readily derived from the measured TOA, given the mic location

• x can be estimated from multiple measurements

Similar expressions can be derived for the location of the receiver, given the location of the source

In the ray space 66

fed

ecb

dba

x

xT

2/2/

2/2/

2/2/

1

,0 2

1

CxCxx


Time Of Arrival on indirect path

• Total TOA (time of flight)

• What constraints do we derive on the geometric primitives?

In the geometric domain 67

PMSP ttt

SPt PMt

M

S

P

t



• The point xP2 is a candidate reflection point for the acoustic path iff

i.e. x must lie on an ellipse whose foci are source and receiver, respectively

• The parameters can be easily obtained from the knowledge of the microphone position and of the TOA

In the ray space 68

fed

ecb

dba

x

x

xT

2/2/

2/2/

2/2/

,0

3

2

1

CxCxx



• The previous constraint not yet involves the geometric primitives of interest, as P (of hom. coordinates x) is just a reflection point of the reflector

• In order to express it directly in terms of the reflector l we need to rewrite the constraint in its adjoint form

where l are the hom. coords. of the line that the reflector lies on, and C* is the adjoint matrix of the ellipse C

In the ray space 69

0* lClT

11* )det( CCCC

MS

Pl

References: [Antonacci2010], [Antonacci2012], [Canclini2011],[Filos2010]


TOA on indirect paths

• Note: the constraint derived by the TOA measured on an indirect path can be used for localizing reflectors (environment inference): we will see an example later

In the ray space 70


TDOA on direct paths

• HP – Multiple receivers

– Visibility between source and receivers

– First arrivals correspond to direct path from source to receivers

(source beams directly illuminate the receivers)

• We have no information on the source time of emission, therefore all that we can measure are Time Differences Of Arrival (TDOAs)

71

21 ttt

S

1R

2R2t

1t t


TDOA on direct paths

• Given t, R1 and R2, the source S is bound to lie on a hyperbola with foci in R1 and R2

72

S

1R

2R2t

1t t


• The point of homogeneous coords x corresponds to the location of the source iff

where C is the matrix representation of a hyperbola det(C’)<0 with foci in R1 and R2

• The (scalable) parameters (a, b, c, d, e, f) can be readily derived from the measured TDOA, given the location of the mics

• x can be estimated from multiple measurements

TDOA on direct paths 73

fed

ecb

dba

x

x

xT

2/2/

2/2/

2/2/

,0

3

2

1

CxCxx


TDOA on indirect paths

• Combination of two constraints

– Homography (linear projective mapping) that embodies the specular reflection

– Constraint obtained from TDOA on direct path from x’ to receivers

74

1

22

0

nnnIH

Hxx'

dT

'

'

'

' ,0''

3

2

1

x

x

xT xCxx

x

x’

d n

R1 R2


TDOA on indirect paths

• The information on the reflector is carried by the homography H

• If we want to nail the reflector given x and the location of the receivers, we need to combine the above constraints wisely

75


DOA on indirect paths

• Consider having a compact array that acquires the signal from the acoustic source xs along with the source from the image xs’

• Geometry:

• If we make an hypothesis on xs , the line l is the axis of the segment xs x’s and the triangle xp xs x’s is isosceles.

Definitions 76

xs : real source r: line on which xs lies q: DOA of xs x’s : image source r’: line on which x’s lies q’: DOA of x’s

xr:reference point of the array


DOA on indirect paths

• Focal property: the tangent line at xP is the bisector of the angle formed by the line joining xS and xP and the perpendicular to r’ through xP

Constraint in the geometric space 77

xS

x

l

l is bound to be tangential to the parabola with focus in xS and directrix r’ [Canclini 2011a]

Same kind of constraint obtained for TOA on indirect paths


Summary of constraints 78

Direct Applications:source loc. and self-calibration

Indirect Applications: reflector localization

TOA

TDOA

DOA

R S

tc 0Cxx

TConstraint

Hp: Source-receiver synchronization

R

S

Hp: Source-receiver synchronization

0* lClT

Constraint

R1

R2

0CxxT

Constraint

Hp: Synchronization among receivers

x

x

’

d n R1

R2

Hp: Synchronization among receivers

M

S

0CHxHxTT

Constraint Line

Constraint

0* lClT

Constraint

Hp: Sources in the farfield

Hp: Sources in the farfield

S

r’


Acoustic images

• An acoustic image is an estimate of the distribution of the power incoming at a microphone array from different angles of arrival.

• Several techniques in the literature:

– Delay and sum Beamformer

– Minimum Variance Distortionless Response Beamformer

– MUltiple SIgnal Classification, etc.

