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 < 2 π 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 object r s ’ V ob a r O aperture r’ V ap u in reflectors V rf D N. Mehrotra and A. Sabharwal ISIT 2020 4 / 21
System Model object r s ’ V ob multipath a r O aperture r’ V ap LOS u in reflectors V rf D 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 � � s , r ′ � � r ′ � ˆ d r ′ E ( s ) = G · I V ob � �� � � �� � Green’s induced function current N. Mehrotra and A. Sabharwal ISIT 2020 5 / 21
System Model Single frequency illumination - no dependence of results on time Measured (backscattered) field � � s , r ′ � � r ′ � ˆ d r ′ E ( s ) = G · I V ob � �� � � �� � Green’s induced function current Combined Green’s function (multipath channel) � s , r ′ � = G ˆ � s , r ′ � + ˜ � s , r ′ � G G � �� � � �� � LOS NLOS N. Mehrotra and A. Sabharwal ISIT 2020 5 / 21
System Model Single frequency illumination - no dependence of results on time Measured (backscattered) field � � s , r ′ � � r ′ � ˆ d r ′ E ( s ) = G · I V ob � �� � � �� � Green’s induced function current Combined Green’s function (multipath channel) � s , r ′ � = G ˆ � s , r ′ � + ˜ � s , r ′ � G G � �� � � �� � LOS NLOS � s , r ′ � = � � r ′ s � � s , r ′ s � � r ′ s , r ′ � ˜ d r ′ s G h G G V rf � �� � � �� � � �� � aperture to reflectivity reflectors reflectors to scene of reflectors N. Mehrotra and A. Sabharwal ISIT 2020 5 / 21
System Model Single frequency illumination - no dependence of results on time Measured (backscattered) field � � s , r ′ � � r ′ � ˆ d r ′ E ( s ) = G · I V ob � �� � � �� � Green’s induced function current Combined Green’s function (multipath channel) � s , r ′ � = G ˆ � s , r ′ � + ˜ � s , r ′ � G G � �� � � �� � LOS NLOS � s , r ′ � = � � r ′ s � � s , r ′ s � � r ′ s , r ′ � ˜ d r ′ s G h G G V rf � �� � � �� � � �� � aperture to reflectivity reflectors reflectors to scene of reflectors E m ≡ 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 E m 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 E m 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 X N of X , � � N : d 2 N ǫ ( A ) = min N ( A ) ≤ ǫ , d N ( A ) = X N ⊆X sup inf g ∈X N � f − g � . inf f ∈A N. Mehrotra and A. Sabharwal ISIT 2020 6 / 21
Problem Formulation DoF ≡ dimensions of smallest basis set that approximates E m 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 X N of X , � � N : d 2 N ǫ ( A ) = min N ( A ) ≤ ǫ , d N ( A ) = X N ⊆X sup inf g ∈X N � f − g � . inf f ∈A 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 2 WT theorem for 1D time-frequency signals A = B Ω ≡ set of unit energy, bandlimited signals ( ω ∈ [ − Ω , +Ω]) X ≡ bandlimited PSWF basis (maximally concentrated to [ − T 2 , + T 2 ]) � · � ≡ L 2 [ − T 2 , + T 2 ] N ǫ ( A ) = min { N : λ N ≤ ǫ } � � Ω T �� = Ω T + O ln π π N. Mehrotra and A. Sabharwal ISIT 2020 7 / 21
Space of Measured Fields object V ob a O aperture V ap u in reflectors V rf cut-set D N. Mehrotra and A. Sabharwal ISIT 2020 8 / 21
Space of Measured Fields object V ob a O m Ω ap Ω ap aperture V ap u in reflectors V rf cut-set D N. Mehrotra and A. Sabharwal 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 S u = S l ∪ S i , where S l is the set of azimuth angles corresponding to physical aperture, S l = { s : s = s ( r ) , r ∈ V ap } , and S i is the set of azimuth angles corresponding to reflector locations in V rf that result in scattering towards V ap , � . � s ′ : s ′ = s ′ ( r ′ s ) , r ′ s ∈ V rf , ap ⊆ V rf S i = N. Mehrotra and A. Sabharwal ISIT 2020 9 / 21
Space of Measured Fields Spatial bandwidth ≡ smallest ’effective’ bandwidth s.t. E m ≈ B W 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. E m ≈ B W Spatial bandwidth remains same for given cut-set Definition (Spatial Bandwidth [BF87]) The spatial bandwidth W of the fields in E satisfies D B w ( E ) ≤ δ ∀ w ≥ W . N. Mehrotra and A. Sabharwal ISIT 2020 10 / 21
Space of Measured Fields Spatial bandwidth ≡ smallest ’effective’ bandwidth s.t. E m ≈ B W Spatial bandwidth remains same for given cut-set Definition (Spatial Bandwidth [BF87]) The spatial bandwidth W of the fields in E satisfies D B w ( E ) ≤ δ ∀ w ≥ W . Lemma (Spatial Bandwidth) � V G 1 ( s , r ′ ) · I ( r ′ ) d r ′ with spatial bandwidth W i , i = 1 , 2 , Given E i ( s ) = the spatial bandwidth for the sum E 1 ( s ) + E 2 ( s ) is W = max { W 1 , W 2 } . Corollary (Spatial Bandwidth) For our multipath system model, W = max { W 1 , W 2 } = k 0 a = 2 π λ a. N. Mehrotra and A. Sabharwal ISIT 2020 10 / 21
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