Patent Publication Number: US-10768339-B1

Title: Imaging hidden objects

Description:
TECHNICAL FIELD 
     This disclosure relates to imaging and, more particularly, to identifying information of objects hidden from view. 
     BACKGROUND 
     Image sensing and acquisition technologies are often optimized for line-of-sight operations. Image sensing devices normally cannot discern objects that are beyond the line-of-sight or hidden from view. For example, a warfighter may not use an image sensing device to discern information of an armed assailants hidden behind a wall that obstructs the line-of-sight view. 
     Technologies for sensing hidden objects normally rely on a scattering surface, such as a wall to scatter light that can illuminate the hidden objects, and exploit light that returns from the hidden objects to recover information (such as geometric shape or image) of the hidden objects. Those technologies normally require some movements of the scattering surface to gain enough information diversity, which may not be practical in some situations. 
     SUMMARY 
     The present disclosure describes a system, apparatus, and method for identifying an image of an object hidden from view. 
     In some implementations, an imaging system for identifying information of a hidden object includes a light source that can emit narrow-band light. The light source may be a laser source, one or more light emitting diodes, or a broadband light source that emits light filtered by a narrow-band filter. In some cases, the narrow-band light may be coherent and have a linewidth of the narrow-band light is less than, or equal to, one nanometer. In some cases, the narrow-band light may have a frequency range in a visible portion of an electromagnetic spectrum, an infrared, an ultraviolet, or a millimeter wave spectrum. In some cases, the hidden object may be transmissive, translucent, opaque, or have a surface roughness greater than a wavelength of light. 
     In some implementations, the light source may generate an illumination beam of narrow-band light and a second beam of narrow-band light. The second beam may have temporal fluctuations correlated with the first beam. In some cases, a first beam splitter may be positioned along a propagation direction of the narrow-band light. In such cases, the first beam splitter can split the narrow band light into an illumination beam and a probe beam. In some cases, the illumination beam may be directed by the first beam splitter toward a first scattering surface that scatters the illumination beam and directs the scattered illumination beam toward a hidden object. Alternatively or additionally, the illumination beam or the probe beam may be may be directed or relayed by a frequency modulator positioned along the propagation direction of the illumination beam or the probe beam. In some implementations, the frequency modulator may shift the temporal frequency of at least one of the first beam or the second beam. The frequency modulator may be an acousto-optic modulator, an electro-optic modulator, or a moving mirror. 
     In some cases, the probe beam may interfere with the light field and create a first interference pattern on a second scattering surface. In some cases, the first scattering surface at least partially overlaps with the second scattering surface. In some cases, the first scattering surface and the second scattering surface may be surfaces of a wall, a door, a metallic structure, a frosted glass, or a rough mirror. 
     In some implementations, a lock-in camera may detect an irradiance of the interference pattern, monitor temporal variations of the irradiance caused by the temporal frequency shift introduced by the frequency modulator, and identifies a complex-valued light field that represents information of the hidden object based on the temporal variations of the irradiance. The information of the hidden object may include a geometry, image, or hologram of the hidden object. Alternatively or additionally, the lock-in camera may perform numerical approximation to calculate the information of the hidden object. The lock-in camera may calculate the geometry of the hidden object based on a phase component of the complex-valued light field and the image of the hidden object based on a magnitude component of the complex-valued light field. Alternatively or additionally, the lock-in may perform a field propagation, a Fourier transform, or a numerical approximation of the complex-valued light field. In some cases, lock-in camera may include a single-pixel detector or a focal plane array detector. In some cases, the single-pixel detector or the focal plane array detector perform synchronous demodulation of the temporal variations of the irradiance. 
     In some implementations, the phase modulator may modulate the phase of the probe beam to introduce a first phase shift of the probe beam before the probe beam interferes with the light field of the scattered illumination beam diffracted by the hidden object. In some cases, the phase modulator may similarly modulate the phase of the probe beam to introduce a second phase shift of the probe beam and/or a third phase shift of the probe beam. Each phase shifted probe beam may create a different interference pattern with the light field of the scattered illumination beam diffracted by the hidden object. In some cases, the image sensor may detect the more than one interference pattern and the image processor may use the more than one interference pattern to calculate the complex-valued light field. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  illustrates an example imaging system for recovering a hologram of an object hidden from view. 
         FIG. 2A  illustrates another example imaging system for recovering an image of an object hidden from view. 
         FIG. 2B  illustrates an example workflow of computing an image information of the object hidden from view. 
         FIG. 3A  illustrates an example experimental imaging system for recovering an image of an object hidden from view. 
         FIG. 3B  shows example apparatuses of the example experimental imaging system. 
         FIG. 3C  illustrates an example experiment for recovering an image and the corresponding experiment result. 
         FIG. 3D  illustrates another example experiment for recovering an image and the corresponding experiment result. 
         FIG. 3E  illustrates yet another example experiment for recovering an image and the corresponding experiment result. 
         FIG. 4  illustrates an example single-shot imaging system for recovering an image of an object hidden from view. 
         FIG. 5  illustrates an example experimental single-shot imaging system for recovering an image and the corresponding experiment result. 
         FIG. 6A  illustrates another example experimental single-shot imaging system for recovering an image and the corresponding experiment apparatuses. 
         FIG. 6B  illustrates the experiment result of the example experimental single-shot imaging system. 
         FIG. 7  is a flowchart illustrating an example process of recovering information of an object hidden from view. 
     
