Three-dimensional ghost imaging lidar via sparsity constraint

Three-dimensional (3D) remote imaging attracts increasing attentions in capturing a target’s characteristics. Although great progress for 3D remote imaging has been made with methods such as scanning imaging lidar and pulsed floodlight-illumination imaging lidar, either the detection range or application mode are limited by present methods. Ghost imaging via sparsity constraint (GISC), enables the reconstruction of a two-dimensional N-pixel image from much fewer than N measurements. By GISC technique and the depth information of targets captured with time-resolved measurements, we report a 3D GISC lidar system and experimentally show that a 3D scene at about 1.0 km range can be stably reconstructed with global measurements even below the Nyquist limit. Compared with existing 3D optical imaging methods, 3D GISC has the capability of both high efficiency in information extraction and high sensitivity in detection. This approach can be generalized in nonvisible wavebands and applied to other 3D imaging areas.


CS sparse reconstruction
CS sparse reconstruction is a method to finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations 8,9 . The measurement model can be expressed as (1) where A is an M×N sensing matrix (M<<N), y (an M×1 vector) is the measurement signals and e (an M×1 vector) denotes the detection noise.
where τ is a nonnegative parameter, 2 V and 1 V represent the Euclidean norm and the 1 -norm of V, respectively. By now, there are many sparse reconstruction algorithms to solve the convex optimization program described in Eq.

The measurement framework of 3D GISC lidar
As shown in Supplementary Figure 1, the measurement process of 3D GISC lidar can be expressed as follows in the framework of CS sparse reconstruction: (1) When the laser emits a pulse, the rotating diffuser will modulate the laser and produce a random speckle pattern. The CCD in the reference path will record the speckle pattern's 2D gray distribution I i (m×n pixels, =1 (2) After M independent modulations, we can obtain M frames of independent 2D gray distributions, then the sensing matrix A (M×N) of CS sparse reconstruction can be obtained. Correspondingly, the M independent time-resolved one-dimensional distributions are used to form the measurement data Y (Y=[y 1 , y 2 , …, y q , … , y Q ], y q is an (3) Based on the CS sparse reconstruction model described in Eq. (1), the measurement process of 3D GISC lidar can be expressed as

Image reconstruction of 3D image
Considering the structure property of 3D images in depth direction, the target's 3D

The design of the system's spatial transverse resolution
In accordance to the principle of ghost imaging, the system's transverse resolution is determined by the transverse size of speckle pattern on the object plane, which is given

The depth resolution for 3D GISC lidar
For 3D GISC lidar, similar to the Rayleigh criterion of spatial transverse resolution, when the difference of the photon's fight time from two objects is smaller than the laser's pulse width, the reflected signal, like the curve shown in Fig.2a, can not be resolved, which will cause the phenomenon that the information of two objects appear in the same tomographic image. However, the relative intensities of the two objects' tomographic images are different in different depths. In the process of 3D image reconstruction, except for exploiting the general assumption of the image's sparsity, we have also used the priori knowledge that the target's images at different depths have no spatial overlap for a

Supplementary Figures
Supplementary Figure 1: The measurement framework of 3D GISC lidar system with pseudo-thermal light. a 1 -a M are the speckle pattern's 2D gray distributions recorded by the reference CCD in different modulations. b 1 -b M are the corresponding timeresolved one-dimensional intensity distribution recorded by the time-resolved singlepixel bucket detector in the object path.