• Example of acoustic image (acquired in a 3D geometry)

79

In the context of geometrical ST audio processing acoustic images are used for (see later on): -self-calibration of microphone arrays -estimation of the reflective properties of walls


Plenacoustic images

• Result from a combination of multiple acoustic images, which are mapped to the ray space. Each acoustic image is acquired by a subset of the microphones in the array [Markovic2012].

• In the context of geometrical acoustics plenacoustic images are used for:

– Source localization

– Reflector localization (along with its endpoints)

– …

80



Part 5:

Applications


Analysis and synthesis

• In the context of space-time audio processing geometrical acoustics is beneficial for

– Soundfield analysis: analysis of the acoustic scene

– Soundfield synthesis: rendering through loudspeaker arrays

– Modeling of the soundfield: tracing of the visibility in the ray space

– Mixed scenarios : e.g. applications where a preliminary identification of the acoustic scene enables an improved rendering

82


Soundfield analysis

• Goals

– Source localization;

– Self-calibration of microphone arrays;

– Localization of reflectors in the environment;

– Estimation of the reflective properties of walls

• The estimation of acoustic and geometric properties of the environment is accomplished through

– Joint analysis of multiple measurements (TOAs, TDOAs, DOAs);

– Processing on acoustic and plenacoustic images.

Goals and contextualization 83


Self-calibration

• Given TOA measurements at each sensor we want to infer the position of the microphone array [Contini2012]

• In case of unknown array geometry, we need both intrinsic and extrinsic array calibration

• In case of known array geometry, we only need extrinsic calibration (estimate array position and orientation)

Using TOAs 84

• Each TOA tmn

generates a constraint on the mic position xm

• xm can be estimated from (at least 3) constraints tmn, n=1,…,N

S1

S2

Sn

SN

Rm

tmn

R1

R2

RM

0

0

0

2

1

mNmmT

mmmT

mmmT

xCx

xCx

xCx

N

n

mnm

T

mmJ1

2)()( xCxx

Cost function


Self-calibration

• In order to obtain a constraint that acts directly on the rotation and translation of the whole array, we need to nest the TOA-related constraint tmn with that associated to the homography H

Using TOAs 85

Constraint on mic position

Merge the homography

Solve for H to obtain rotation matrix R and translation vector t

Nn

Mmmmnm

T

,...,1

,...,1,0''

xCx

0

0

0

0

1121

1111

MMN

TM

T

mmn

Tm

T

TT

TT

HxCHx

HxCHx

HxCHx

HxCHx


Self-calibration

• Goal: estimate position and orientation of multiple arrays (extrinsic calibration) from individual acoustic images [Redondi2009]

Using DOAs (farfield) 86

We can use TDOA-based constraints on the acoustic images for robust peak localization

Conventional stereo-imaging constraints (epipolar geometry) can now be used for array extrinsic calibration

B

A

X X’

R, t


Self-calibration from TDOAs

• Multiple array geometry

– Relationship between x and x’ through the fundamental matrix F

– The acquisition of N>7 acoustic images enables estimation of R and t

(up to a scaling factor) • Robust estimation methods (e.g. RanSaC) can be adopted for more accurate results

Using DOAs (farfield) 87

0Fxx'T

Demo Calibration error as a function of the number of sources


Self-calibration

• From L>=5 correspondences Xl X’l

we find the homography that best fuses the coordinate system of the first acoustic camera with that of the second one

• Mutual positions estimated through Rigid Body Motion [Valente2010 ]or Maximum Likelihood analysis [Valente2010a] on the set of correspondences Xl X’l

Using TDOAs (near field) 88

Two (or more) arrays acquire volumetric acoustic maps and localize the same source in space in their local coordinates systems, Xl and X’l.


Self-calibration

• Comparison of the accuracy of the estimation for different source distances for Rigid Body Motion (RBM), Maximum Likelihood Estimation (MLE) and DOA-based technique

Using TDOAs (near-field) 89

Setup Results

Maximum likelihood analysis allows an accurate estimation in both the near- and in the far-field


Source localization

• A step backward: redefining the geometric space

Using TDOAs 90

Representation adopted: (x,y) coordinates (x,y,z) coordinates z: distance covered by the wavefront from the source position

Source localization: finding the apex of the cone, given a set of noisy measurements, possibly coming from unsynchronized arrays [Compagnoni2012].

Metrics

Cost function:

N

i

aeaeJ1

/2

/

222 )()()( ssse zzyyxx

)()()( 22

sssa zzyyxx

Cone equation

Cone aperture


Source localization

• Setup – 81 source positions on a square grid.

