Three-Dimensional Terahertz Coded-Aperture Imaging Based on Back Projection

Terahertz coded-aperture imaging (TCAI) can overcome the difficulties of traditional radar in forward-looking and high-resolution imaging. Three-dimensional (3D) TCAI relies mainly on the reference-signal matrix (RSM), the large size and poor accuracy of which reduce the computational efficiency and imaging ability, respectively. According to the previous research on TCAI, traditional TCAI cannot reduce the heavy computational burden while the improved TCAI achieve reconstructing the target parts of different ranges in parallel. However, large-sized RSM still accounts for the computational complexity of traditional TCAI and the improved TCAI. Therefore, this paper proposes a more efficient imaging method named back projection (BP)-TCAI (BP-TCAI). Referring to the basic principle of BP, BP-TCAI can not only divide the scattering information in different ranges but also project the range profiles into different imaging subareas. In this way, the target parts in different subareas can be reconstructed simultaneously to synthesize the whole 3D target and thus decomposes the computational complexity thoroughly. During the pulse compression and projection processes, the signal-to-noise ratio (SNR) of BP-TCAI is also improved. This present the imaging method, model and procedures of traditional TCAI, the improved TCAI and the proposed BP-TCAI. Numerical experimental results prove BP-TCAI to be more effective and efficient than previous imaging methods of TCAI. Besides, BP-TCAI can also be seen as synthetic aperture radar (SAR) imaging with coding technology. Therefore, BP-TCAI opens a future gate combining traditional SAR and coded-aperture imaging.


Introduction
Traditional radar imaging method relies too much on the relative motion between the radar and target to achieve forward looking and staring imaging. Fortunately, terahertz coded-aperture imaging (TCAI) [1][2][3] can overcome the difficulties with a high resolution. Based on the imaging principles of both optical coded-aperture imaging [4,5] and radar coincidence imaging (RCI) [6,7], TCAI achieves reconstructing the target by producing spatiotemporal independent signals with the coded aperture. Moreover, terahertz waves (0.1-10 THz) allow visualization of hidden object at millimeter level [8][9][10] with stronger penetration capability than light and higher resolution over microwave.
Because of the promising ability in flexible manipulation on terahertz and millimeter waves, metasurfaces have been applied into areas of high-resolution computational imaging [11,12] and some scanning devices [13,14]. Besides, the Harvard Robotics Laboratory (HRL) manufactures an economic The signal arriving at the coded aperture is assumed as plane wave. Therefore, the time delay terms for each transmitting element of the coded aperture are the same and they can be set as zeroes. As the coded aperture contains I transmitting elements, the radiating signal after modulation is deduced as: to strike a balance of high-resolution imaging and beam forming. Then, the radiating signal illustrates the 3D imaging area. High-resolution imaging relies on strong spatiotemporal independence of the radiation field on the 3D imaging area. Therefore, we hope to improve the phase modulation degree for high-resolution imaging. However, excessive modulation of traditional TCAI will damage the beam forming and thus reduce the imaging distance. The proposed BP-TCAI achieves better imaging performance with no requirements of shortening the imaging distance, which will be further described in Sections 2.3. Reflected by the 3D target, the echo signal arriving at the single detector is written as: where k  is the scattering coefficient corresponding to the k-th grid-cell, , , i k o t is the total time delay passing though the i-th transmitting element, the k-th grid cell and the receiver. K is the grid-cell number of the 3D imaging area.

Signal Propagation
The transmitting signal before modulation is linear frequency modulation (LFM) signal and it is formulated as: where s t (t) is the transmitting signal at time t, f is the signal frequency and j denotes the imaginary unit. Besides, f = f 0 + 0.5γt, where f 0 and γ are the center frequency and chirp rate, respectively. The signal arriving at the coded aperture is assumed as plane wave. Therefore, the time delay terms for each transmitting element of the coded aperture are the same and they can be set as zeroes. As the coded aperture contains I transmitting elements, the radiating signal after modulation is deduced as: where ϕ i (t) is the random phase-modulation term for the i-th transmitting element at time t. ϕ i (t) changes randomly over fast time within the range of [p l , p h ], where p l and p h are the minimum and maximum value of ϕ i (t). Phase modulation range is proportional to high-resolution imaging, while excessive modulation will damage the beam forming. [p l , p h ] should be set as proper value to strike a balance of high-resolution imaging and beam forming. Then, the radiating signal illustrates the 3D imaging area. High-resolution imaging relies on strong spatiotemporal independence of the radiation field on the 3D imaging area. Therefore, we hope to improve the phase modulation degree for high-resolution imaging. However, excessive modulation of traditional TCAI will damage the beam forming and thus reduce the imaging distance. The proposed BP-TCAI achieves better imaging performance with no requirements of shortening the imaging distance, which will be further described in Section 2.3. Reflected by the 3D target, the echo signal arriving at the single detector is written as: where β k is the scattering coefficient corresponding to the k-th grid-cell, t i,k,o is the total time delay passing though the i-th transmitting element, the k-th grid cell and the receiver. K is the grid-cell number of the 3D imaging area. Apparently, the echo signal cannot be sampled directly at THz band. The same as the transmitting signal written in Equation (1), the reference signal is also generated for sampling the echo signal. By mixing the echo signal in Equation (3) with the reference signal, the sampled signal is formulated as Compared with Equation (3), the item of exp(j2π f t) is eliminated. Therefore, the echo signal is down converted into baseband signal and it can be sampled directly in a general way.

