Sampling Without Time: Recovering Echoes of Light via Temporal Phase Retrieval
Ayush Bhandari Aurelien Bourquard Ramesh Raskar








Sampling Without Time: 

Recovering Echoes of Light via Temporal Phase Retrieval










Ayush Bhandari^†, Aurélien Bourquard ^ and Ramesh Raskar^†




^† Media Laboratory and ^ Research Laboratory of Electronics 
 Massachusetts Institute of Technology Cambridge, MA 02139–4307 USA. 

 
^†,ayush@MIT.edu  ∙  aurelien@MIT.edu  ∙  raskar@MIT.edu



 To Appear in the Proceedings of IEEE ICASSP, 2017[The 42^nd IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP).]. 

This work expands on the ideas discussed in <cit.> and <cit.>.






150mm

Abstract

This paper considers the problem of sampling and reconstruction of a continuous-time sparse signal without assuming the knowledge of the sampling instants or the sampling rate. This topic has its roots in the problem of recovering multiple echoes of light from its low-pass filtered and auto-correlated, time-domain measurements. Our work is closely related to the topic of sparse phase retrieval and in this context, we discuss the advantage of phase-free measurements. While this problem is ill-posed, cues based on physical constraints allow for its appropriate regularization. We validate our theory with experiments based on customized, optical time-of-flight imaging sensors. What singles out our approach is that our sensing method allows for temporal phase retrieval as opposed to the usual case of spatial phase retrieval. Preliminary experiments and results demonstrate a compelling capability of our phase-retrieval based imaging device. 







 1.5


§ INTRODUCTION

During a recent visit to meet a collaborator, the author (who happens to be an avid photographer) saw a stark reflection of a local monument on the window panes of the Harvard science center. This is shown in Fig. <ref>. This is a common place phenomenon at a macro scale as well as a micro scale (for example microscopy). At the heart of this problem is a fundamental limitation, that is, all of the conventional imaging sensors are agnostic to the time information. Alternatively stated, the image–formation process is insensitive to the potentially distinct times that the photons can spend traveling between their sources and the detector.


To elaborate, consider the Gedankenexperiment version of Fig. <ref> described in Fig. <ref> (a). Let us assume that a light source is co-located with the imaging sensor (such as a digital camera). The reflection from the semi-reflective sheet (such as a window pane) arrives at the sensor at time t_1 = 2d_1/c while the mannequin is only observable on or after t_2 = 2d_2/c. Here, c is the speed of light. While we typically interpret images as a two dimensional, spatial representation of the scene, let us for now, consider the photograph in Fig. <ref> along the time-dimension. For the pixel ł x_0, y_0 $̊, this time-aware image is shown in Fig. <ref> (b). Mathematically, our time-aware image can be written as a2–sparse signal (and as aK–sparse signal in general), 


    m( r,t) = ( λ _1Γ _1)( r)δ( t - t_1 (r)) + ( λ _2Γ _2)( r)δ( t - t_2(r))

where{ Γ_k( r ) }are the constituent image intensities,{ λ_k( r ) }are the reflection coefficients,r = łx , y^̊⊤is the 2D spatial coordinate, andδ(·)is the Dirac distribution.






A conventional imaging sensor produces images by marginalizing time information, resulting in the 2D photograph, 

    ∫_0^Δ≫t_2m( r,t)dt  = ( λ _1Γ _1)( r) + ( λ _2Γ _2)( r) ≡ Iłr.̊

Recovering{ Γ_k }, k = 1,2givenI, more generally,Kechoes of light, 

    m( r,t) = ∑_k = 0^K - 1( λ _kΓ _k)( r)δ( t - t_k(r))

is an ill-posed problem. Each year, a number of papers <cit.> attempt to address this issue by using regularization and/or measurement-acquisition diversity based on image statistics, polarization, shift, motion, color, or scene features. Unlike previous works, here, we ask the question: Can we directly estimate{ Γ_k }_k = 0^K - 1in (<ref>)? In practice, sampling (<ref>) would require exorbitant sampling rates and this is certainly not an option. Also, we are interested in{ Γ_k }_k = 0^K - 1only, as opposed to{ Γ_k,t _k }_k = 0^K - 1<cit.> wheret_kis a non-linear argument in (<ref>). As a result, our goal is to recover the intensities of a sparse signal. For this purpose, we explore the idea of sampling without time—a method for sampling a sparse signal which does not assume the knowledge of sampling instants or the sampling rate. 
In this context, our contributions are twofold: 



  *  For the general case of K-echoes of light, our work relies on estimating the constituent images {Γ _k}_k = 0^K - 1 from the filtered, auto-correlated, time-resolved measurements. This is the distinguishing feature of our approach and is fundamentally different from methods proposed in literature which are solely based on spatio-angular information (cf. <cit.> and references therein). 


