Optimized Stochastic Resonance Method for Signal Detection and Image Processing

• US 20070171964A1
• Filed: 10/20/2006
• Published: 07/26/2007
• Est. Priority Date: 10/20/2005
• Status: Active Grant

• Current Assignee: Syracuse University
• Original Assignee: Syracuse University
• Inventors: Chen, Hao, Michels, James, Varshney, Pramod
• Granted Patent: US 7,668,699 B2
• Time in Patent Office: Days
• US Class Current: 375/227
• CPC Class Codes: H04B 17/26 using historical data, averaging values or statistics

Abstract

Apparatus and method for improving the detection of signals obscured by noise using stochastic resonance noise. The method determines the stochastic resonance noise probability density function in non-linear processing applications that is added to the observed data for optimal detection with no increase in probability of false alarm. The present invention has radar, sonar, signal processing (audio, image and video), communications, geophysical, environmental, and biomedical applications.

Claims

1. A method of increasing the probability of detecting at least one signal embedded in non-Gaussian noise, said method comprising the steps of:

(a) recording an observed data process;

(b) calculating stochastic resonance noise; and

(c) adding said stochastic resonance noise to said data, thereby improving the detection of said at least one signal in said non-Gaussian noise.

2. The method of claim 1, wherein the step of calculating stochastic resonance noise comprises the step of determining the stochastic resonance noise probability density function that does not increase the probability of false alarm.

3. The method of claim 2, wherein the step of calculating stochastic resonance noise further comprises the step of calculating the stochastic resonance noise data probability density function from a known probability density function for said observed data process, wherein the stochastic resonance noise probability density function equals λ

δ

(n−

n₁)+(1−

λ

)δ

(n−

n₂), with values n₁ and n₂ equal to two delta function locations having probabilities of λ and

(1−

λ

), respectively.

4. The method of claim 3, wherein the step of calculating the stochastic resonance noise probability density function further comprises the steps of:

(i) determining F_i(x)=∫

_RNφ

(y) p_i(y−

x)dy i=0,1 using known critical function φ

(y) and known data probability density functions p_i(•

), i=0,1;

(ii) determining the three unknown quantities n₁, n₂, and λ

using the known values k₀, ƒ

₀₁ and ƒ

₀₂ using equations,

$\begin{array}{c}\frac{d<br><br>J}{d<br><br>f_0}<br><br>\left(f_{01}<br><br>\left(k_0\right)\right)=\frac{d<br><br>J}{d<br><br>f_0}<br><br>(f_{02}<br><br>\left(k_0\right),\\ \frac{d<br><br>J}{d<br><br>f_0}<br><br>\left(f_{02}<br><br>\left(k_0\right)\right)=k_0,\end{array}$

and J(ƒ

₀₂(k₀))−

J(ƒ

_01(k0)=k₀(ƒ

_02(k0)−

ƒ

₀₁(k₀)); and

(iii) determining the probability of occurrence for n₁ and n₂ as λ and

1−

λ

, respectively, using the equation

$\mathrm{\lambda <br><br>}=\frac{f_{02}<br><br>\left(k_0\right)-P_{\mathrm{FA}}^x}{f_{02}<br><br>\left(k_0\right)-f_{01}<br><br>\left(k_0\right)}.$

5. The method of claim 4, wherein the step of adding said stochastic resonance noise to said data comprises the steps of:

adding said stochastic resonance noise n₁ and n₂ with probability λ and

1−

λ

, respectively, to said data to form a data process; and

applying a fixed detector to said data process;

6. The method of claim 2, further comprising the steps of:

determining from said observed data process the stochastic resonance noise probability density function that consists of two delta functions;

determining stochastic resonance noise consisting of two random variables n₁ and n₂ with values equal to those of said two delta function locations and having probabilities equal to those of said stochastic resonance noise probability density function

adding said stochastic resonance noise to said data; and

applying a fixed detector to the resulting data process.