Imaging Model
Based on time discretion of Equation (4), the mathematical model of traditional TCAI is deduced as: where Sr and S are the echo vector (EV) and reference-signal matrix (RSM), respectively. β is the scattering-coefficient vector (SCV). w is the measurement noise vector (MNV). N and K are the sampling-time and grid-cell numbers, respectively. s(t n , k), the array element of S, is formulated as: where t n denotes the n-th sampling time. Table 1 has shown the imaging procedure of traditional TCAI as below. Table 1. Imaging procedure of traditional terahertz coded-aperture imaging (TCAI).

Imaging process
Step 1: Obtain the echo vector (EV) Sr by the following procedures.
(1) The computer controls the transmitter to send signal.
(2) Controlled by the computer, the coded aperture randomly modulates the transmitting signal.
(3) The single detector receives the echo signal, which carries the 3D target information.

GM-Based TCAI
With the knowledge of solving linear equations, Equation (5) can be solved by compressed sensing (CS) algorithm. However, when the SNR is too low, the 3D target becomes difficult to be reconstructed in time domain. Due to large amount of meshed grid cells, the large-scaled RSM results in heavy computational burden. To solve problems of low SNR and increasing computational complexity, the improved TCAIs proposed in Refs. [19,20] transforms the EV and RSM into range domain. Essentially, the transformation is the process of pulse compression, which can improve the SNR in a certain degree and distinguish the target information in different ranges [17]. However, the range information is difficult to be detected when the SNR is under −10 dB. Therefore, GM-TCAI proposed in Ref. [18] adopts GM to find the range cells containing scattering information. Both Refs. [17,18] belong to range-domain TCAIs while Ref. [18] is an improved version of Ref. [17]. Therefore, this paper only shows the basic imaging procedures and model of GM-TCAI and more details are presented in Ref. [18].

EV Extraction of GM-TCAI
As we know, both matched filtering and dechirping belong to pulse compression. Matched filtering is a more general way of pulse compression designed for common signals, where the correlation is utilized. Because of the particular characteristics, LFM signal can also be compressed by dechirping, which obtain the range by processing the baseband echo signal with Fourier transformation. Because the transmitting signal is LFM signal, the baseband echo signal in Equation (5) can be compressed by simple Fourier transformation. Then, the echo after pulse compression is shown as: where F (·) denotes the Fourier transformation, which can be operated fast with current computational technology. S r ( f t ) is the echo in frequency domain. To avoid confusion of the signal frequency, we adopt f t to describe the frequency symbol corresponding to t. S r ( f t ) presents spike pulses in the range cells containing target information. The scattering information within the same range gathers in the same spike pulse. However, when the SNR is too low, the target-containing range cells are difficult to recognize. Ref. [19] adopts GM to project S r ( f t ) into manifold and find the useful range cells by Kullback-Leibler divergence (KLD) [17][18][19]. Detailed detecting procedure of GM-TCAI has been presented enough in Ref. [19]. Herein, FSr is defined as the original EV of GM-TCAI, which is obtained by frequency discretion of S r ( f t ).
The 3D imaging area in Figure 1 has two imaging planes in different ranges. As each imaging plane is about in one range cell, FSr will show two spike pulses. Thus, FSr 1 and FSr 2 can be extracted from FSr, which is shown in Figure 2. FSr 1 and FSr 2 are two GM-TCAI EVs related to the two spike pulses. Besides, we adopt r 1 and r 2 to index the corresponding row positions of FSr 1 and FSr 2 in FSr. K 1 and K 2 are the grid-cell numbers of the two imaging planes, respectively. proposed in Ref. [18] adopts GM to find the range cells containing scattering information. Both Refs. [17,18] belong to range-domain TCAIs while Ref. [18] is an improved version of Ref. [17]. Therefore, this paper only shows the basic imaging procedures and model of GM-TCAI and more details are presented in Ref. [18].

