Method and apparatus for remote object sensing employing compressive sensing

1. A method for remote object sensing on-board a vehicle, comprising:

operating an analog-to-digital (A/D) converter at a sampling rate proportional to a compressibility of sensed objects to generate a digitized form of an on-vehicle low-resolution radar imaging system and reflected from a remote object;

within a signal processor:

employing compressive sensing to analyze the digitized form of the waveform originating from an on-vehicle low-resolution radar imaging system and reflected from the remote object, said compressive sensing including generating a matrix comprising a temporal projection which is predetermined in an off-line environment, a Fourier transform, and an integral term configured to analyze the reflected waveform; and

identifying leading and trailing edges of the remote object by employing a norm minimization procedure to reconstruct a range profile based upon the reflected waveform analyzed by said compressive sensing;

wherein employing a norm minimization procedure includes finding a minimum of the reflected waveform employing linear programming; and

providing an estimate of an actual location of the remote object based on the identified leading and trailing edges of the remote object.

2. The method of claim 1 , wherein said waveform originating from the on-vehicle low-resolution radar imaging system comprises a reflected linear frequency modulation (LFM) waveform.

3. The method of claim 2 , wherein said radar imaging system comprises a low bandwidth radar device, and said LFM waveform comprises a high frequency waveform that is less than 200 MHz.

4. The method of claim 1 , wherein the norm minimization procedure comprises an Ll norm minimization procedure.

5. The method of claim 1 , wherein said radar imaging system comprises a low bandwidth radar device configured to monitor a field of view relative to the vehicle.

6. The method of claim 5 , wherein said field of view comprises a front view relative to the vehicle.

7. The method of claim 5 , wherein said field of view comprises a side view relative to the vehicle.

8. The method of claim 1 , wherein compressive sensing including generating a matrix comprising the temporal projection, the Fourier transform, and the integral term configured to analyze the reflected waveform comprises employing compressive sensing to determine a measurement matrix Φ applied to the reflected waveform, wherein the measurement matrix Φ is represented by the following relationship:

Φ=[0 I 0]*[ F]*[It]

wherein [0 I 0] is a temporal projection matrix,

[F] is a Fourier transform matrix, and

[It] is an integral term.

9. The method of claim 8 , wherein said measurement matrix Φ applied to the reflected waveform is determined in a derivative space.

10. The method of claim 1 , wherein identifying leading and trailing edges of the remote object comprises employing an Ll norm minimization procedure including finding a minimum of the reflected waveform ( ) employing linear programming executing in accordance with the following relationship:

min

x

^

∈

R

N

⁢



x

^



1

1

subject to

∥ξ∥ 1 2 ≦ε

wherein ε is a small number less than one and approaching zero, and

ξ= y−Φ .

11. A method for remote object sensing on-board a vehicle employing a low-resolution radar imaging system, comprising:

operating an analog-to-digital (A/D) converter at a sampling rate proportional to a compressibility of sensed objects to generate a digitized form of a linear frequency modulation (LFM) waveform generated by the low-resolution radar imaging system and reflected from a remote object;

within a signal processor:

employing compressive sensing to analyze the digitized form of the LFM waveform generated by the low-resolution radar imaging system and reflected from the remote object, said compressive sensing including determining a measurement matrix for the remote object in a derivative space based upon a temporal projection which is predetermined in an off-line environment, a Fourier transform, and an integral term; and

identifying leading and trailing edges of the remote object by employing a norm minimization procedure to reconstruct a range profile based upon the measurement matrix for the remote object in the derivative space;

wherein employing a norm minimization procedure includes finding a minimum of the reflected waveform employing linear programming; and

providing an estimate of an actual location of the remote object based on the identified leading and trailing edges of the remote object.

12. The method of claim 11 , wherein the norm minimization procedure comprises an Ll norm minimization procedure.

13. The method of claim 11 , wherein determining the measurement matrix for the remote object comprises determining a measurement matrix Φ applied to the reflected LFM waveform wherein the measurement matrix Φ is represented by the following relationship:

Φ=[0 I 0]*[ F]*[It]

wherein [0 I 0] is a temporal projection matrix,

[F] is a Fourier transform matrix, and

[It] is an integral term.

14. The method of claim 11 , wherein identifying leading and trailing edges of the remote object comprises employing an Ll norm minimization procedure including finding a minimum of the reflected waveform ( ) employing linear programming executing in accordance with the following relationship:

min

x

^

∈

R

N

⁢



x

^



1

1

subject to

∥ξ∥ 1 2 ≦ε

wherein ε is a small number less than one and approaching zero, and

ξ= y−Φ .