DoF Analysis for Multipath-Assisted Imaging: Single Frequency - - PowerPoint PPT Presentation

DoF Analysis for Multipath-Assisted Imaging: Single Frequency Illumination

Nishant Mehrotra and Ashutosh Sabharwal

Department of Electrical and Computer Engineering Rice University

ISIT 2020

• N. Mehrotra and A. Sabharwal

ISIT 2020 1 / 21

Problem Setting

Active, coherent imaging in multipath

Active ≡ dedicated illumination source. Ex: Camera flash Coherent ≡ phase available at aperture. Ex: mm-wave/THz

• N. Mehrotra and A. Sabharwal

ISIT 2020 2 / 21

Problem Setting

Active, coherent imaging in multipath

Active ≡ dedicated illumination source. Ex: Camera flash Coherent ≡ phase available at aperture. Ex: mm-wave/THz

Multipath-assisted radar imaging, NLOS imaging

See around corners - [CB05, KY11, GS13, LAAZ13, LWO19], etc. Image source: [AAT19]

• N. Mehrotra and A. Sabharwal

ISIT 2020 2 / 21

Problem Setting

Active, coherent imaging in multipath

Active ≡ dedicated illumination source. Ex: Camera flash Coherent ≡ phase available at aperture. Ex: mm-wave/THz

Imaging through scattering media

See through skin/fog - [AKH+18, WSV19], etc. Image source: [AKH+18]

• N. Mehrotra and A. Sabharwal

ISIT 2020 2 / 21

Objective

’Can multipath/scattering result in super-resolution?’

• N. Mehrotra and A. Sabharwal

ISIT 2020 3 / 21

Objective

’Can multipath/scattering result in super-resolution?’ ’Does the degrees of freedom increase with multipath/scattering?’

Imaging - DoF determines spatial resolution of image Communication - DoF determines spatial multiplexing capability

• N. Mehrotra and A. Sabharwal

ISIT 2020 3 / 21

Objective

’Can multipath/scattering result in super-resolution?’ ’Does the degrees of freedom increase with multipath/scattering?’

Imaging - DoF determines spatial resolution of image Communication - DoF determines spatial multiplexing capability

Prior work:

[Jan11, FMMS15] - no super-resolution gain for circular cut-sets [CB05, XJ06, KY11, GS13] - super-resolution for finite apertures

• N. Mehrotra and A. Sabharwal

ISIT 2020 3 / 21

Objective

’Can multipath/scattering result in super-resolution?’ ’Does the degrees of freedom increase with multipath/scattering?’

Imaging - DoF determines spatial resolution of image Communication - DoF determines spatial multiplexing capability

Prior work:

[Jan11, FMMS15] - no super-resolution gain for circular cut-sets [CB05, XJ06, KY11, GS13] - super-resolution for finite apertures

This Work

Yes, DoF increases with multipath under certain conditions

Finite apertures with angular extent < 2π Highly reflective scatterers surrounding imaging target Known and static channel between aperture and scene

• N. Mehrotra and A. Sabharwal

ISIT 2020 3 / 21

System Model

D uin r’ rs’ r

• bject

Vob reflectors Vrf aperture Vap a O

• N. Mehrotra and A. Sabharwal

ISIT 2020 4 / 21

System Model

D uin r’ rs’ r multipath LOS

• bject

Vob reflectors Vrf aperture Vap O a

• N. Mehrotra and A. Sabharwal

ISIT 2020 4 / 21

System Model

Single frequency illumination - no dependence of results on time

• N. Mehrotra and A. Sabharwal

ISIT 2020 5 / 21

System Model

Single frequency illumination - no dependence of results on time Measured (backscattered) field E (s) =

• Vob

ˆ G

s, r′

• Green’s

function

· I

r′

induced current

dr′

• N. Mehrotra and A. Sabharwal

ISIT 2020 5 / 21

System Model

Single frequency illumination - no dependence of results on time Measured (backscattered) field E (s) =

• Vob

ˆ G

s, r′

• Green’s

function

· I

r′

induced current

dr′ Combined Green’s function (multipath channel) ˆ G

s, r′ = G s, r′

• LOS

+ ˜ G

s, r′

• NLOS
• N. Mehrotra and A. Sabharwal

ISIT 2020 5 / 21

System Model

Single frequency illumination - no dependence of results on time Measured (backscattered) field E (s) =

• Vob

ˆ G

s, r′

• Green’s

function

· I

r′

induced current

dr′ Combined Green’s function (multipath channel) ˆ G

s, r′ = G s, r′

• LOS

+ ˜ G

s, r′

• NLOS

˜ G

s, r′ =

• Vrf

h

r′s

• reflectivity
• f reflectors

G

s, r′s

• aperture to

reflectors

G

r′s, r′

• reflectors

to scene

dr′s

• N. Mehrotra and A. Sabharwal

ISIT 2020 5 / 21

System Model

Single frequency illumination - no dependence of results on time Measured (backscattered) field E (s) =