EV Extraction of GM-TCAI
As we know, both matched filtering and dechirping belong to pulse compression. Matched filtering is a more general way of pulse compression designed for common signals, where the correlation is utilized. Because of the particular characteristics, LFM signal can also be compressed by dechirping, which obtain the range by processing the baseband echo signal with Fourier transformation. Because the transmitting signal is LFM signal, the baseband echo signal in Equation (5) can be compressed by simple Fourier transformation. Then, the echo after pulse compression is shown as:  [17][18][19]. Detailed detecting procedure of GM-TCAI has been presented enough in Ref. [19]. Herein, FSr is defined as the original EV of GM-TCAI, which is obtained by frequency discretion of ( ) The 3D imaging area in Figure 1 has two imaging planes in different ranges. As each imaging plane is about in one range cell, FSr will show two spike pulses. Thus, where S( f t , k) is the Fourier transformation of s(t, k). By frequency discretion on S( f t , k), we get the k-th column of FS. Therefore, FS 1 and FS 2 are constructed by extracting corresponding rows of FS o 1 and FS o 2 , which are deduced by combining the related K 1 and K 2 columns of FS.

Imaging Model of GM-TCAI
Taken an imaging plane named x for example, the imaging model of GM-TCAI is deduced as: where FSr x , FS x , β x and w GM x are the EV, RSM, SCV and MNV of GM-TCAI, respectively. N x and K x are the extracted-frequency and grid-cell numbers of imaging plane x, respectively.
Based on Equation (9), each imaging plane in Figure 1 can be reconstructed in parallel to decompose the global computational complexity and then are combined together to reconstruct the 3D target. Table 2 has shown the imaging procedure of GM-TCAI as below. Table 2. Imaging procedure of geometric measures (GM)-TCAI.

Input
The Time − Domain EV Sr.

Imaging process
Step 1: parfor x = 1:X (parfor denotes the for loop in parallel, X means the total imaging-plane numbers) (1) Extract EV FSr x from FSr, which is transformed from Sr via Equation (7). Besides, the row positions of FSr x in FSr is indexed as r x . (2) According to the detailed processes and Equation (8)  (3) Reconstruct β x via Equation (9) end Step 2: Obtain the 3D imaging resultβ GM in combination of β x , x = 1 : X.

Output
Return the GM-TCAI imaging resultβ GM .

BP-Based TCAI
When the imaging plane is too large or the resolution cell is too small, the computational burden is still pretty heavy for traditional TCAI and GM-TCAI. This paper proposes an effective and efficient TCAI method based on BP. In this approach, BP helps to transforms the coding imaging from time domain into space domain, where different target parts in different sub-areas can be reconstructed independently and simultaneously with high SNR. There are two factors that can improve the SNR of BP-TCAI. On one hand, the echo signal of the i-th receiver is transformed from time domain into range profiles via Equation (13). As shown in Section 2.2, the SNR can be improved by extracting the range cells containing scattering information. On the other hand, BP helps project the range-domain signals into space domain. Thus, the space areas containing target information can be extracted for further imaging and thus improve the SNR again. Figure 3, the BP-TCAI is mainly composed of a coded-aperture transceiver antenna and a computer. Similar to Figure 1, the red and blue dashed lines denote the transmitting and receiving processes, respectively. Unlike the traditional TCAI architecture, the proposed BP-TCAI is a SIMO system, which denotes a single transmitting antennas and multiple receiving antennas. The red square in the center of the coded aperture denotes the single transmitter. The colorful circles describing the multiple receivers are located in the coded aperture in array form. Different colors mean different amplitude or phase modulation. Different from the traditional TCAI, the modulation operation is achieved in the receiving terminal with no requirements of reducing the working distance. The computer can control the single transmitter to send signal and the multiple receivers to receive the echo added with modulation. Finally, the computer processes all the echo signals for high-resolution BP-TCAI. The detailed signal propagation process of BP-TCAI is shown as below. Similar to Section 2.1, we suppose the coded aperture only modulates the signal phase rather than the amplitude. range cells containing scattering information. On the other hand, BP helps project the range-domain signals into space domain. Thus, the space areas containing target information can be extracted for further imaging and thus improve the SNR again. As shown in Figure 3, the BP-TCAI is mainly composed of a coded-aperture transceiver antenna and a computer. Similar to Figure 1, the red and blue dashed lines denote the transmitting and receiving processes, respectively. Unlike the traditional TCAI architecture, the proposed BP-TCAI is a SIMO system, which denotes a single transmitting antennas and multiple receiving antennas. The red square in the center of the coded aperture denotes the single transmitter. The colorful circles describing the multiple receivers are located in the coded aperture in array form. Different colors mean different amplitude or phase modulation. Different from the traditional TCAI, the modulation operation is achieved in the receiving terminal with no requirements of reducing the working distance. The computer can control the single transmitter to send signal and the multiple receivers to receive the echo added with modulation. Finally, the computer processes all the echo signals for highresolution BP-TCAI. The detailed signal propagation process of BP-TCAI is shown as below. Similar to Section 2.1, we suppose the coded aperture only modulates the signal phase rather than the amplitude. Firstly, the transmitting signal from the single transmitter illustrates the 3D imaging area directly. The signal form is the same as that in Equation (1).