– 1000 realizations

– RMS localization error, expressed in meters.

Using TDOAs 91

Cone equation Cone aperture SRD-LS [Beck2008]


The line parameters of the reflector are found as the global minimum of the cost function, i.e.

Inference of the geometry

• When multiple TOA measurements are conducted on indirect paths coming from the same reflector, we can reconstruct its location by finding the common tangent to all the ellipses, which corresponds to solve the system

Based on TOAs 92

0

...

0

0

*

2*

1*

lCl

lCl

lCl

NT

T

T

N

i

iTJ

1

2* )()( lCllCost function

)(minargˆ lll

J



Using TOAs 93

• If l is a minimum of J(l) , then also kl is mimimum need of de-homogenization to obtain a single global minimum

• Without loss of generality the global minimum is found by minimizing the cost function on the reduced ray spaces l1=1 and l3=1



• When only TDOAs are available (i.e. use of uncontrolled acoustic sources), the geometry of the environment is estimated through a three-steps procedure [Filos2010]:

1. TDOAs related to the direct path are used to localize the acoustic source

2. TDOAs related to indirect path are converted into TOAs once the location of the acoustic source is available.

3. Reflectors are localized by combining the resulting TOAs

Using TDOAs 94



• Extension to multiple reflectors: Hough transform [Antonacci2012, Filos2011]

Hough transform 95

Accuracy on a per-wall basis as a function of the number of sources used for the estimation of the TOAs



• DOAs related to reflective paths constrain the reflector to be tangent to parabolas, specified by the location of the acoustic source and by the

measured DOA [Canclini2011].

• When the source moves in space, the same reflector is observed from different viewpointsthe reflector is the common tangent to all the parabolas

Using DOAs 96

Hand-measured reflector line

Estimated reflector line

Parabolas associated to DOAs

Microphone array

Loudspeaker



• The reflector line is estimated from the combination of multiple constraints, as for the TOA case [Canclini2011].

• Average reflector location error (angle and distance) as a function of the number of DOAs:

Using DOAs 97

An accurate localization of the reflector is possible even for N=4 sources

Note: extension to multiple reflectors is possible using the Hough transform


Estimation of the reflective properties of walls

• Goal: estimate the reflection coefficient of the walls in an enclosure

• Idea: – model the acoustic propagation and generation of the spatial pseudospectrum

– only unknowns are the reflection coefficients

– match the simulated pseudospectrum with the measured one acquired by a microphone array

98

Assumptions: • room geometry is known (or

estimated in a preliminary step) • Source and array locations are known

(or estimated) • Directive properties of the

microphone array are known

Data model



Step 1: matching of the pseudospectra

99

Model

Measured



Step 1: matching of the pseudospectra • q: amplitude factor of the pseudospectrum and depends on the reflection coefficient of

the walls unknown

• M: models the contributions to the pseudospetrum, up to a scale factor it is known with the a-priori information

100



Step 2: estimation of the reflection coefficient

In each element of q the reflection coefficients are in a multiplicative relationship the logarithm of q is taken:

101

Example



Step 2: estimation of the reflection coefficients

102

Least-squares solution:

counts the number of occurrences of variables in m for each of the components of q

= Reflection coefficients:


Estimation of the reflective properties of walls 103

Experimental setup Matching of the pseudospectra

wooden panels

sem

i-a

nec

ho

ic r

oo

m

Experimental results


Estimation of the reflective properties of walls 104

Sub-band reflection coefficient estimates

Typical reflection coefficients for wooden panels


Scene inference from plenacoustic images

• Source and reflector localization from plenacoustic images [Markovic2012]

105



• Source localization

106

Source localization error: below 3 cm for both real and image sources



• Room estimation

107


Scene inference from plenacoustic images 108

Estimated walls

Real walls

Wall with finite extension

Microphone array


Soundfield synthesis

• Goals:

– Render a virtual acoustic source using a loudspeaker array

– Render the acoustics of a virtual environment

– Compensate for the early reflections of the environment in which the loudspeaker array is operating (“room compensation”)

– Exploit the early reflections of the real environment, e.g. for a virtual home theater system

109


Beamshaping Rendering virtual acoustic sources 110

Goal:

Propagation matrix

Desired response

LS solution

• Rendering of a virtual source along with its radiation pattern in a listening region, by means of a loudspeaker array [Antonacci2009]

• We focus on directional sources, modeled as beams; we need to control:

• Source position

• Direction of emission

• Beam aperture Formulation as an inverse problem:

virtual source

loudspeakers

listening region


Beamshaping Rendering virtual acoustic sources 111

Why a LS solution (and not conventional beamforming)?