• Vob

ˆ G

s, r′

• Green’s

function

· I

r′

induced current

dr′ Combined Green’s function (multipath channel) ˆ G

s, r′ = G s, r′

• LOS

+ ˜ G

s, r′

• NLOS

˜ G

s, r′ =

• Vrf

h

r′s

• reflectivity
• f reflectors

G

s, r′s

• aperture to

reflectors

G

r′s, r′

• reflectors

to scene

dr′s Em ≡ space of all measured fields E (s)

• N. Mehrotra and A. Sabharwal

ISIT 2020 5 / 21

Problem Formulation

DoF ≡ dimensions of smallest basis set that approximates Em Ex: 1D signals, DoF corresponds to PSWF basis reconstruction

• N. Mehrotra and A. Sabharwal

ISIT 2020 6 / 21

Problem Formulation

DoF ≡ dimensions of smallest basis set that approximates Em Ex: 1D signals, DoF corresponds to PSWF basis reconstruction

Definition (Degrees of Freedom [BF89])

The DoF of a set A is the smallest dimension N such that A can be approximated up to accuracy ǫ by N-dimensional subspaces XN of X, Nǫ (A) = min

• N : d2

N (A) ≤ ǫ

• ,

dN (A) = inf

XN⊆X sup f ∈A

inf

g∈XN f − g .

• N. Mehrotra and A. Sabharwal

ISIT 2020 6 / 21

Problem Formulation

DoF ≡ dimensions of smallest basis set that approximates Em Ex: 1D signals, DoF corresponds to PSWF basis reconstruction

Definition (Degrees of Freedom [BF89])

The DoF of a set A is the smallest dimension N such that A can be approximated up to accuracy ǫ by N-dimensional subspaces XN of X, Nǫ (A) = min

• N : d2

N (A) ≤ ǫ

• ,

dN (A) = inf

XN⊆X sup f ∈A

inf

g∈XN f − g .

Three key choices:

Set A i.e. what set does measured data belong to? Set X i.e. what basis is being used for sampling and/or interpolation? Norm · i.e. what is the interval over which approximation is made?

• N. Mehrotra and A. Sabharwal

ISIT 2020 6 / 21

Example: 1D Time-Frequency DoF

Shannon’s 2WT theorem for 1D time-frequency signals

A = BΩ ≡ set of unit energy, bandlimited signals (ω ∈ [−Ω, +Ω]) X ≡ bandlimited PSWF basis (maximally concentrated to [− T

2 , + T 2 ])

· ≡ L2[− T

2 , + T 2 ]

Nǫ (A) = min {N : λN ≤ ǫ} = ΩT π + O

• ln

ΩT

π

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ISIT 2020 7 / 21

Space of Measured Fields

D uin

• bject

Vob reflectors Vrf aperture Vap a O cut-set

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ISIT 2020 8 / 21

Space of Measured Fields

D uin

• bject

Vob reflectors Vrf aperture Vap a O cut-set Ωap Ωap

m

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ISIT 2020 8 / 21

Space of Measured Fields

Spatial support ≡ azimuth angles of signals flowing through cut-set Spatial support increases due to NLOS signal support on cut-set Net support ≡ non-overlapping union of LOS and NLOS supports

• N. Mehrotra and A. Sabharwal

ISIT 2020 9 / 21

Space of Measured Fields

Spatial support ≡ azimuth angles of signals flowing through cut-set Spatial support increases due to NLOS signal support on cut-set Net support ≡ non-overlapping union of LOS and NLOS supports

Definition (Spatial Support Set)

The spatial support set is defined as Su = Sl ∪ Si, where Sl is the set of azimuth angles corresponding to physical aperture, Sl = {s : s = s(r), r ∈ Vap} , and Si is the set of azimuth angles corresponding to reflector locations in Vrf that result in scattering towards Vap, Si =

s′ : s′ = s′(r′s), r′s ∈ Vrf ,ap ⊆ Vrf .

• N. Mehrotra and A. Sabharwal

ISIT 2020 9 / 21

Space of Measured Fields

Spatial bandwidth ≡ smallest ’effective’ bandwidth s.t. Em ≈ BW Spatial bandwidth remains same for given cut-set

• N. Mehrotra and A. Sabharwal

ISIT 2020 10 / 21

Space of Measured Fields

Spatial bandwidth ≡ smallest ’effective’ bandwidth s.t. Em ≈ BW Spatial bandwidth remains same for given cut-set

Definition (Spatial Bandwidth [BF87])

The spatial bandwidth W of the fields in E satisfies DBw (E) ≤ δ ∀w ≥ W .