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     The present disclosure describes techniques for recovering information, such as a hologram, an image, or any geometry of an object obstructed or hidden from an observer&#39;s view. In some implementations, the techniques can exploit the fact that real-world object surfaces are intrinsically rough at optical scale and consequently scatter light. The scattered light can traverse the scene and illuminate the object hidden from view. The scattered light acts as a virtual source of illumination for the object hidden from view. A fraction of the light striking the hidden object can be redirected towards a scattering surface, such as a wall, and subsequently, towards an image sensor used by an observer to sense the hidden object. The scattered light field intercepted by this surface may include obscured object information, so that the scattering surface may be viewed as a virtual detector. This suggests that light scattered by an optically rough surface that is visible to both the observer and the hidden object may be used to detect latent information of the hidden object. 
     In practice, the scattering surface may scramble the phase of the light from the hidden object. As a result, the image on the rough scattering surface, as observed by the image sensor, can be heavily corrupted by multiplicative noise arising from unknown scattering properties (such as complex reflectance) of the surface, and fails to yield discernible information on the hidden object. In some implementations, the disclosed techniques also include methods for circumventing the phase scrambling at the scattering surface to recover the hologram. The light used for recovering the hologram may be a narrow-band light, such as a laser. The light may be split into an illumination beam and a reference beam. At least one of the illumination beam or the reference beam may be spatially or temporally modulated to create diversity to the scattering properties of the surface. The diversity can be exploited to isolate the latent information of the hidden object from corrupting noise. 
       FIG. 1  illustrates an example imaging system  100  for recovering a hologram of an object  150  hidden from view. At a high level, the example imaging system  100  includes a narrow-band light source  110 , a first beam splitter  120 , a spatial light modulator (SLM)  130 , a second beam splitter  160 , an absorber  180 , and an imager  170 . An object  150  is blocked from the view of the imaging system  100  by a diffuser  140 . A diffuser  140  serves as the scattering surface that is simultaneously visible to the imager  170  and the object  150 . An example of the diffuser  140  can be an opaque wall. The light scattered by the diffuser can illuminate the hidden object  150 , and at least a portion of the scattered light may be diffracted towards the diffuser  140 . As a result, the field of the diffracted light incident on the diffuser  140  may include information about the hidden object  150 . The imager  170  that can observe the diffuser  140  may be able to detect the field. 
     In some implementations, the light source  110  can emit narrow-band light, such as a laser. A first beam splitter  120  positioned at the propagation direction of the emitted light can split the light into an illumination beam and a probe beam. The illumination beam and the probe beam can both illuminate on the diffuser  140 . The probe beam can serve the purposes of: (1) converting the phase-variations in the diffracted field into intensity variations that can be sensed by a square law detector (not shown), and (2) heterodyning unresolved spatial frequencies in the diffracted field into a limited passband of an imager observing the diffuser. The diffuser  140  may scatter the illumination beam into a scattered light beam and direct the scattered light beam towards the hidden object  150 . At least a portion of the scattered light beam may be diffracted by the hidden object  150 . The diffracted light includes information of the hidden object  150  to be recovered by the imager  170 . 
     In some implementations, the probe beam may be split by a second beam splitter  160  into a first split beam and a second split beam. The first split beam can be absorbed by an absorber  180 . The second split beam can be directed to the diffuser  140  such that the incident field of the second split beam overlaps with the incident field of the light diffracted by the hidden object  150 . This overlap can record an interference pattern or interferogram on the diffuser  140 . In some cases, the probe beam may be directed by the first beam splitter  120  toward the diffuser  140  without being split by the second beam splitter  160  and overlaps with the light diffracted by the hidden object  150  to create the interferogram on the diffuser  140 . In some cases, the probe beam may be spatially modulated by the SLM  130  to create a phase shift and/or directed by the SLM  130  to the diffuser. In such cases, the second beam splitter may be omitted. 
     The imager  170  may sense the intensity of the interferogram through scattered light from the diffuser  140  redirected by the second beam splitter  160 . In some cases, the imager  170  may be positioned to directly sense the intensity of the interferogram without using the second beam splitter  160 . Based on the intensity of the interferogram, the imager  170  may numerically resolve the complex-valued fields of the light, including the hologram through back propagation. In some implementations, the sensed intensity of the interferogram at the imager  170  can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
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     The term t int  represents the time interval over which the detector integrates photons. The term t cam  represents the average time of flight of light paths from the diffuser to the detector. The term Δt represents the difference in the travel time of the probe beam and the backscattered field incident on the diffuser. The terms r and x denote the transverse coordinates of points on the diffuser and detector, respectively. The term    diffuser  represents the multiplicative noise arising from the unknown scattering properties of the diffuser. The phase offset φ 0  represents the phase sift imparted to the probe beam. 
     In some implementations, the desired hologram i 3 (x;Δt) may be assembled by digitally recombining one or more interferogram images acquired using differently phased probe beams. The phase shifting of the probe beam may help in isolating the hologram term from the unwanted zeroth-order component and twin component. The expression for the hologram can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
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     In some cases, the hologram may be heavily corrupted by multiplicative noises arising from unknown scattering properties (e.g., complex reflectance) of the diffuser  140 . To combat this, a randomized wavefront error may be deliberately introduced to the imager  170 . As a result, the optical point spread function (PSF) may exhibit stochastic character and may be largely uncorrelated with shifted copies of the PSF. This phenomenon can be expressed as:
 
(∫ A     CCD     dx h   cam ( x−r ) h   cam *( x−ŕ ))∝δ( r−ŕ )  (5)
 
     The use of stochastic optical blur may allow tailoring the second-order statistics of the multiplicative noise term    diffuser , and mitigating its influene by aggregating the field amplitude of the hologram i 3 (x;Δt). It follows from Equations (4)-(5) that
 
∫ A     CCD     dx[i   3 ( x;Δt )]=   dr|     diffuser ( r )| 2 ×[   probe *( r,t−t   cam   −Δt ) ( r,t−t   cam )]  (6)
 