As shown in
Secondly, convolved with the 3D target, the radiating signal is deduced as: o k t is time delay between the transmitter and the kth grid cell. Thirdly, the echo signal arriving at the i-th coded-aperture receiver is written as: where , t i  is the random phase-modulation term for the i-th coded-aperture receiver at time t, , , o k i t is the total time delay though the single transmitter, the k-th grid cell and the i-th coded-aperture receiver.
Finally, the baseband echo signal is sampled by mixing 0 ( ) i BP r S t with the reference signal. The reference signal is the same as Equation (1), while the baseband echo signal is formulated as, Firstly, the transmitting signal from the single transmitter illustrates the 3D imaging area directly. The signal form is the same as that in Equation (1).
Secondly, convolved with the 3D target, the radiating signal is deduced as: where S BP rad (t) is the radiating signal at time t, t o,k is time delay between the transmitter and the k-th grid cell.
Thirdly, the echo signal arriving at the i-th coded-aperture receiver is written as: where ϕ t,i is the random phase-modulation term for the i-th coded-aperture receiver at time t, t o,k,i is the total time delay though the single transmitter, the k-th grid cell and the i-th coded-aperture receiver. Finally, the baseband echo signal is sampled by mixing S BP r0 i (t) with the reference signal. The reference signal is the same as Equation (1), while the baseband echo signal is formulated as, Similar to Equation (5), S BP r i (t) can be sampled directly at baseband frequency. Moreover, the imaging model and procedures are illustrated in detail as below.

EV Extraction of BP-TCAI
According to the BP theory in SAR imaging [21,22], the target scattering information is obtained by projecting the range profile to the space position of each resolution cell. The resolution cell denotes the grid cell in coded-aperture imaging.
Firstly, similar to the principle of Equation (7), the range profile of S BP r i (t) is deduced from: where Then, the scattering coefficient of the k-th grid cell is deduced from: where ϕ k,i = 2π f c t o,k,i is the phase compensation term corresponding to the k-th grid cell and the i-th coded-aperture receiver. f c is the center frequency of the transmitting signal. BS r (s k ) is the scattering coefficient of the k-th grid cell. Apparently, BS r (s k ) is obtained by coherent superposition of all the I range profiles, which is deduced from Equation (11). s k describes the k-th space-domain frequency of f t . Therefore, the grid cell containing target information can be extracted for further imaging. As described in Section 2.2.3, the imaging plane x contains K x grid cells. Therefore, we combine the K x scattering coefficients BS r (s k x ), k x = 1, · · · K x together to construct the EV of BP-TCAI, which is written as: where [·] T describes the transposition of vector or matrix. As shown in Figure 3, each imaging plane is subdivided into four subareas, each of which has a character-shaped target. The subareas of imaging plane x are named as x 1 , x 2 , x 3 and x 4 respectively. Thus, the stronger scattering coefficients are extracted independently from BSr x to construct BSr x 1 , BSr x 2 , BSr x 3 and BSr x 4 ,which are shown in Figure 4. Besides, r x 1 , r x 2 , r x 3 and r x 4 are indexed as the corresponding row positions of BSr x 1 , BSr x 2 , BSr x 3 and BSr x 4 in BSr x . K x 1 , K x 2 , K x 3 and K x 4 are the grid-cell numbers of the subareas, respectively.  , e x p 2 Referring to Equations (11)- (13), the x k -th column of x BS is formulated as: where ( , )

RSM Conformation of BP-TCAI
As shown in Figure 4, BS x is the BP-based RSM corresponding to the EV BSr Apparently, BS x should be deduced to obtain BS x 1 , BS x 2 , BS x 3 and BS x 4 . The detailed deduction process is shown as below.
Firstly, we define the reference signal related to the i-th coded-aperture detector and the k x -th grid cell as: Referring to Equations (11)-(13), the k x -th column of BS x is formulated as: where is the k m -th vector element of BS k x and BS k x is the k x -th column of BS x . Moreover, k m = 1, 2, · · · , K x , ϕ k m ,i = 2π f c t o,k m ,i is the phase compensation term corresponding to the k m -th grid cell and the i-th coded-aperture receiver. t o,k m ,i is the total time delay through the single transmitter, the k m -th grid cell and the i-th coded-aperture receiver.
Eventually, BS x is formulated as:

Imaging Model of BP-TCAI
A subarea subdivided from imaging plane x is named as x a . The imaging model of BP-TCAI for x a is deduced as: where SSr x a , SS x a , β x a and w BP x a are the EV, RSM, SCV and MNV of BP-TCAI, separately. N x a and K x a are the extracting and the entire grid-cell numbers of subarea x a , respectively.
According to Equation (19), subareas of each imaging plane in Figure 3 are resolved in parallel and then are synthesized together to get the 3D target. In this method, both the computational complexity and noise power are much less than the traditional TCAI and GM-TCAI. Table 3 has shown the imaging procedure of BP-TCAI as below. Table 3. Imaging procedure of back projection (BP)-TCAI.

Requirement
A Computer and a Coded-Aperture Array Transceiver.