• no constraints on loudspeaker and control point positions

• easy for handling near-field

• better control of parameters

• efficiency: propagation matrix and its (pseudo) inverse stored in advance

Fact: conditioning of could be a problem, especially at low frequencies

• SVD regularization:

• SVD is preferable w.r.t. other methods (e.g., Tikhonov):

• More controllable

• Its solution depends only on

)( GGH

rUDVh

UDVG

H

H

1~ˆ

G

)0,0,/1,/1(diag~

1

1 K D


Virtual environment rendering 112

Virtual acoustics: rendering the effect of a source in a virtual environment (in free-field) [Markovic 2010] Assumption: a complex acoustic wave field can be modeled as the superposition of elementary beams (geometrical acoustics)

Example: simple environment

Idea: use loudspeakers for synthesizing elementary beams, and obtain the complex wave field through superposition

The source in the room generates a set of virtual image sources according to the geometry of the environment we want to render



What if the environment is more complex?

• Image sources are suitable only in simple convex environments

• Visibility tests are very time demanding

• A convenient solution is beam tracing, which compute visibility very efficiently



Examples

NMSE (%):

500 Hz 7.4 %

1000 Hz 7.9 %

NMSE (%):

500 Hz 5.1 %

1000 Hz 5.3 %


Room compensation Contextualization 115

Fact: from analysis methodologies we can extract the room geometry

Idea: we use again the beam tracing engine for predicting the room reverberation (at least early reflections)

• Using reflective TOAs

• Using reflective DOAs


Room compensation 116 Early reflections modeling

early reflections

room compensated beam-shaping:

We now consider the rendering system operating in an arbitrary room:

• Use beam-tracing for computing the set of image loudspeakers visible at control points

• Modify Green’s functions as

• The propagation matrix becomes


Room compensation 117

Desired wave field Free-field beam shaping

Non-compensated beam shaping

Room-compensated beam shaping

Normalized MSE at a reference

point •The non-compensated response exhibits relevant peaks at resonant modes

•Room compensation dampens most of that peaks


Environment-aware virtual environment rendering

Goal: render the acoustics of a virtual environment when the array operates in another environment

118

• Loudspeakers operate in an arbitrary reverberant room

• Image loudspeakers determined through beam-tracing

• Room compensation enables a free-field behaviour of the beam shaping

• Design of a virtual environment to be rendered…

• … and rendering through superposition of beams


Environment-aware virtual environment rendering

119

Desired wave field Free-field wf rendering

Non-compensated wf rendering

Room-compensated wf rendering


Exploiting the environment 120

So far we limited ourselves to room compensation…

… is it possible to think at the environment as an “augmented” rendering system?

image array (first order)

image array (first order)

image array (second order)

image array (second order)

IDEA: use the walls (i.e., image loudspeakers) to increase the array

Application: simulation of a 5.1 surround system [Canclini2012]



Environment-aware virtual 5.1 surround

What about undesired reflections?

We can introduce, again, room compensation (slightly modified):

Desired response Actual response

CGG roomdes

desroomT

roomT

roomT

GGGGC1)(ˆ

ffcomp hCh ˆCompensated loudspeaker gains

listening area

C

L R RL RR



Example: reproduction of the rear-right channel

Desired Non-compensated Room-compensated

listening area

RR



Part 6:

Conclusions and perspectives


Conclusions

• Projective geometry and oriented projective geometry offer powerful tools for organizing the information coming from measurements in a layered fashion

– Many nonlinear relationships become linear in projective space

– Seamless interworking with Computer Vision methodologies (e.g. camera self-calibration)

– Same approach for a variety of applications, from calibration to rendering

• Extension to 3D geometries is possible

– Examples: • inference of the geometry [Nastasia2011], [Filos2012], [Canclini2012a]

• Modeling of propagation in 3D geometries

125


Perspectives

• Areas of growth

– Environment geometry from plenacoustic imaging

– WF reconstruction from plenacoustic images

– Source separation/characterization from plenacoustic images

– Data-driven plenacoustic rendering

– Model-driven plenacoustic super-resolution

– Extension to algebraic geometry

– Joint calibration and synchronization of spatial distributions of unsynchronized sensors

126



References


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133


Contributors

• Paolo Bestagini

• Alberto Calatroni

• Antonio Canclini

• Marco Compagnoni

• Alessio Contini

• Andrea Galbiati

• Dejan Markovic

• Erica Nastasia

• Giorgio Sandrini

• Daniele Valente

eusipco 2012 - eurasip.org · geometric space-time audio processing a. sarti and f. antonacci...

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