     In some cases, the transmissivity of the diffuser  140 , denoted as |   diffuser (r)| 2 , may remain unchanged over the spatial extent of the probe beam. The expression for the integrated field amplitude may then be reduced to an exact linear projection expressed as    |   probe * . In such cases, a hologram may be assembled from the projections    |   probe * , by selecting probe beams from an orthonormal basis set. Examples of basis sets may include spatially uniform fields (planar/spherical) and stochastic fields. The assembly process can mirror the reconstruction scheme in a single pixel compressed sensing imager, so that the machinery of compressive sensing can be exploited to assemble the hologram. In some implementations, a lattice of light spots may be used to sample the diffracted field at the diffuser  140 , at resolutions commensurate with the transverse extent of a single light spot. 
       FIG. 2A  illustrates another example imaging system  200 A for recovering an image of an object  260  hidden from view. The example imaging system  200 A includes a laser source  210 , a first beam splitter  220 , a first SLM  230 , a second beam splitter  270 , a second SLM  280 , and a camera  290 . An object  260  is obstructed from view of the camera  290  by an occluder  250 . A scattering enclosure  240  is simultaneously visible to the camera  290  and the object  260 . 
     The laser source  210  can generate coherent, narrow-band laser light. Coherent light can be a beam of photons that have substantially the same frequency and are all at substantially the same frequency. The first beam splitter  220  may be positioned at the propagation direction of the light source to split the laser light to an illumination beam and a probe beam. The illumination beam may traverse to the scattering enclosure  240  and is scattered by the enclosure  240  to illuminate the hidden object  260 . The portion of the scattering enclosure  240  surface (surface    1 ) that scatters the illumination beam can be considered as a virtual source  240 A of light. At least a portion of the light scattered by the virtual source  240 A may be diffracted by the hidden object  260  and traverses back to illuminate another portion of the scattering enclosure (surface    2 )  240 B. Since the light incident on surface    2  is diffracted by the hidden object  260 , it includes hologram information about the hidden object  260 . Therefore, surface    2  may serve as a virtual detector  240 B that records information about the hologram of the hidden object  260 . The camera  290  may image the virtual detector  240 B surface and use the captured data to recover the hologram. In some implementations, the imaging optics within the camera module  290  may serve the purpose of relaying the light distribution at the virtual detector  240 B onto a remote physical detector (not shown) for image recovery. In some implementations, a first and/or a second SLMs  230 ,  280  may be used to spatially and/or temporally modulate the light beams to tailor the light distribution at the virtual source  240 A and/or the virtual detector  240 B. In some cases, the first SLM  230  may be positioned at the propagation direction of the illumination beam to relay or direct the illumination beam towards the scattering enclosure surface    1  or the virtual source  240 A. In some implementations, the second SLM  280  may be positioned at the propagation direction of the probe beam to relay or direct the probe beam towards the scattering enclosure surface    2    240 B. In some cases, a second beam splitter  270  may be used to relay or direct the probe beam towards the scattering enclosure surface    2    240 B and/or relay or direct the light distributed on the scattering enclosure surface    2    240 B to the camera  290 . The camera  290  may include an image sensor (not shown) to detect or sense light, and an image processor (not shown) to perform digital image processing. 
     As aforementioned, the light scattered by the virtual source  240 A can illuminate the hidden object  260 , and a portion of the light is directed towards the virtual detector  240 B. A hologram can be identified by detecting the light field incident on the virtual detector  240 B. In some implementations, the process of detecting and recovering a hologram can be performed through numerical computations. The light field may be represented by complex-valued variables. The hologram may then be calculated by resolving the complex values of the latent field. 
     In some cases, the example imaging system  200 A can use coherent detection to determine the complex-valued field. Since the illumination beam and the probe beam are originated from the same laser source  210 , their temporal fluctuations are highly correlated. Therefore, the probe beam can be used to mix with the latent field created by light diffracted from the illumination beam for coherent detection. The mixture of the probe beam and illumination beam can create the interferogram on the virtual detector  240 B. The camera  290  may capture the interferogram information and use it to numerically compute the latent scene information of the hidden object  260 . 
       FIG. 2B  illustrates an example workflow  200 B for computing the image information of the object  260  hidden from view. For the purpose of illustration, it is assumed that the example workflow is based on the example imaging system  200 A illustrated in  FIG. 2A . It is to be understood that the example workflow can be used for computing latent scene information for other implementations. One or more elements of the imaging system  200 A are intentionally hidden from the workflow  200 B diagram for simplicity of illustration. 
     As discussed in the illustration of  FIG. 2A , the camera  290  may include an image sensor for detecting the interferogram recorded at the virtual detector  240 B. The image sensor may then convey the detected information to an image processor to perform digital image processing. In some cases, the image processing may be delegated to be performed by a remote detector separate from the camera  290 . In some implementations, assume that the camera  290  has a space-invariant blur h cam , a magnification factor m, and an active sensor area   meter 2 . The expression for the captured image intensity from the interferogram can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       i 
                       cam 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         ; 
                         
                           φ 
                           0 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         
                           t 
                           int 
                         
                       
                       ⁢ 
                       
                         dt 
                         ⁢ 
                         
                           
                              
                             
                               
                                 ∫ 
                                 𝒜 
                               
                               ⁢ 
                               
                                 dr 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       
                                         ℛ 
                                         diffuse 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         r 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   𝒰 
                                                   
                                                     l 
                                                     ⁢ 
                                                     a 
                                                     ⁢ 
                                                     t 
                                                     ⁢ 
                                                     e 
                                                     ⁢ 
                                                     n 
                                                     ⁢ 
                                                     t 
                                                   