Imaging process
Step 1: Obtain the time domain echo signal by the following procedures.
(1) The computer controls the single transmitter to send signals.
(2) Multiple coded-aperture detectors randomly modulate and receive the echo signals.
(3) The modulated echo signals are transported into the computer for imaging.
Step 2: parfor x = 1:X parfor a = 1:A (A describes the imaging-area numbers in imaging plane x) (1) Extract SD-EV BSr x from BSr x , which is deduced from Equation (15). Besides, the row positions of BSr xa is indexed as r xa . (2) Construct the RVM of BP-TCAI BS xa corresponding to BSr xa according to the detailed processed in Section 2.3.2 and Equations (16)- (20).

Output
Return the BP-TCAI imaging resultβ BP .

Comparisons of Computational Complexity
As for the computational complexity comparisons, the most time-consuming part are the reconstruction algorithms according to Equations (4), (9) and (21) for traditional TCAI, GM-TCAI and BP-TCAI, respectively. For example, SBL is adopted as the reconstruction algorithm for all the three imaging methods. Furthermore, the most time-consuming steps of SBL are matrix inversion and matrix-vector multiplication, the costs of the two operations are O K 3 and O K 2 , respectively. K is the number of the grid cell. As shown in Equations (4), (9) and (21), the grid-cell numbers for traditional TCAI, GM-TCAI and BP-TCAI are K, K x and K x a , respectively. Besides, the number of iteration for SBL can be set as M. Therefore, for the reconstruction part, the computational complexities of the three imaging methods are O M K 3 + K 2 , O M K x 3 + K x 2 and O M K x a 3 + K x a 2 , separately.
Because of the extracting processing, K x a is much less than K and K x . Therefore, the computational burden of BP-TCAI is reduced more significantly than traditional TCAI and GM-TCAI. Besides the reconstruction algorithm, GM-TCAI and BP-TCAI transform the received time-domain signals into range profiles and space domain according to Equations (7) and (14), separately. The transformation complexities of GM-TCAI and BP-TCAI can be calculated as O(K log K) and O(IK), where I is the number of coded-aperture receivers. Apparently, compared with the reconstruction part, the transformation step of GM-TCAI and BP-TCAI only cost little time. From the theoretical analysis, it can be concluded that BP-TCAI consumes much less time than traditional TCAI and GM-TCAI.

Experimental Results
In this section, firstly, we analyze the point spread function (PSF) of BP-TCAI with different phase-modulation degrees. Essentially, the PSF is the imaging result of a point target located in the center of an imaging plane. Secondly, the range profiles under different SNRs are presented to detect the locations of the two imaging planes. Besides, to find the influence of phase modulation on the BP imaging, we present the BP projecting results without and with phase modulation, respectively. Finally, sparse Bayesian learning (SBL) [23] is used to compare the imaging results of traditional TCAI, GM-TCAI and BP-TCAI. Besides, both traditional TCAI and GM-TCAI calculate the radiating signals and echo signals according to Equations (2) and (6), respectively. Besides, BP-TCAI calculate the radiating and echo signals according to Equations (10) and (12), respectively.
Moreover, to evaluate the imaging performance of TCAI, GM-TCAI and BP-TCAI, we adopt the relative imaging error (RIE) and probability of successful imaging (PSI) [24], which are defined as is inverse proportional to the imaging quality while PSI is proportional to the imaging performance. The primary parameters used in the simulations are given in Table 4. As shown in Figure 3, the 3D imaging area has two imaging planes, each of which contains four subareas. Traditional TCAI and GM-TCAI requires random phase modulation while it may damage the beam formation and thus reduce the maximum imaging range. BP-TCAI applies the modulation operation into the receiving terminal, which has no influence on the original imaging distance. Besides, the parameters of the coded aperture for traditional TCAI, GM-TCAI and BP-TCAI are the same. Unlike TCAI and GM-TCAI, the coded aperture of BP-TCAI functions as a SIMO system. The experiments are performed on a computer with Intel Core CPU i7-8700U at 3.2 GHz and 16 GB of memory.

PSF Analysis
As we know, traditional BP imaging for SAR or inverse synthetic aperture imaging (ISAR) is performed without phase modulation, which is the guarantee of successful imaging for coded-aperture imaging. The BP-TCAI method can also be seen as coding-array SAR. To analyze the imaging performance of coding-array SAR, Figure 5 presents the PSF with different modulation degrees, including no modulation, [−0.25π, 0.25π], [−0.5π, 0.5π] and [−π, π] continuous phase modulations. The PSF presented in Figure 5 is simply the projection results of BP without further imaging. Figure 5a-d, e-h and i-l present the vertical view, x-axis and y-axis cross-section views of the PSFs, respectively. As shown in Figure 5a, the target point is in the center of the imaging plane. Apparently, the grid cell containing scattering information shows distinct spike pulses, which is shown in Figure 5e,i. From Figure 5b to Figure 5d, the BP imaging results become more and more randomly with the increasing phase-modulation ranges. However, when the phase-modulation range is [−π, π], the PSF image becomes blurred and is difficult to detect the target point. As for Figure 5c, firstly, the BP projecting result shows the right position of the target, which is helpful for the EV extraction of BP-TCAI. Secondly, the random condition around the target point provides enough arbitrary measurement modalities for high-resolution imaging. Therefore, to compare the imaging performance of traditional TCAI, GM-TCAI and BP-TCAI, we set the phase-modulation range as [−0.5π, 0.5π], which strike a and high resolution. TCAI, the coded aperture of BP-TCAI functions as a SIMO system. The experiments are performed on a computer with Intel Core CPU i7-8700U at 3.2 GHz and 16 GB of memory.