• N. Mehrotra and A. Sabharwal

ISIT 2020 10 / 21

Space of Measured Fields

Spatial bandwidth ≡ smallest ’effective’ bandwidth s.t. Em ≈ BW Spatial bandwidth remains same for given cut-set

Definition (Spatial Bandwidth [BF87])

The spatial bandwidth W of the fields in E satisfies DBw (E) ≤ δ ∀w ≥ W .

Lemma (Spatial Bandwidth)

Given Ei (s) =

• V G1 (s, r′) · I (r′) dr′ with spatial bandwidth Wi, i = 1, 2,

the spatial bandwidth for the sum E1 (s) + E2 (s) is W = max {W1, W2} .

Corollary (Spatial Bandwidth)

For our multipath system model, W = max {W1, W2} = k0a = 2π

λ a.

• N. Mehrotra and A. Sabharwal

ISIT 2020 10 / 21

Space of Measured Fields

Em characterized by spatial and spatial frequency support sets

s W 2k0a Sl Si Su Ωap

• N. Mehrotra and A. Sabharwal

ISIT 2020 11 / 21

DoF Analysis

DoF ≡ dimensions of smallest subspace that approximates Em A = Em ≈ BW ≡ set of unit energy signals bandlimited to [−W , +W ] X ≡ bandlimited PSWF basis (concentrated to S), · ≡ L2(S) S = Sl (lower bound) and S = Su (upper bound)

• N. Mehrotra and A. Sabharwal

ISIT 2020 12 / 21

DoF Analysis

DoF ≡ dimensions of smallest subspace that approximates Em A = Em ≈ BW ≡ set of unit energy signals bandlimited to [−W , +W ] X ≡ bandlimited PSWF basis (concentrated to S), · ≡ L2(S) S = Sl (lower bound) and S = Su (upper bound)

Theorem (Multipath DoF)

Nǫ (Em) is upper and lower bounded as Nl ≤ Nǫ (Em) ≤ Nu, where Nl = N0,l + 1 π2 ln

• 1 − ǫ2

ǫ2

• ln

πN0,l

2

• + O (ln (N0,l)) ,

Nu = N0,u + κ π2 ln

• 1 − ǫ2

ǫ2

• ln

πN0,u

2

• + O (ln (N0,u)) ,

N0,l = Wm(Sl)

π

, N0,u = Wm(Su)

π

, W = 2πa

λ

and κ disjoint intervals in Su.

• N. Mehrotra and A. Sabharwal

ISIT 2020 12 / 21

DoF Analysis

Ex: Continuous, closed aperture surrounding object volume

Ωap = 2π = ⇒ m(Sl) = Ωap = 2π = ⇒ Su = Sl ∪ Si = Sl No effect of scattering as Su = Sl Nl = Nu = 2W + O (ln (2W )) i.e. no super-resolution gain Recovers result of [FMMS15]

D uin

• bject

Vob reflectors Vrf aperture Vap a O

• N. Mehrotra and A. Sabharwal

ISIT 2020 13 / 21

DoF Analysis

Ex: Finite aperture with reflectors of reflectivity h(·) < 1

Heavy tail decay of singular values beyond LOS DoF, N = N0,l Singular values ≈ 1 for N ≤ N0,l and ≈ h (r′s) , r′s ∈ Su \ Sl o/w

𝜏n 1 h1 n Nl Nu

• (Nu)
• (Nu)

h2 ⋱ h𝜆-1

• (Nu)

⋮ LOS Multipath average reflectivity

• f reflectors in ith

interval in Su\Sl

• N. Mehrotra and A. Sabharwal

ISIT 2020 13 / 21

Numerical Evaluation

2D scattering SIMO setup with large specular reflectors Known and static channel between aperture and scene

𝜄 z x Lob Lap aperture Vap

• bject

Vob reflectors Vrf (xtx,-D) (xrx,-D) (x’,z’) multipath LOS Drf Drf Lrf

• N. Mehrotra and A. Sabharwal

ISIT 2020 14 / 21

Numerical Evaluation

Specular reflectors = ⇒ mirroring of physical aperture

𝜄 z x Lap (x’,z’) Drf Drf virtual aperture Vap,v

(2)

⋮ ⋮ virtual aperture Vap,v

(1)

Ωap Ωap,v

(2)

Ωap,v

(1)

Lap,v

(1)

Lap,v

(2)

• N. Mehrotra and A. Sabharwal

ISIT 2020 14 / 21

Numerical Evaluation

Specular reflectors = ⇒ mirroring of physical aperture DoF analysis = ⇒ 2× to 3× increase compared to LOS DoF W = 1 2k0Lob cos θ, Ωap = 2Lap