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         r 
                                                         ′ 
                                                       
                                                       ⁢ 
                                                       t 
                                                     
                                                     - 
                                                     
                                                       t 
                                                       
                                                         c 
                                                         ⁢ 
                                                         a 
                                                         ⁢ 
                                                         m 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 e 
                                                 
                                                   
                                                     - 
                                                     i 
                                                   
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   
                                                     φ 
                                                     0 
                                                   
                                                 
                                               
                                               ⁢ 
                                               
                                                 
                                                   𝒰 
                                                   
                                                     p 
                                                     ⁢ 
                                                     r 
                                                     ⁢ 
                                                     o 
                                                     ⁢ 
                                                     b 
                                                     ⁢ 
                                                     e 
                                                   
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     r 
                                                     , 
                                                     
                                                       t 
                                                       - 
                                                       
                                                         t 
                                                         
                                                           c 
                                                           ⁢ 
                                                           a 
                                                           ⁢ 
                                                           m 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   } 
                                 
                                 ⁢ 
                                 
                                   
                                     h 
                                     
                                       c 
                                       ⁢ 
                                       a 
                                       ⁢ 
                                       m 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       x 
                                       - 
                                       r 
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                           
                           2 
                         
                         ⁢ 
                         where 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         r 
                       
                     
                     = 
                     
                       m 
                       ⁢ 
                       
                         v 
                         d 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The terms t int , t cam  represent the integration time and average time of flight of light paths from the virtual detector  240 B to the physical detector of the camera  290 . The terms r=mv d  and x denote the transverse coordinates of points on the virtual detector  240 B and physical detector of the camera  290 , respectively. The term    diffuse  represents the complex-valued reflectance associated with the virtual detector surface  240 B. 
     Expanding equation (7) can yield a mixture of three components that represent the captured camera image of the interferogram: (1) A zeroth-order component expressed as |   diffuse   ⊕h cam | 2 +|   diffuse     probe ⊕h cam | 2 , (2) a twin component expressed as conj (   diffuse   ⊕h cam )×(   diffuse     probe ⊕h cam ), and (3) a hologram component of interest expressed as (   diffuse   ⊕h cam )×conj(   diffuse     probe ⊕h cam ). The task then becomes isolating the hologram component from the zeroth-order component and the twin component. 
     As shown in equation (7), a phase shift φ 0  can be introduced to the probe beam field represented by    probe . This phase shifting may be enabled by modulating the probe beam using the second SLM  280 . In some implementations, a phase shift may also be introduced by modulating the illumination beam using the first SLM  230 . The phase shift may help isolate the hologram field denoted by    3  from the interfering zeroth-order component and twin components. The hologram field    3  may be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             𝒰 
                             3 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
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                             ⁢ 
                             
                               drd 
                               ⁢ 
                               
                                 
                                   r 
                                   ′ 
                                 
                                 [ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
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                                                   ⁢ 
                                                   a 
                                                   ⁢ 
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                                               ⁡ 
                                               
                                                 ( 
                                                 
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                                                   - 
                                                   r 
                                                 
                                                 ) 
                                               
                                             
                                             × 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
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                                                 ⁢ 
                                                 a 
                                                 ⁢ 
                                                 m 
                                               
                                               ⋆ 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 x 
                                                 - 
                                                 
                                                   r 
                                                   ′ 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               
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                                               ⁡ 
                                               
                                                 ( 
                                                 r 
                                                 ) 
                                               
                                             
                                             × 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               ℛ 
                                               diffuse 
                                               ⋆ 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 r 
                                                 ′ 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   × 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           ( 
                           
                             
                               ∫ 
                               0 
                               
                                 t 
                                 int 
                               
                             
                             ⁢ 
                             
                               dt 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 𝒰 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     r 
                                     , 
                                     
                                       t 
                                       - 
                                       
                                         t 
                                         
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                                           ⁢ 
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                                           ⁢ 
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                                     p 
                                     ⁢ 
                                     r 
                                     ⁢ 
                                     o 
                                     ⁢ 
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                                     ⁢ 
                                     e 
                                   
                                   ⋆ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       r 
                                       ′ 
                                     
                                     , 
                                     
                                       t 
                                       - 
                                       
                                         t 
                                         
                                           c 
                                           ⁢ 
                                           a 
                                           ⁢ 
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                           ) 
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           { 
                           
                             ∫ 
                             
                               
                                 ∫ 
                                 𝒜 
                               
                               ⁢ 
                               
                                 dr 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 d 
                                 ⁢ 
                                 
                                   
                                     r 
                                     ′ 
                                   
                                   [ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   h 
                                                   
                                                     c 
                                                     ⁢ 
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                                                     m 
                                                   
                                                 
                                                 ⁡ 
                                                 
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                                                     ′ 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
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                                           diffuse 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           r 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
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                                         ⁡ 
                                         
                                           ( 
                                           r 
                                           ) 
                                         
                                       
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                                           ⋆ 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             r 
                                             ′ 
                                           
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                                             p 
                                             ⁢ 
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                                           * 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             r 
                                             ′ 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             ] 
                           
                           } 
                         
                         × 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           { 
                           
                             
                               ∫ 
                               0 
                               
                                 t 
                                 int 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           exp 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 - 
                                                 i 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
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                                               ⁢ 
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                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 
                                                   v 
                                                   _ 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     t 
                                                     - 
                                                     
                                                       t 
                                                       
                                                         c 
                                                         ⁢ 
                                                         a 
                                                         ⁢ 
                                                         m 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         × 
                                       
                                     
                                   
                                   
                                     
                                       
                                         exp 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             i 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             2 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             π 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               v 
                                               _ 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 
                                                   t 
                                                   