PSF Analysis
As we know, traditional BP imaging for SAR or inverse synthetic aperture imaging (ISAR) is performed without phase modulation, which is the guarantee of successful imaging for codedaperture imaging. The BP-TCAI method can also be seen as coding-array SAR. To analyze the imaging performance of coding-array SAR, Figure 5 presents the PSF with different modulation degrees, including no modulation, [−0.25π, 0.25π], [−0.5π, 0.5π] and [−π, π] continuous phase modulations. The PSF presented in Figure 5 is simply the projection results of BP without further imaging. Figure 5a-d, e-h and i-l present the vertical view, x-axis and y-axis cross-section views of the PSFs, respectively. As shown in Figure 5a, the target point is in the center of the imaging plane. Apparently, the grid cell containing scattering information shows distinct spike pulses, which is shown in Figure 5e,i. From Figure 5b to Figure 5d, the BP imaging results become more and more randomly with the increasing phase-modulation ranges. However, when the phase-modulation range is [−π, π], the PSF image becomes blurred and is difficult to detect the target point. As for Figure 5c, firstly, the BP projecting result shows the right position of the target, which is helpful for the EV extraction of BP-TCAI. Secondly, the random condition around the target point provides enough arbitrary measurement modalities for high-resolution imaging. Therefore, to compare the imaging performance of traditional TCAI, GM-TCAI and BP-TCAI, we set the phase-modulation range as [−0.5π, 0.5π], which strike a and high resolution.

Range Profile Analysis
As shown in Table 1, the number of the coded-aperture array elements is 625 and the sampling times within a pulse is 2000. Therefore, the coded-aperture receives 625 pulses with the length of 2000. To obtain the range profiles of the echo signals, the 625 pulses are processed by IFT operation according to Equation (13).

Projection Results of BP
According to Equation (13), the range-profile signals in Figure 6d-f are projected into R1 and R2, which are shown in Figure 7a-c. Besides, Figure 7d-f illustrates the projection results without modulation. The "C," "A," "B" and "P" shape targets are distributed in R1, while "N," "U," "D" and "T" shape targets are located in R2. The "CABP" and "NUDT" denote coded aperture using BP and

Range Profile Analysis
As shown in Table 1, the number of the coded-aperture array elements is 625 and the sampling times within a pulse is 2000. Therefore, the coded-aperture receives 625 pulses with the length of 2000. To obtain the range profiles of the echo signals, the 625 pulses are processed by IFT operation according to Equation (13).

Range Profile Analysis
As shown in Table 1, the number of the coded-aperture array elements is 625 and the sampling times within a pulse is 2000. Therefore, the coded-aperture receives 625 pulses with the length of 2000. To obtain the range profiles of the echo signals, the 625 pulses are processed by IFT operation according to Equation (13).

Projection Results of BP
According to Equation (13), the range-profile signals in Figure 6d-f are projected into R1 and R2, which are shown in Figure 7a-c. Besides, Figure 7d-f illustrates the projection results without modulation. The "C," "A," "B" and "P" shape targets are distributed in R1, while "N," "U," "D" and "T" shape targets are located in R2. The "CABP" and "NUDT" denote coded aperture using BP and

Projection Results of BP
According to Equation (13), the range-profile signals in Figure 6d-f are projected into R1 and R2, which are shown in Figure 7a-c. Besides, Figure 7d-f illustrates the projection results without modulation. The "C," "A," "B" and "P" shape targets are distributed in R1, while "N," "U," "D" and "T" shape targets are located in R2. The "CABP" and "NUDT" denote coded aperture using BP and National University of Defense Technology, respectively. The eight subareas of R1 and R2 are named as A1-A8. When the SNRs are 30 dB and 0 dB, both the BP projection results with and without modulation can display the basic profile of the 3D target. Although the projection results with modulation are a little blurred, the seemed ambiguous images actually contain arbitrary coding information, which can be exploited by BP-TCAI. In contrast, BP-TCAI cannot achieve further imaging with the projection results without modulation in Figure 7d-f. Under −30 dB, the SNR is too low for both Figure 7c,f, to resolve the scattering distribution of the target. However, the GM method used in GM-TCAI 19 can also be applied into BP-TCAI to detect the primary target positions. With the GM tool, regardless of the SNRs, Figure 7g-i show the scattering points of the target in the right positions, which can be used for EV extraction. National University of Defense Technology, respectively. The eight subareas of R1 and R2 are named as A1-A8. When the SNRs are 30 dB and 0 dB, both the BP projection results with and without modulation can display the basic profile of the 3D target. Although the projection results with modulation are a little blurred, the seemed ambiguous images actually contain arbitrary coding information, which can be exploited by BP-TCAI. In contrast, BP-TCAI cannot achieve further imaging with the projection results without modulation in Figure 7d-f. Under −30 dB, the SNR is too low for both Figure 7c,f, to resolve the scattering distribution of the target. However, the GM method used in GM-TCAI 19 can also be applied into BP-TCAI to detect the primary target positions. With the GM tool, regardless of the SNRs, Figure 7g-i show the scattering points of the target in the right positions, which can be used for EV extraction. The "C," "A," "B" and "P" shape targets are distributed in R1, while "N," "U," "D" and "T" shape targets are located in R2. The "CABP" and "NUDT" denote coded aperture using BP and National University of Defense Technology, respectively. R1 contains four subareas named as A1-A4 and R2 has four subareas marked as A5-A8.