  *  As will be apparent shortly, our work is intimately linked with the problem of (sparse) phase retrieval (cf. <cit.> and references therein). Our “sampling without time” architecture leads to an interesting measurement system which is based on time-of-flight imaging sensors <cit.> and suggests that K^2-K+1 measurements suffice for estimation ofK echoes of light. To the best of our knowledge, neither such a measurement device nor bounds have been studied in the context of image source separation <cit.>.




For the sake of simplicity, we will drop the dependence ofmandΓon spatial coordinatesr. This is particularly well suited for our case because we do not rely on any cross-spatial information or priors. Also, scaling factorsλ_kare assumed to be absorbed inΓ_k. 


 Significance of Phase-free or Amplitude-only measurements: It is worth emphasizing that resolving spikes from a superposition of low-pass filtered echoes, is a problem that frequently occurs in many other disciplines. This is a prototype model used in the study of multi-layered or multi-path models. Some examples include seismic imaging <cit.>, time-delay estimation <cit.>, channel estimation <cit.>, optical tomography <cit.>, ultrasound imaging <cit.>, computational imaging <cit.> and source localization <cit.>. Almost all of these variations rely on amplitude and phase information or amplitude and time-delay information. However, recording amplitude-only data can be advantageous due to several reasons. Consider a common-place example based on pulse-echo ranging. Let pł t=̊sin( ωt) be the emitted pulse. Then, the backscattered signal reads r( t ) = Γ_0sin( ω t - θ _0). In this setting, on-chip estimation of phase (θ_0) or time delay (t_0)<cit.>,



  *  can either be computationally expensive or challenging since t_k is a non-linear parameter in (<ref>), and hence, in r( t ). 


  *  requires more measurements. For instance, 2 measurements suffice for amplitude-only estimation (Γ_0) while amplitude-phase pair requires 4 measurements <cit.>. This aspect of phase estimation is an important bottleneck as the the frame rate of an imaging device is inversely proportional to number time-domain measurements. 


  *  is prone to errors. In many applications, multiple-frequency measurements, ω  = kω _0, are acquired <cit.> assuming that phase and frequency, θ _0 = ωt_0 are linearly proportional. However, this is not the case in practice. The usual practice is to oversample <cit.>.


In all such cases, one may adopt our methodology of working with intensity-only measurements. 




§ IMAGE–FORMATION MODEL FOR TOF SENSORS

Optical ToF sensors are active devices that capture 3D scene information. We adopt the generic image–formation model used in <cit.> which is common to all ToF modalities such as lidar/radar/sonar, optical coherence tomography, terahertz, ultrasound, and seismic imaging. In its full generality, and dropping dependency on the spatial coordinate for convenience, one can first formalize this ToF acquisition model as:

    p → h → r →ϕ→ yy↦m =  y_

where, 



  * p( t ) >0 is a T-periodic probing function which is used to illuminate the scene. This may be a sinusoidal waveform, or even a periodized spline, Gaussian, or Gabor pulse, for instance.



  * hł t , τ$̊ is the scene response function (SRF). This may be a filter, a shift-invariant functionh_ SIł t , τ=̊ h ł t - τ$̊, or a partial differential equation modeling some physical phenomenon <cit.>. 
  * r( t ) = ∫p( τ)h( t,τ)dτ is the reflected signal resulting from the interaction between the probing function and the SRF. 


  * y( t ) = ∫r( τ)ϕ( t,τ)dτ is the continuous-time signal resulting from the interaction between the reflected function and the instrument response function (IRF), or ϕ, which characterizes the electro-optical transfer function of the sensor. 


  * y is a set of discrete measurements of the form. y( t )|_t = nΔ. 