                                                     c 
                                                     ⁢ 
                                                     a 
                                                     ⁢ 
                                                     m 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               d 
                               ⁢ 
                               t 
                             
                           
                           } 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ℛ 
                                 diffuse 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   𝒰 
                                   latent 
                                 
                                 ⊗ 
                                 
                                   h 
                                   
                                     c 
                                     ⁢ 
                                     a 
                                     ⁢ 
                                     m 
                                   
                                 
                               
                             
                             ) 
                           
                           × 
                           
                             conj 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ℛ 
                                   diffuse 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     𝒰 
                                     
                                       p 
                                       ⁢ 
                                       r 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       b 
                                       ⁢ 
                                       e 
                                     
                                   
                                   ⊗ 
                                   
                                     h 
                                     
                                       c 
                                       ⁢ 
                                       a 
                                       ⁢ 
                                       m 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     The integrated amplitude of the complex-valued hologram    3 (x) can be computed as: 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       𝒜 
                     
                     ⁢ 
                     
                       
                         
                           𝒰 
                           3 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       dx 
                     
                   
                   = 
                   
                     
                       
                         t 
                         
                           i 
                           ⁢ 
                           n 
                           ⁢ 
                           t 
                         
                       
                       × 
                       
                         { 
                         
                           ∫ 
                           
                             
                               ∫ 
                               𝒜 
                             
                             ⁢ 
                             
                               dr 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   r 
                                   ′ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           ∫ 
                                           𝒜 
                                         
                                         ⁢ 
                                         
                                           dx 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               h 
                                               
                                                 c 
                                                 ⁢ 
                                                 a 
                                                 ⁢ 
                                                 m 
                                               
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 x 
                                                 - 
                                                 r 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               h 
                                               
                                                 c 
                                                 ⁢ 
                                                 a 
                                                 ⁢ 
                                                 m 
                                               
                                               ⋆ 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 x 
                                                 - 
                                                 
                                                   r 
                                                   ′ 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     × 
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ℛ 
                                                   diffuse 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   r 
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 𝒰 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   r 
                                                   ) 
                                                 
                                               
                                               × 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   ℛ 
                                                   diffuse 
                                                   ⋆ 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     r 
                                                     ′ 
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 
                                                   𝒰 
                                                   
                                                     p 
                                                     ⁢ 
                                                     r 
                                                     ⁢ 
                                                     o 
                                                     ⁢ 
                                                     b 
                                                     ⁢ 
                                                     e 
                                                   
                                                   * 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     r 
                                                     ′ 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         } 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         
                           
                             t 
                             
                               i 
                               ⁢ 
                               n 
                               ⁢ 
                               t 
                             
                           
                           × 
                           
                             { 
                             
                               ∫ 
                               
                                 
                                   ∫ 
                                   𝒜 
                                 
                                 ⁢ 
                                 
                                   dr 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   d 
                                   ⁢ 
                                   
                                     
                                       r 
                                       ′ 
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           
                                             χ 
                                             
                                               c 
                                               ⁢ 
                                               a 
                                               ⁢ 
                                               m 
                                             
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 r 
                                                 ′ 
                                               
                                               - 
                                               r 
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       ℛ 
                                                       diffuse 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       r 
                                                       ) 
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                     𝒰 
                                                     ⁡ 
                                                     
                                                       ( 
                                                       r 
                                                       ) 
                                                     
                                                   
                                                   × 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       ℛ 
                                                       diffuse 
                                                       ⋆ 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         r 
                                                         ′ 
                                                       
                                                       ) 
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                     
                                                       𝒰 
                                                       
                                                         p 
                                                         ⁢ 
                                                         r 
                                                         ⁢ 
                                                         o 
                                                         ⁢ 
                                                         b 
                                                         ⁢ 
                                                         e 
                                                       
                                                       * 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         r 
                                                         ′ 
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                             } 
                           
                         
                         | 
                         
                             
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             w 
                             ⁢ 
                             here 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               χ 
                               
                                 c 
                                 ⁢ 
                                 a 
                                 ⁢ 
                                 m 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   r 
                                   ′ 
                                 
                                 - 
                                 r 
                               
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       = 
                       
                         
                           ( 
                           
                             
                               ρ 
                               max 
                             
                             
                               
                                 r 
                                 ′ 
                               
                               - 
                               r 
                             
                           
                           ) 
                         
                         × 
                         
                           
                             J 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                 
                                   
                                     r 
                                     ′ 
                                   
                                   - 
                                   r 
                                 
                                 
                                   ρ 
                                   max 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     The term ρ max  in equation (9) can represent the cutoff frequency of the imaging optics for circular pupils. If the camera blur is compactly supported, i.e., (x cam ( r −r)≈δ( r −r)), the expression for integrated amplitude of the hologram may be approximated as: 
     
       
         
           
             
               
                 
                   w 
                   ⁢ 
                   
                     = 
                     def 
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           ∫ 
                           𝒜 
                         
                         ⁢ 
                         
                           
                             
                               𝒰 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           dx 
                         
                       
                       ) 
                     
                     ≈ 
                     
                       
                         t 
                         
                           i 
                           ⁢ 
                           n 
                           ⁢ 
                           t 
                         
                       
                       × 
                       
                         { 
                         
                           
                             ∫ 
                             𝒜 
                           
                           ⁢ 
                           
                             dr 
                             ⁢ 
                             
                               
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     It can be shown from equation (10) that the integrated amplitude (w) is invariant to phase distortions arising from surface roughness at the virtual detector  240 B. Furthermore, w∝   |   probe *  for virtual detector surface  240 B with constant reflectivity. As such, the hologram image field    recon  may be assembled from inner product measurements associated with an orthonormal basis set. Assume that the probe beam field    probe  is obtained by paraxial propagation of a spatial pattern (E k,l ) defined on the P×P grid of the SLM  280  with pixel pitch Δμm. This implies that    probe =E k,l ⊕h fsp   z , where h fsp   z  is the free-space blur associated with a z meter path from the SLM  280  to the virtual detector  240 B. Let w k,l  denote the integrated amplitude associated with the spatial pattern E k,l , the hologram of the hidden scene elements may be assembled from the P 2  inner product measurements associated with the P 2  basis patterns, as shown in equation (11), which in turn can be calculated to yield the expression in equation (12). 
     