Imaging Results Analysis
As shown in Figure 7, the projection results are divided to construct the EVs of different subareas via Equation (14). According to the extracted EVs and Equations (16)- (20), the corresponding RSMs are built for high-resolution imaging of BP-TCAI. To verify the superiority of BP-TCAI, Figure 8 compares the 3D imaging results of BP-TCAI and the previous algorithms, including traditional TCAI and GM-TCAI. The imaging models of traditional TCAI, GM-TCAI and BP-TCAI are based on 30 dB, respectively. The "C," "A," "B" and "P" shape targets are distributed in R1, while "N," "U," "D" and "T" shape targets are located in R2. The "CABP" and "NUDT" denote coded aperture using BP and National University of Defense Technology, respectively. R1 contains four subareas named as A1-A4 and R2 has four subareas marked as A5-A8.

Imaging Results Analysis
As shown in Figure 7, the projection results are divided to construct the EVs of different subareas via Equation (14). According to the extracted EVs and Equations (16)- (20), the corresponding RSMs are built for high-resolution imaging of BP-TCAI. To verify the superiority of BP-TCAI, Figure 8 compares the 3D imaging results of BP-TCAI and the previous algorithms, including traditional TCAI and GM-TCAI. The imaging models of traditional TCAI, GM-TCAI and BP-TCAI are based on Equations (5), (9) and (21), respectively. The imaging procedures of the three imaging methods are described in Tables 1-3 Figure 8a,d,g, when the SNR is 30 dB, all the three TCAI methods can reconstruct the 3D target. For 0 dB condition presented in Figure 8b,e,h, both BP-TCAI and GM-TCAI reconstruct the precise target while the traditional TCAI resolves the target with some background noise. When the SNR is greater than or equal to 0 dB, all the three imaging methods can reconstruct the 3D target regardless of the imaging performance. However, traditional TCAI resolves a cluster of disordered scattering points in the 3D imaging area under −30 dB, which is shown in Figure 8c. As for BP-TCAI and GM-TCAI, it is difficult to judge the image quality from Figure 8f,i, both of which resolve the clear 3D target. Therefore, we use RIE and PSI to compare the imaging performance of them.  Figure  8a,d,g, when the SNR is 30 dB, all the three TCAI methods can reconstruct the 3D target. For 0 dB condition presented in Figure 8b,e,h, both BP-TCAI and GM-TCAI reconstruct the precise target while the traditional TCAI resolves the target with some background noise. When the SNR is greater than or equal to 0 dB, all the three imaging methods can reconstruct the 3D target regardless of the imaging performance. However, traditional TCAI resolves a cluster of disordered scattering points in the 3D imaging area under −30 dB, which is shown in Figure 8c. As for BP-TCAI and GM-TCAI, it is difficult to judge the image quality from Figure 8f,i, both of which resolve the clear 3D target. Therefore, we use RIE and PSI to compare the imaging performance of them.  Figure 9a,b presents the RIE and PSI comparisons of traditional TCAI, GM-TCAI and BP-TCAI. As described in the third paragraph of Section 3, the smaller RIE and the larger PSI denote better imaging performance. Illustrated by the blue lines in Figure 9, the largest RIE and the least PSI under different SNRs indicate that the imaging performance of traditional TCAI is the worst among the three imaging methods. The red and green lines in Figure 9 denote the imaging evaluations of BP-TCAI and GM-TCAI, respectively. Apparently, the imaging performance of BP-TCAI is slightly better than the GM-TCAI.  Figure 9a,b presents the RIE and PSI comparisons of traditional TCAI, GM-TCAI and BP-TCAI. As described in the third paragraph of Section 3, the smaller RIE and the larger PSI denote better imaging performance. Illustrated by the blue lines in Figure 9, the largest RIE and the least PSI under different SNRs indicate that the imaging performance of traditional TCAI is the worst among the three imaging methods. The red and green lines in Figure 9 denote the imaging evaluations of BP-TCAI and GM-TCAI, respectively. Apparently, the imaging performance of BP-TCAI is slightly better than the GM-TCAI. As shown in Table 4, the 3D imaging areas has two imaging planes, both of which contain 60 × 60 grid cells. Furthermore, each imaging plane has four subareas with 30 × 30 grid cells. When the number of sampling time and grid cells are the same, the RSM sizes of traditional TCAI, GM-TCAI and BP-TCAI are 7200 × 7200, 3600 × 3600 and 900 × 900, respectively. According to the theory analysis in Section 2.4, the computational complexity of TCAI relies mainly on the size of the RSM. Therefore, the computational burden of BP-TCAI is much less than traditional TCAI and GM-TCAI. Table 5 compares the runtime of the three imaging methods. Apparently, BP-TCAI achieves the fast imaging while traditional TCAI is the most time-consuming. Moreover, BP-TCAI can divide each imaging plane containing scattering information into more than four subareas. Therefore, the RSM of BP-TCAI can be further downsized to improve the computational efficiency. Actually, the shapes and sizes of the sub-areas can change according to the distribution of the target. Therefore, the computational time can be optimized with proper shapes and sizes. In conclusion, BP-TCAI is the most effective and efficient imaging method in contrast to traditional TCAI and GM-TCAI.

