# Research on the 3D imaging algorithm of spin target based on the Hough transform

- Jin Li
^{1}Email author and - Yiming Pi
^{1}

**2013**:90

https://doi.org/10.1186/1687-1499-2013-90

© Li and Pi; licensee Springer. 2013

**Received: **5 December 2012

**Accepted: **28 February 2013

**Published: **28 March 2013

## Abstract

As one of the most typical characteristics in space target motion, spin phenomenon has good 3D imaging application potential. Conventional target imaging algorithm fails to make full use of the rotating features of the target to obtain the characteristics of target space, and the speedy spin of targets will cause the dramatic changes in the positions of scattering center within short observation session, which may lead to the failure of imaging algorithm. Aiming at such a special phenomenon of the space target, the time frequency distribution curve of echoes in each scattering center could be mapped onto the parameter space to obtain the position of each scattering center by taking advantage of Hough transformation, thus the 3D features of spin target could be obtained. In this article, the 3D imaging algorithm was studied on the basis of Hough transformation, and its effectiveness was tested with simulation. Meanwhile, the translational motion and shielding effect of space target were discussed, and favorable imaging results were achieved.

### Keywords

3D imaging Spin target Hough transform ISAR## 1. Introduction

Study on the 3D inverse synthetic aperture radar (ISAR) imaging techniques has been attracting more and more attention [1–6]. Compared to 2D ISAR imaging techniques, more detailed information about the target can be provided by 3D ISAR imaging. The current 3D ISAR imaging algorithm mainly consists of two types. The first type takes advantage of various reception channels and receives the echo signals of the target on the basis of phase interference, and it can conduct 2D image for the signals received by each antenna with conventional imaging algorithm [7–9]. As a result, the three-dimensional spatial information of each scattering point can be extracted from the differences of 2D imaging interference phase. In the second type, the target 3D imaging is constructed with the 2D image sequence obtained from different observation angle through a receiving antenna [10]. In both algorithms, a spatial freedom degree is added to obtain three-dimensional resolution.

The traditional ISAR imaging algorithm is based on slow turntable model. High-speed rotating targets, such as airplane propeller, spin precession-guided missile warhead, space debris, etc., often fail to meet the requirements of slow turntable model. However, for spin target, its rotating angular velocity can be estimated, which in actually provides a degree of freedom for 3D ISAR imaging [11, 12]. As in [13, 14], on the condition of the turntable model with high-speed rotating target and its spin angular velocity is known, the methods general radon transform and extended Hough transform were used to three-dimensional space information extraction for rotating targets. The basic idea of these algorithms is taking advantage of sine envelope of spin target to estimate the scattering points’ three-dimensional location by using curvilinear integral under range-compress domain. The operand of these algorithms is always huge because the energy accumulation along curve is four-dimensional curve detection process. In this article, sine envelope of spin target was used to estimate the 3D position of scattering point in the form of curvilinear integral within distance compressed domain, thus the extraction of 3D spatial information of the spin target could be realized.

## 2. Target and echo signal model

*O*of radar coordinate

*O-XYZ*, the projection of radar sight axis on

*OXY*plane was in the same direction and coincided with

*Y*-axis, and the included angle of radar sight axis and

*Z*-axis was α. Reference frame

*o*-UVW paralleled to coordinate system

*O-XYZ*, and both corresponding coordinate axes pointed to the same direction. The origin

*o*was seated in the target rotation axis, axis

*W*was the direction of angular velocity vector of target rotation, axes

*U, V*, and

*W*formed right-handed rectangular coordinate system. In

*O-XYZ*coordinate, the coordinate of point

*o*was (

*X*

_{0},

*Y*

_{0},

*Z*

_{0}), and obviously

*X*

_{0}= 0. The distance between two coordinate origins

*O*and

*o*was

*R*

_{0}, and then

*o*-

*xyz*and reference coordinate system

*o-UVW*shared the same origin, both the axes

*z*and

*W*pointed to the same direction, and the target coordinate system

*o-xyz*rotated with the targets. Initially,

*t*

_{0}= 0, the target coordinate system

*o-xyz*and reference coordinate system

*o-UVW*coincided, and at moment

*t*, the included angle of the two coordinate systems was

*θ*(

*t*), namely

*P*in coordinate system

*o-xyz*was (

*x*

_{ P },

*y*

_{ P },

*z*

_{ P }), while it was (

*u*

_{ P },

*v*

_{ P },

*w*

_{ P }) in coordinate system, obviously

*P*and the origin

*o*was

*r*

_{ P }, the coordinate of

*P*in coordinate system

*O-XYZ*was (

*X*

_{ P }

*, Y*

_{ P }

*, Z*

_{ P }), and its distance with origin

*O*was

*R*

_{ P }(

*t*). Then

*u*

_{ P }<<

*R*

_{0},

*v*

_{ P }<<

*R*

_{0}, when

*w*

_{ P }<<

*R*

_{0}, Equation (5) could be approximated as the following equation:

*oUV*plane, as shown in Figure 2, then there was the following coordinate transformation relation:

*ω*(

*t*) =

*ω*, was a constant, then

*T*is the pulse width,

*f*

_{0}is carrier frequency, γ is time-frequency rate, $\mathrm{rect}\left(\frac{\tilde{t}}{T}\right)$ is rectangular window function, defined as

In which, *B* = *γT* was the signal bandwidth.

*P*within observation time was constant σ

_{ P }, then the radar echo of this point could be represented as

*t*is the observation time,

*T*

_{obs}is the total time-bandwidth of the observation, and

*c*is the speed of light. Substitute Equation (12) into Equation (16), and compensate for the constant phase term brought by

*R*

_{0}, it could be obtained

*K*scattering points, its corresponding coordinate and reflection coefficient were (

*x*

_{ k },

*y*

_{ k },

*z*

_{ k }) and

*σ*

_{ k }(

*k*= 1, 2,…,

*K*), then the target echo could be represented as

*f*in Equation (18), the data after range compression could be obtained, namely the range residence time domain data, as shown in the following equation:

### 3.3. D imaging algorithm

On the premise of given sine curve cycle (namely *ω* was given), its position and shape were determined by three spatial coordinate parameters *x*_{
k
}^{'}, *y*_{
k
}^{'}, and *z*_{
k
}^{'}. Therefore, the 3D position information of the scattering point could be obtained by extracting the information about sine curve.

However, it could be seen from the above analysis that, owning to the influence of the included angle α between the radar sight and target axis, the actually extracted position information of scattering point was not authentic position information, but the compression of real position, namely, the position coordinate extracted along the axis direction (namely along axis *z*) was about cosα times of the real position coordinate, while the position coordinate which was vertical to spin axis (namely along axes *x* and *y*) was about sinα times of the real position coordinate. Therefore, the smaller α was, the closer the coordinate along axis *z* would be. The better the resolution along axis *z* was, the worse the resolution along axes *x* and *y* would be, and vice versa.

For a target formed from one or several scattering points, the target echo was the sum of each scattering point. Therefore, sine curves might tangle with each other. In addition, with imaging treatment period, each scattering point may not always be shone by the radar beam owning to the shield, namely the scattering point may not have echo. Therefore, there might be discontinuities in sine curve. As a result, it was a little difficult to directly detect the sine curve within range residence time domain, also the image domain. Each sine curve in the image domain corresponded to a peak in the parameter domain, and the parameter corresponding to the peak was the 3D position coordinate of the scattering point. Consequently, with the help of Hough transformation, the detection for the global curve in the image domain could be converted to detection of peaks in easily realized parameter domain.

In the above equation, *d*(Φ) was the Hough transformation result of image *D*(*m,n*), Φ was the multi-dimensional vectors formed by related curve parameters, *n* = *f*(*m*; *Φ*) was the curve to be detected. Here, *m* was the corresponding residence time *t*, the curve parameter vector Φ consisted of three positional parameters, namely ${x}_{k}^{\text{'}}$, ${y}_{k}^{\text{'}}$, and ${z}_{k}^{\text{'}}$, *f*(*m*; *Φ*) corresponded to the curve *r* = *x*_{
k
}^{'} sin *ωt* + *y*_{
k
}^{'} cos *ωt* + *z*_{
k
}^{'} in the image for range residence time domain.

After Hough transformation, the corresponding curve of each scattering point in range residence time domain would produce a peak in the parameter domain. The estimation for the spatial position of scattering point could easily be realized through detecting the peaks in parameter domain.

*X*(

*r,t*) of the scattering point with unit-strength scattering coefficient at this position was constructed according to Equation (19) as follows:

According to certain standard and taking advantage of *S*(*r,t*) and *X*(*r,t*), the scattering coefficient of scattering point in this position could be estimated.