  * y = y * y is the cyclic auto-correlation of y, where * and ł· denote the convolution and time reversal operators, respectively. 


The interplay betweenp,h, andϕresults in variations on the theme of ToF imaging <cit.>. In this work, we will focus on an optical ToF setting. Accordingly:
  *  The probing function corresponds to a time-localized pulse.

  
  *  The SRF, accounting for the echoes of light, is a K-sparse filter, 

    h_( t,τ)≡ h_Kł t - τ=̊∑_k = 0^K - 1Γ _kδ( t - τ -  t_k).

  *  The IRF is fixed by design asϕ _SI( t,τ) = p( t + τ). This implements the so-called homodyne, lock-in sensor <cit.>. 

Due to this specific shift-invariant structure of the SRF and IRF, we have,y = p*p *h _K ≡p*h _K.




Finally, the measurements read,  



    m = y  = łφ*h_K,̊  φ = p.


§ SAMPLING ECHOES OF LIGHT


 §.§ Bandlimited Approximation of Probing Function


Due to physical constraints inherent to all electro-optical systems, it is reasonable to approximate the probing signal as a bandlimited function <cit.>. We useL–term Fourier series with basis functionsu_n(ω_0 t ) e^ω_0nt,ω_0 T = 2π, to approximatepwith,

    p( t ) ≈p( t ) = ∑_| ℓ| ⩽ Lp_ℓu_ℓ(  ω_0 t) ,

where thep_ℓare the Fourier coefficients. Note that there is no need to computepłt $̊ explicitly as we only require the knowledge of p (cf. (<ref>)). In Fig. <ref>, we plot the calibrated p and its Fourier coefficients. In this case, T = 232.40 ns. We also plot its approximation, φ = p with L = 25 together with the measured p and φ.     



 §.§ Sampling Theory Context

The shift-invariant characterization of the ToF image–formation model allows to re-interpret the sampled version of (<ref>) as the filtering of a sparse signal h_K with a low-pass kernel ψ = φ (cf. <cit.>). 

    m( t )    acmh_K*φ≡h_K*ψ.


Note that the properties of the auto-correlation operation imply that the sparsity of h_K is K^2 - K + 1, unlike h_K that is K-sparse and is completely specified by  due to symmetry. Based on the approximation (<ref>) and the properties of convolution and complex exponentials, and defining ψ_ℓ = |p_ℓ|^4, we rewrite m(t) as,  

    m ł t  =  ∑_|ℓ|≤ Lψ_ℓ u_ℓłω_0t %̊s̊∫h_Kłτů_ℓ^*łω_0τ dτ_Fourier Integral,










Finally, the properties of the Fourier transform imply that sampling the above signal m(t) at time instants t = nΔ, results in discrete measurements of the form m = 𝐔𝐃_ψ̂ŝ, which corresponds to the available acquired data in our acquisition setting. Combining all the above definitions, it follows that:




  * m∈ℝ^N is a vector of filtered measurements (cf. (<ref>)). 


  * 𝐔∈ℂ^N ×( 2L + 1) is a DFT matrix with elements [ u_n łω_0ℓΔ]_n,ℓ.


  * 𝐃_ψ̂∈ℂ^ł 2L+1×̊ł 2L+1 is a diagonal matrix with diagonal elements ψ_ℓ. These are the Fourier-series coefficients of φ. 


  * s∈ℝ^2L+1 is a phase-less vector containing the Fourier coefficients of h_K, which is obtained by sampling the Fourier transform s( ω) of h_K at instants ω = ℓω_0. The signal s( ω) directly depends on the quantities |Γ_k| of interest to be retrieved and is expressed as


    s( ω)     = | ∑_k = 0^K - 1Γ _ku_t_kłω|^2≡ |h_K łω|̊^2  
        =  ∑_k = 0^K - 1|Γ _k|^2_a_0 + 2∑_k = 0^K - 1∑_m = k + 1^K - 1| Γ _k|| Γ _m|_a_k, mcos( ωt_k,m + ∠Γ _k,m)

where t_k,m = t_k - t_m and ∠Γ _k,m = ∠Γ _k - ∠Γ _m. 



It is instructive to note that the relevant unknowns {Γ _k}_k = 0^K - 1 can be estimated from s∈ℝ^2L+1 in (<ref>) which in turn depend only on L as opposed to sampling rate and sampling instants. This aptly justifies our philosophy of sampling without time.