Conclusions
This paper proposes an imaging method of TCAI based on BP to reduce the computational burden and achieve high-resolution imaging for 3D imaging. Actually, BP helps to transforms the coding imaging from time domain into space domain, where different target parts can be reconstructed independently and simultaneously with higher SNR. Numerical experiments demonstrate that BP-TCAI is an effective and efficient imaging method. Furthermore, the BP-TCAI can also be seen as a combination of the traditional SAR imaging and the coding strategy. Therefore, BP-TCAI not only can be used to improve the imaging performance of TCAI but also provide a prospective approach for traditional SAR. Though this proposed imaging method is verified to be more effective and efficient than the previous works, there are still some limitations that can be studied by other researchers. Firstly, the imaging performance of BP-TCAI is only slightly better GM-TCAI, which is the best improved TCAI proposed in our previous paper. With the same reconstruction algorithm, the imaging performance is mainly decided by the SNR. The SNR of BP-TCAI can be improved during the pulse compression and projection processes while the SNR of As shown in Table 4, the 3D imaging areas has two imaging planes, both of which contain 60 × 60 grid cells. Furthermore, each imaging plane has four subareas with 30 × 30 grid cells. When the number of sampling time and grid cells are the same, the RSM sizes of traditional TCAI, GM-TCAI and BP-TCAI are 7200 × 7200, 3600 × 3600 and 900 × 900, respectively. According to the theory analysis in Section 2.4, the computational complexity of TCAI relies mainly on the size of the RSM. Therefore, the computational burden of BP-TCAI is much less than traditional TCAI and GM-TCAI. Table 5 compares the runtime of the three imaging methods. Apparently, BP-TCAI achieves the fast imaging while traditional TCAI is the most time-consuming. Moreover, BP-TCAI can divide each imaging plane containing scattering information into more than four subareas. Therefore, the RSM of BP-TCAI can be further downsized to improve the computational efficiency. Actually, the shapes and sizes of the sub-areas can change according to the distribution of the target. Therefore, the computational time can be optimized with proper shapes and sizes. In conclusion, BP-TCAI is the most effective and efficient imaging method in contrast to traditional TCAI and GM-TCAI.

Conclusions
This paper proposes an imaging method of TCAI based on BP to reduce the computational burden and achieve high-resolution imaging for 3D imaging. Actually, BP helps to transforms the coding imaging from time domain into space domain, where different target parts can be reconstructed independently and simultaneously with higher SNR. Numerical experiments demonstrate that BP-TCAI is an effective and efficient imaging method. Furthermore, the BP-TCAI can also be seen as a combination of the traditional SAR imaging and the coding strategy. Therefore, BP-TCAI not only can be used to improve the imaging performance of TCAI but also provide a prospective approach for traditional SAR. Though this proposed imaging method is verified to be more effective and efficient than the previous works, there are still some limitations that can be studied by other researchers. Firstly, the imaging performance of BP-TCAI is only slightly better GM-TCAI, which is the best improved TCAI proposed in our previous paper. With the same reconstruction algorithm, the imaging performance is mainly decided by the SNR. The SNR of BP-TCAI can be improved during the pulse compression and projection processes while the SNR of GM-TCAI can also be improved by pulse compression. Therefore, the SNR or the imaging performance is only improved in a certain degree by the projection operation of BP. Secondly, the imaging efficiency can still be improved even BP-TCAI is more efficient than before. According to the basic principle of SAR imaging, BP algorithm is an effective imaging method but consumes much time in the projection procedure. In the future work, some other useful imaging theories rather than BP can be used to project the signals into space areas with more effective and efficient approach.