       
         
           
             
               
                 
                   
                     
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     The rect( . . . ) function in equations (11) and (12) emerges from the orthogonality of the basis patterns. As expressed in equation (12), the image of the hidden object  260  may be recovered by numerical back-propagation of the complex-valued image field    recon . 
     In some implementations, the speckle-like appearance of the secondary illumination  295  may arise from scattering at the virtual source  240 A. This scattering may not impact image recovery, but it may introduce speckle artifacts in the reconstruction. In some cases, the complex-valued reflectivity of the scattering surface    diffuse  as experienced by the field and the probe beam may be identical, as the speckle fields produced by the plane wave components constituting the illumination beam and the probe beam can exhibit a large degree of spatial correlation in accord with the “memory effect.” The “memory effect” (in reflection mode) may stem from near identical optical path length differences experienced by two mutually coherent beams with small angular separation. 
       FIG. 3A  illustrates an example experimental imaging system  300 A for recovering an image of an object hidden from view. In this example system  300 A, object  330 A to be detected is embedded between two diffusers. The technologies used by the example imaging system  300 A to recover the image may include interferometry, holography and super-resolution. 
     As discussed previously, the example imaging systems disclosed in the present disclosure can recover images by sensing the interferogram created by the light scattered by the obscured objects and the probe beam, as viewed on a remote scattering surface. Therefore, an interferometer may be used in the experiment of the example imaging system  300 A. In the present example, a free-space Mach-Zehnder interferometer may be used. An example specification  350 A of the interferometer is shown on the right hand side of  FIG. 3A . 
       FIG. 3B  shows example apparatuses  300 B of the example experimental imaging system  300 A. The example apparatuses  300 B include a non-polarizing cube beam splitter  320 B, endowed with a laser notch filter thin film coating at 532 nm. The beam splitter  320 B can be used to derive an illumination/object beam  322 B and a probe/reference beam  324 B from a single collimated laser source  310 B. The propagation length of each arm of the interferometer may exceed two meters to mimic the observation of a hidden object at standoff. The hidden object  330 B can be a transmissive element that is embedded between two ground glass diffusers  350 B and  360 B. The diffusers may serve as surrogates for the scattering wall. Light emerging from diffuser  350 B may act as virtual source. Diffuser  360 B may intercept light scattered by a hidden object  330 B, thereby acting as a virtual detector. 
     A pellicle beam splitter  340 B may be used to combine the scattered light from the hidden object with the probe beam and direct the light towards the camera using a folding mirror. The use of a pellicle beam splitter  340 B can avoid path length distortions in the plane wave components that make up the latent field. The angular separation of the scattered field and the probe beam can be controlled by adjusting the orientation of the pellicle beam splitter  340 B. 
     A diffraction limited imager or camera  380 B may be used to record the interference patterns observed at the virtual detector surface. The reflective SLM  370 B in the reference arm may provide the phase diversity required to isolate the hologram term from the zeroth order and twin components. The use of a SLM  370 B may allow engineering probe beams from an orthonormal basis, while introducing precise phase-retardation/path length differences between the interferometer arms. Liquid Crystal on Silicon (LCoS) devices SLM such as the Holoeye PLUTO may afford such capabilities at real-time rates. In some cases, the Parallel-Aligned (PA-LCoS) variety is of special interest in that it permits phase-only modulation of the incident light without residual amplitude modulation. A half-wave plate beam splitter  390 B in the laser beam path may be used to help in aligning the polarization state of the laser beam with the slow axis of the phase-only SLM  370 B. This can avoid undesired coupling between the fast and slow (phase-shifted) components of the light emerging from the SLM  370 B. 
     A series of increasingly sophisticated experiments using experimental system  300 A and apparatuses  300 B are illustrated in  FIGS. 3C-3E  to show the feasibility of image recovery techniques described in the present disclosure. The development of the experiments may generally include two phases. The first phase may demonstrate the feasibility of image recovery, identify design requirements, and explore practical limits. The second phase may examine approaches to circumvent those limits. 
       FIG. 3C  illustrates an example experiment  300 C for recovering an image and the corresponding experiment result  310 C. In the illustrated experiment  300 C, the goal is to acquire a “gold-standard” hologram of the hidden object  360 C. The camera  330 C is repositioned to enable the acquisition of the hologram, as sensed at the virtual detector  350 C. The hologram is associated with a two-meter focal length planoconvex lens embedded between the two diffusers that also serve as virtual source  340 C and virtual detector  350 C. 
     The “gold-standard” hologram  310 C may be recovered by processing four images of the fringe patterns  320 C obtained by stepping through four phase shifts on the reflective SLM  370 C. The radial structure of the fringes is consistent with the interference of a collimated probe beam with the converging beam emerging from the hidden lens. The phase of the estimated hologram is shown in the phase inset  390 C. Numerical propagation of the estimated hologram shows the converging/focusing structure of the phasefront emerging from the plano-convex lens. Furthermore, it is observed that the minimum spot size can be obtained for a propagation distance of 1.7 meters, consistent with the separation between the phase object  340 C and the camera  330 C. The experiment result of the “gold-standard” hologram  310 C may have reaffirmed the notion that scattering at the virtual detector corrupts the phase of the hidden object field, while still preserving hidden object information. 