 





§ RECONSTRUCTION VIA PHASE RETRIEVAL


Given data model m = 𝐔𝐃_ψ̂ŝ (<ref>), we aim to recover the image parameters |Γ_k| at each sensor pixel. For this purpose, we first estimate samples of ŝ(ω) as s = 𝐃_ψ^-1𝐔^+m, where 𝐔^+ is the matrix pseudo-inverse of 𝐔. This is akin to performing a weighted deconvolution, knowing that 𝐔 is a DFT matrix. Next, we solve the problem of estimating |Γ_k| in two steps (1) First we estimate a_k,m, and then, (2) based on the estimated values, we resolve ambiguities due to |·|.



Parameter Identification via Spectral Estimation: Based on the coefficients s, one can then retrieve the amplitude and frequency parameters that are associated with the oscillatory terms as well as the constant value in (<ref>). The oscillatory-term and constant-term parameters correspond to {a_k, m, t_k, m} and a_0, respectively. All parameter values are retrievable from s through spectral estimation <cit.>; details are provided in <cit.> for the interested reader.

Note that, given the form of (<ref>) and our acquisition model, the sparsity level of the sequence s—corresponding to the total amount of oscillatory and constant terms—is (K^2 - K)/2 + 1. The magnitude values a_k, m and a_0 can thus be retrieved if at least L≥ (K^2 - K) + 1 autocorrelation measurements are performed <cit.>, which is the case in the experiments described in Section <ref>. 





Resolving Ambiguities:
Based on the aforementioned retrieved parameters, one wishes to then deduce {|Γ_k|}_k=0^K-1. The estimated cross terms {a_k, m, t_k, m} allow to retrieve the values of |Γ_k| through simple pointwise operations. Here we will focus on the case, K=2. Due to space limitations, we refer the interested readers to our companion paper <cit.> for details on the general case (K>2). The case when K=2, 



    s( ω) mag   | Γ _0u_t_0^*( ω) + Γ _1u_t_1^*( ω)|^2 + ε _ω( Γ _k,t_k)_k > 2_≈ 0
    
    =    | Γ _0|^2 + | Γ _1|^2_a_0 + 2| Γ _0|| Γ _1|_a_01cos( ωt_0,1 + ∠Γ _0,1),




may be a result of two distinct echoes (Fig. <ref>) or due to approximation of higher order echoes by two dominant echoes (due to inverse-square law). In this case, we effectively estimate two terms: a_0 and a_01 (<ref>). The set of retrieved magnitude values then amount to solving<cit.>: 

    | | Γ _0| ±| Γ _1|| = √(a_0±a_01),     a_0,a_01 > 0.

Thanks to the isoperimetric property for rectangles: a_0≥ a_01, the r.h.s above is always positive unless there is an estimation error in which case, an exchange of variables leads to the unique estimates,

    {|Γ_k|}_k = 1,2 = √(a_0 + a_01)±√(a_0 - a_01)/2.




§ EXPERIMENTAL VALIDATION




 §.§ Simulation

Noting that measurements m = 𝐔𝐃_ψ̂ŝ and ŝ are linked by an invertible, linear system of equations, knowing m amounts to knowing ŝ. We have presented detailed comparison of simulation results in <cit.> for the case where one directly measures ŝ. In this paper, we will focus on practical setting where the starting point of our algorithm is (<ref>).


[!t]
    
    < g r a p h i c s >

    Experimental results on acquisition and reconstruction of echoes of light in a scene. (a) Single pixel, time-stamped samples y = p*h _2 serve as our ground truth because the phase information is known. We also plot the estimated, 2-sparse SRF (<ref>) in red ink. (b) Fourier domain frequency samples of the SRF, h_2 łℓω_0$̊ withL = [-20,20]andω_0 = 2π/Δ, Δ= 70ps. We plot the measured data in green and the fitted data in black. The frequencies are estimated using Cadzow's method. (c) Same pixel, time-stampless, low-pass filtered, auto-correlated samples (<ref>) together with the estimated, auto-correlated SRFh_2in red ink. (d) Fourier domain samplessłℓω_0 $̊ (<ref>) and its fitted version (black ink). (e) Ground truth images for the experiment. (f) Estimated images using temporal phase retrieval.
    