• L2

ap + 4D2

Ωm

ap = 2 sin

• tan−1
• 2Drf + Lap,v

2

D

• − 2 sin
• tan−1
• 2Drf − Lap,v

2

D

• Corollary (Multipath DoF)

2LapLob cos θ λ

• L2

ap + 4D2

• LOS

≤ Nǫ (Em) ≤ 6LapLob cos θ λ

• 9L2

ap + 4D2

• multipath
• N. Mehrotra and A. Sabharwal

ISIT 2020 15 / 21

Numerical Evaluation

Same reflectivity of both reflectors = ⇒ step-like behavior

increasing reflectors’ reflectivities

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ISIT 2020 16 / 21

Numerical Evaluation

Different reflectivity of both reflectors = ⇒ multi-step behavior

multi-step behavior

• N. Mehrotra and A. Sabharwal

ISIT 2020 16 / 21

Numerical Evaluation

Same reflectivity of both reflectors, validity across θ and effect of ǫ

• N. Mehrotra and A. Sabharwal

ISIT 2020 16 / 21

Discussions and Conclusion

This work: DoF for finite apertures increases with multipath

Finite apertures with angular extent < 2π Highly reflective scatterers surrounding imaging target Known and static channel between aperture and scene

Main result: Lower and upper bounds on DoF for finite apertures

Consistent with results of [FMMS15] for closed, continuous apertures Heavy tail decay of singular values between DoF bounds Singular value amplitudes governed by reflectors’ reflectivities

Future work: Performance evaluation of multipath-assisted algorithms

• N. Mehrotra and A. Sabharwal

ISIT 2020 17 / 21

References I

• M. Aladsani, A. Alkhateeb, and G. C. Trichopoulos, Leveraging

mmwave imaging and communications for simultaneous localization and mapping, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019,

• pp. 4539–4543.

Nick Antipa, Grace Kuo, Reinhard Heckel, Ben Mildenhall, Emrah Bostan, Ren Ng, and Laura Waller, Diffusercam: lensless single-exposure 3d imaging, Optica 5 (2018), no. 1, 1–9.

• O. Bucci and G. Franceschetti, On the spatial bandwidth of scattered

fields, IEEE Transactions on Antennas and Propagation 35 (1987),

• no. 12, 1445–1455.
• O. M. Bucci and G. Franceschetti, On the degrees of freedom of

scattered fields, IEEE Transactions on Antennas and Propagation 37 (1989), no. 7, 918–926.

• N. Mehrotra and A. Sabharwal

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References II

Margaret Cheney and Robert J. Bonneau, Imaging that exploits multipath scattering from point scatterers, SPIE 5808, Algorithms for Synthetic Aperture Radar Imagery XII (Edmund G. Zelnio and Frederick D. Garber, eds.), vol. 5808, International Society for Optics and Photonics, May 2005, p. 142.

• M. Franceschetti, M. D. Migliore, P. Minero, and F. Schettino, The

information carried by scattered waves: Near-field and nonasymptotic regimes, IEEE Transactions on Antennas and Propagation 63 (2015),

• no. 7, 3144–3157.
• G. Gennarelli and F. Soldovieri, A Linear Inverse Scattering Algorithm

for Radar Imaging in Multipath Environments, IEEE Geoscience and Remote Sensing Letters 10 (2013), no. 5, 1085–1089.

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References III

• R. Janaswamy, On the em degrees of freedom in scattering

environments, IEEE Transactions on Antennas and Propagation 59 (2011), no. 10, 3872–3881. V Krishnan and B Yazici, Synthetic aperture radar imaging exploiting multiple scattering, Inverse Problems 27 (2011), no. 5, 055004.

• M. Leigsnering, F. Ahmad, M. G. Amin, and A. M. Zoubir,

Compressive sensing based specular multipath exploitation for through-the-wall radar imaging, 2013 IEEE International Conference

• n Acoustics, Speech and Signal Processing, May 2013,
• pp. 6004–6008.

David B. Lindell, Gordon Wetzstein, and Matthew O’Toole, Wave-based non-line-of-sight imaging using fast f-k migration, ACM

• Trans. Graph. 38 (2019), no. 4.
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References IV

Yicheng Wu, Manoj Sharma, and Ashok Veeraraghavan, Wish: Wavefront imaging sensor with high resolution, Light: Science & Applications 8 (2019).

• J. Xu and R. Janaswamy, Electromagnetic degrees of freedom in 2-d

scattering environments, IEEE Transactions on Antennas and Propagation 54 (2006), no. 12, 3882–3894.

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