7. The method of claim 6, wherein the step of determining from said observed data process the stochastic resonance noise probability density function that consists of two delta functions comprises the steps of:

estimating the stochastic resonance noise consisting of two random variables n₁ and n₂ by estimating the unknown data probability density functions;

applying estimated probability density functions;

determining F_i(x)=∫

_RNφ

(y) p_i(y−

x)dy i=0,1, using known critical function φ

(y) and known data probability density functions p_i(•

), i=0,1;

determining the three unknown quantities n₁, n₂, and λ

using the known value k₀, and estimated values for {circumflex over (ƒ

)}₀₁, {circumflex over (ƒ

)}₀₂, and Ĵ

in the equations

$\begin{array}{c}\frac{d<br><br>\hat{J}}{d<br><br>f_0}<br><br>\left({\hat{f}}_{01}<br><br>\left(k_0\right)\right)=\frac{d<br><br>\hat{J}}{d<br><br>f_0}<br><br>({\hat{f}}_{02}<br><br>\left(k_0\right),\\ \frac{d<br><br>\hat{J}}{d<br><br>f_0}<br><br>\left({\hat{f}}_{02}<br><br>\left(k_0\right)\right)=k_0,\end{array}$

and Ĵ

({circumflex over (ƒ

)}₀₂(k₀))−

Ĵ

({circumflex over (ƒ

)}₀₁(k₀)=k₀({circumflex over (ƒ

)}₀₂(k₀)−

{circumflex over (ƒ

)}₀₁(k₀)); and

determining the probability of occurrence for n₁ and n₂ as λ and

1−

λ

, respectively, using the equation

$\mathrm{\lambda <br><br>}=\frac{{\hat{f}}_{02}<br><br>\left(k_0\right)-P_{\mathrm{FA}}^x}{{\hat{f}}_{02}<br><br>\left(k_0\right)-{\hat{f}}_{01}<br><br>\left(k_0\right)}.$

8. A method in reducing the probability of error in non-Gaussian noise, comprising the steps of:

recording an observed data process;

determining stochastic resonance noise; and

adding stochastic resonance noise to the data of the recorded data process.

9. The method of claim 8, wherein the step of determining stochastic resonance noise comprises the steps of

identifying a known data probability density function for said data process;

determining from said known probability density function of said observed data process the stochastic resonance noise probability density function that consists of a single delta function, δ

(n−

n₀) with value n₀ equal to the delta function location with probability one.

10. The method of claim 9, further comprising the step of determining a minimum probability of error using

$\begin{array}{c}{P}_{e,\min }={\pi }_{}\end{array}$

1 ⁡

[ 1 - max f 0 ⁢

G ⁡

( f 0 , π

0 π

1 ) ] , where ⁢

 

⁢

G ⁡

( f 0 , k ) = J ⁡

( f 0 ) - kf 0 = P D - kP FA .

11. The method of claim 10, wherein the stochastic resonance noise probability density function consists of a single delta function located at n₀ and is calculated according to n₀=F₀⁻

1(f₀), where ƒ

₀ is the value the maximizes

$G<br><br>\left(f_0,\frac{{\mathrm{\pi <br><br>}}_0}{{\mathrm{\pi <br><br>}}_1}\right).$

12. A method of evaluating functions

$f_1,J$

( f 0 ) , ⅆ

J ⅆ

f 0 ,

comprising the steps of;

solving the equation x=F₀<sup>−

1</sup>(ƒ

₀) for any ƒ

₀;

obtaining the value of ƒ

₁ by ƒ

₁=F₁(x);

solving

$J<br><br>\left(f_0\right)=\underset{f_1}{\max }<br><br>\left(f_1<br><br>\left(f_0\right)\right);<br><br>$

and

solving

$\frac{d<br><br>J}{d<br><br>f_0}=\underset{\mathrm{\Delta <br><br>}\to <br><br>0}{\lim }<br><br>\frac{J<br><br>\left(f_0+\mathrm{\Delta <br><br>}\right)-J<br><br>\left(f_0\right)}{\mathrm{\Delta <br><br>}}.$