       FIG. 3D  illustrates another example experiment  300 D for recovering an image and the corresponding experiment result  310 D. The illustrated example experiment  300 D may be considered a natural progression of example experiment  300 C illustrated in  FIG. 3C . In this example experiment  300 D, diffraction limited imaging optics may be used to relay the fringe pattern observed at the virtual detector  350 D to a remote image detector  330 D. 
     The hologram of the phase object can be recovered by processing four images  320 D of the fringe patterns obtained by stepping through four phase shifts on the reflective SLM  370 D. The radial structure of the fringes is consistent with the interference of a collimated probe beam with the converging beam emerging from the hidden lens. The distinct speckle structure in the fringe patterns arise from scattering at the virtual detector  350 D surface. The phase of the estimated hologram is shown in the phase inset  390 D. Numerical propagation of the estimated hologram shows the converging/focusing structure of the phasefront emerging from the plano-convex lens. Furthermore, it is observed that the minimum spot size is obtained for a propagation distance of 1.7 meters, consistent with the separation between the phase object  340 D and the detector  330 D. The experiment result  310 D reaffirms the notion that scattering at the virtual detector corrupts the phase of the hidden object field, while still preserving object information. 
       FIG. 3E  illustrates yet another example experiment  300 E for recovering an image and the corresponding experiment result  310 E. The example experiment  300 E can be used to examine the impact of scattering at the virtual source  340 E and virtual detector  350 E. Similar to example experiment  300 D, diffraction limited optics may be used to relay the fringe pattern observed at the virtual detector  350 E to the remote detector  330 E. The sections of the apparatus leading up to the virtual detector  350 E resemble the experimental setup of example experiment  300 D. The distinction may lie in computational recording of the hologram on a scattering surface, and relaying of the hologram to the remote detector  330 E. 
     Examination of the phase of the hologram confirms the lack of any discernible structure. Numerical back propagation of the hologram by a distance commensurate with the spacing between the virtual detector  350 E and the hidden phase object (0.28 m)  360 E reveals a distinctly quadratic phase structure. The experiment result  310 E shows that scattering at the virtual source  340 E and virtual detector  350 E may corrupt the phase of the hidden object field, while still preserving object information. 
       FIG. 4  illustrates an example single-shot imaging system  400  for recovering an image of an object  440  hidden from view. The aforementioned implementations generally exploit spatial diversity in the phase of the split beams to perform image recovery. The hologram component is isolated from the corrupting influence of the zeroth-order component and twin component based on knowledge of the phase shift applied to the illumination beam and/or probe beam. The hologram image is assembled by processing a sequence of interference pattern images acquired under the phase shifts of the probe beam. By contrast, the example imaging system  400  illustrated in  FIG. 4  can adopt a single-shot approach to recover the hologram term instead of assembling and processing a sequence of images. The approach is based on an operation of a heterodyne interferometer, which can exploit diversity in the temporal frequency instead of spatial diversity in the phases of the beams. 
     At a high level, the example single-shot imaging system includes a laser source  410 , an Acousto-optic modulator (AOM)  420 , a virtual source  430  surface, a virtual detector  450  surface, a lock-in camera  460 , and a RF driver  470 . The example single-shot imaging system  400  can use the virtual source  430  as scattering surface for diffuse illumination and perform indirect imaging of the hidden objects  440  through the virtual detector  450  surface. The single-shot imaging system  400  can use one or more AOM  420  to upshift the frequency of the illumination beam by v b . In some implementations, the frequency shift v b  may be generated by an RF driver  470 . The difference v b  in the temporal frequencies of the interfering fields may be much less than the frequency of the laser light,  v , so that the wavelength of the light in the two beams is nearly identical. It is to be understood that the AOM  420  may apply different temporal frequency shifts for different implementations. 
     The computation of image of the hidden object  440  using the example single-shot imaging system can begin from using the lock-in camera  460  to capture the instantaneous light intensity of the interferogram distributed on the virtual detector  450 . The light intensity sensed by the lock-in camera  460  can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
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     The terms r=mv d  and x denote the transverse coordinates of points on the virtual detector  450  and the lock-in camera  460 , respectively. The term z, 900   diffuse  represents the complex-valued reflectance associated with the virtual detector  450  surface. 
     In some implementations, it is assumed that the laser source  410  is monochromatic. As such, the expressions for the instantaneous field in illumination beam and probe beam of the interferometer may be expressed in the complex phasor form as:
 