 §.§ Practical Validation


Our experimental setup mimics the setting of Fig. <ref>. A placard that reads “Time–of–Flight” is covered by a semi-transparent sheet, hence K = 2. The sampling rate is Δ≈ 70 × 10^-12 seconds using a our custom designed ToF imaging sensor <cit.>. Overall, the goal of this experiment is to recover the magnitudes {|Γ _k|}_k = 0^k=1 given auto-correlated measurements mł nΔacmył nΔ,̊𝐦∈ℝ^2795. To be able to compare with a “ground truth”, we acquire time-domain measurements ył nΔ$̊ before autocorrelation, whose Fourier-domain phases are intact. In Fig. <ref>(a), we plot the non-autocorrelated measurements𝐲while phase-less measurements𝐦are shown in Fig. <ref>(c) from which we note that the samples are symmetric in time domain due tom = y(cf. <ref>). In the first case (cf. Fig. <ref>(a)) wherey = p *h_K, we have <cit.>,



    y( ω) = | p( ω)|^2h_K^*( ω),  h_K( ω) = ∑_k = 0^K - 1Γ _ku_t_k( ω).


Similar to𝐦in (<ref>), in this case, the measurements read𝐲 = 𝐔𝐃_φ 𝐡and we can estimate the complex-valued vector𝐡. The phase information in𝐡allows for the precise computation of{Γ_k,t_k}_k=0^K-1<cit.>. These “intermediate” measurements serve as our ground truth. The spikes corresponding to the SRF (<ref>) are also marked in Fig. <ref>(a). 

Fourier-domain measurementsh_2^*linked withyłnΔ$̊ are shown in Fig. <ref>(b). Accordingly, h^*_2( ω), ω = ℓω_0, l = {-L, …, L} where ω_0 = 2 π/Δ and L = 20. We estimate the frequency parameters {Γ_k,t_k}_k=0^K-1 using Cadzow's method <cit.>. The resulting fit is plotted in Fig. <ref>(b). 

In parallel to Figs. <ref>(a),(b), normalized, autocorrelated data 𝐦 = 𝐲 are marked in Fig. <ref>(c). The signal 𝐲 is also shown as reference in green dashed line. We also plot the estimated h_2 (autocorrelated SRF). In Fig. <ref>(d), we plot measured and deconvolved vector s = 𝐃_ψ^-1𝐔^+m from which we estimate a_0,a_k,m (<ref>). The result of fitting using Cadzow's method with 41 samples is shown in Fig. <ref>(d).

The reconstructed images, |Γ_0| and |Γ_1|, due to amplitude-phase measurements (our ground truth, Fig. <ref>(a)) are shown in Fig. <ref>(e). Alternatively, the reconstructed images with auto-correlated/intensity-only information are shown in <ref>(f). One can observe the great similarity between the images obtained in both cases, where only a few outliers appear in the phase-less setting. The PSNR values between the maps reconstructed with and without the phase information are of 30.25 dB for |Γ_1| and 48.88 dB for |Γ_0|. These numerical results indicate that, overall, the phase loss that occurs in our autocorrelated measurements still allows for accurate reconstruction of the field of view of the scene. 

Finally, in order to determine the consistency of our reconstruction approach, the PSNR between the original available measurements and their re-synthesized versions—as obtained when reintroducing our reconstructed profiles |Γ_k| into our forward model—are also provided. As shown in Figs. <ref>(a) and (c), the PSNR values in the oracle and phase-less settings correspond to 43.68 dB and 42.41 dB, respectively, which confirms that our reconstruction approach accurately takes the parameters and structure of the acquisition model into account. 

 

§ CONCLUSIONS


In this paper, we have introduced a method to satisfactorily recover the intensities of superimposed echoes of light, using a customized ToF imaging sensor for acquisition and temporal phase retrieval for reconstruction. Up to our knowledge, this is the first method that performs time-stampless sampling of a sparse ToF signal, in the sense that we only measure the amplitudes sampled at uniform instants without caring about the particular sampling times or sampling rates. This innovation can potentially lead to alternative hardware designs and mathematical simplicity as phases need not be estimated in hardware.










1.5





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