*I*(σ), and suppose it as 0, namely

*X**(

*r*,

*t*) was the conjugation of

*X*(

*r,t*). Then the estimation value $\widehat{\sigma}$ of scattering coefficient could be obtained

After the information about 3D position and scattering coefficient of this scattering point was obtained, the information about this scattering point in the echo data was eliminated, namely suppose $S\left(r,t\right)=S\left(r,t\right)-\widehat{\sigma}X\left(r,t\right)$, and the above procedures of parameter estimation were repeated with new data *S*(*r, t*), till $\widehat{\sigma}$ was smaller than the pre-set threshold. At this moment, the information of all scattering points was extracted from echo data, and 3D image of the target could be reconstructed with the information.

## 4. Simulation experiment

*v*

_{ e }= 1/

*s*along

*z*direction.

**Coordinate and scattering coefficient of each scattering point**

Number | Coordinate of 3D position | Scattering coefficientσ | ||
---|---|---|---|---|

x(m) | y(m) | z(m) | ||

1 | $0.3\sqrt{2}$ | 0 | $-0.6\sqrt{2}$ | 1.3 |

2 | 0 | $0.3\sqrt{2}$ | $-0.6\sqrt{2}$ | 1.0 |

3 | $-0.3\sqrt{2}$ | 0 | $-0.6\sqrt{2}$ | 1.0 |

4 | 0 | $-0.3\sqrt{2}$ | $-0.6\sqrt{2}$ | 1.0 |

5 | $0.2\sqrt{2}$ | $0.2\sqrt{2}$ | $0.3\sqrt{2}$ | 1.3 |

6 | $-0.2\sqrt{2}$ | $0.2\sqrt{2}$ | $0.3\sqrt{2}$ | 1.0 |

7 | $-0.2\sqrt{2}$ | $-0.2\sqrt{2}$ | $0.3\sqrt{2}$ | 1.0 |

8 | $0.2\sqrt{2}$ | $-0.2\sqrt{2}$ | $0.3\sqrt{2}$ | 1.0 |

9 | 0 | 0 | $0.8\sqrt{2}$ | 1.0 |

*z*direction, the position of scattering points changed continuously. As a result, the center of sine curve also changed continuously, namely ‘inclined’ sine curve was formed.

One or several scattering points could be used to obtain the estimation value ${\widehat{v}}_{e}$ of translational velocity, which would compensate for the translation component caused by this speed. However, the original 3D parameter domain $\left({x}_{k}^{\text{'}},{y}_{k}^{\text{'}},{z}_{k}^{\text{'}}\right)$ would be added to 4D parameter domain $\left({x}_{k}^{\text{'}},{y}_{k}^{\text{'}},{z}_{k}^{\text{'}},{v}_{e}\right)$. When estimating the parameter, 4D search was needed. Therefore, this method would greatly increase the calculated amount.

*t*= 0) and other echoes. Owning to the good symmetry among each scattering point, the peak would occur at the quarter of each rotation cycle. The range residence time domain after compensation was shown in Figure 6, and obviously the translation component was compensated well. At this moment, subsequent algorithm such as Hough transformation, etc., could be conducted, and 3D imaging could be realized.

*t*= 0 and other moments were shown in Figure 7. Obviously, the maximum peak would occur only when the rotation cycle was in integral multiple. For the unknown rotation cycle, this method could also be used to estimate the spin cycle of the target.

*z*= −0.6, as shown in Figure 8. There were four scattering points in this plane, therefore, it would form four distinct peaks in the parameter domain, and the corresponding parameter information was the spatial position information of the scattering point. Besides, owning to the influences of such factors as the side lobes of each scattering point, there were small fluctuations in the parameter domain.

*α*=

*π*/4, the coordinate value in the direction of axes

*x*,

*y*, and

*z*was about $\sqrt{2}/2$ times of the real value. In the figure, the marked figures were the estimated value of the scattering coefficient of the scattering point at corresponding position obtained from this algorithm. It could be seen from the figure that 3D imaging could be realized from this algorithm.

## 5. Conclusion

In this article, 3D imaging algorithm of spin target based on Hough transformation was proposed. By taking advantage of the sine envelope of spin target, the 3D position of scattering point was estimated in the form of curve integral within range compression domain, thus the extraction of 3D spatial information of the spin target could be realized, and this algorithm was tested to be valid through simulation experiment. In addition, the translational motion and shielding condition existed in the real target were also taken into consideration in the simulation experiment, and corresponding measures were adopted for translational motion compensation. Under the circumstance of incomplete target sine curve, the target parameter was extracted to realize the 3D imaging of spin target.

## Declarations

### Acknowledgement

This study was supported by the National Natural Science Foundation of China (61271287) and the Fundamental Research Funds for the Central Universities (ZYGX2011J020).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.