 ( r,t )=A( r )exp(− i (   v +v   b ) t )
 
   probe ( r,t )=A probe ( x )exp(− i v t )  (14)
 
     Incorporating equation (14) into equation (13) can yield the following revised expression for the instantaneous detector intensity: 
     
       
         
           
             
               
                 
                   
                     
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     The first term in equation (15) represents the zeroth-order field contribution. The second and third terms represent the hologram and twin contributions. It is shown from equation (15) that the temporal frequency associated with the hologram term is distinct from the other terms. As such, a synchronous demodulation strategy that exploits the orthogonality of complex sinusoids may be used in isolating the hologram term from the other terms. Synchronous demodulation may be implemented by the lock-in camera  460 . In some implementations, a single-pixel detector, focal-plane array detector, or other similar devices may be used for synchronous demodulation to recover the hologram of the hidden object  440  from other corrupting fields. 
       FIG. 5  illustrates an example experimental single-shot imaging system  500  for recovering an image and the corresponding experiment result  590 . In this example, the experimental imaging system  500  may be used to recover the hologram of a transmissive resolution target  510  hidden behind a diffuser  520 . The example imaging system  500  can be set up based on a Mach-Zehnder interferometer adapted for hologram recovery. In the particular experiment, light from a 532 nm linearly-polarized collimated laser source  530  (e.g., Vortran Stardus  532 - 40 , beam diameter 0.8 mm) can be used to provide the illumination beam and probe beam for the interferometer. One or more AOMs  540  (e.g., Intraaction AOM-802AF1) may be used to upshift the temporal frequency of the illumination beam by 40 KHz, relative to the temporal frequency of the probe beam. The illumination beam modulated by the AOM  540  may be expanded using a 5X Beam expander  570 . An example United States Air Force (USAF) resolution target (RES-1, Newport) inserted in the illumination path is used as the hidden target  510 . A neutral density (ND) filter  580  can be adopted to introduce a power mismatch between the two beams, where the optical density of the ND  580  equals 1. Light from the two beams is combined using a non-polarizing beam splitter (BS 2 ) before being directed towards a diffuser  520  (e.g., Thorlabs DG100X100-1500) that obscures the view of the hidden target  510 . A lock-in camera  550 A (Heliotis Helicam C3) observing the light distribution at the diffuser  520  can provide a hologram of the hidden target  510 . The real and imaginary components of the acquired hologram are shown in the image insets  590  of  FIG. 5 . It is shown in the result image  560  that the example imaging system  500  can fully resolve the number “3”, following back-propagation from the diffuser  520  to the hidden target  510 . 
       FIG. 6A  illustrates another example experimental single-shot imaging system  600  for recovering an image and the corresponding experiment apparatuses  650 . The example experiment  600  may be used to validate the single-shot hologram recovery concept. The example experimental apparatuses  650  may be set up as a Mach-Zehnder interferometer adapted for hologram recovery. In this experiment, the goal is to detect the hologram in the recombination arm  610  of the example imaging system  600 . The example imaging system  600  may include interferometric apparatuses, a ND filter  620 , and a lock-in camera  630 . In some cases, the lock-in camera  630  can only discern a weak sinusoid against a strong background. In other words, the power of the hologram component can be much weaker than the zeroth-order and twin components, which makes it difficult to be isolated. The inclusion of the ND filter  620  may help examine the impact of power mismatch on the ability to recover the complex fields of the hologram. The results of the example experiment  600  are illustrated in  FIG. 6B . 
       FIG. 6B  illustrates experiment result  690  of the example experimental single-shot imaging system  600 . The fringes observed in the phase of the estimated hologram  690  arise from the deliberate angular misalignment of the illumination beam and the probe beam. As shown in the image insets  690  in  FIG. 6B , the power mismatch does not have an adverse impact on the ability to recover complex-valued fields in a Mach-Zehnder heterodyne interferometer  600 . The single-shot phase retrieval using the example experimental imaging system  600  also shows insensitivity to fringe drift arising from pointing errors in the illumination beam and/or the probe beam. 
       FIG. 7  is a flowchart illustrating an example process  700  of recovering information of an object hidden from view. The example process  700  may be performed by any imaging system or imager in accordance with the disclosure. 
     At  710 , a light source generates a first beam of narrow-band light. 
     At  720 , the light source generates a second beam of narrow-band light. In some cases, the light source may be a laser source, one or more light emitting diodes, or a broadband light source with a narrow-band light filter. In some cases, the light source may include a narrow-band light source that can emit narrow-band light and a first beam splitter that can split the narrow-band light into the first beam and the second beam. In some cases, the narrow-band light may be a coherent light and a linewidth of the narrow-band light may be less than or equal to one nanometer. In some cases, the narrow-band light has a frequency range in a visible portion of an electromagnetic spectrum, an infrared, an ultraviolet, or a millimeter wave spectrum. In some cases, the first beam may be an illumination beam and second beam may be a probe beam. 
     At  730 , a frequency modulator may be positioned along a propagation direction of at least one of the first beam or the second beam to shift a temporal frequency of at least one of the first beam or the second beam. In some cases, the frequency modulator may be an electro-optic modulator or a moving mirror. 
     At  740 , the frequency modulator or the first beam splitter may direct the first beam toward a first scattering surface and direct the second beam is directed towards a second scattering surface. Both the first scattering surface and the second scattering surface may have a surface roughness greater than a wavelength of light. The first scattering surface may scatter the first beam into a scattered light that illuminates a hidden object. The hidden object may reflect at least a portion of the scattered light towards the second scattering surface. The at least a portion of the scattered light may interfere with the second beam and produces an interference pattern on the second scattering surface. 
     At  750 , a lock-in camera may detect an irradiance of the interference pattern and monitor temporal variations of the irradiance caused by the temporal frequency shift introduced by the frequency modulator. 
     At  760 , the lock-in camera may identify a complex-valued light field that represents information of the hidden object based on the temporal variations of the irradiance. The information of the hidden object may include a hologram, an image, or any geometry information of the hidden object. In some cases, the lock-in camera may include a single-pixel detector or a focal plane array detector. The single-pixel detector or the focal plane array detector may perform synchronous demodulation of the temporal variations of the irradiance. 
     In some cases, the lock-in camera may further perform numerical approximation to identify a hologram of the hidden object based on the complex-valued light field. In some cases, the lock-in camera may perform numerical approximation to identify an image of the hidden object based on a magnitude component of the complex-valued light field. In some cases, the lock-in camera may perform numerical approximation to identify a geometry of the hidden object based on a phase component of the complex-valued light field. In some implementations, the lock-in camera may further perform a field propagation, a Fourier transform, or a numerical approximation of the complex-valued light field. 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention.