Linearized static panoramic optical mirror

• US 6,754,614 B1
• Filed: 02/23/2000
• Issued: 06/22/2004
• Est. Priority Date: 02/25/1999
• Status: Expired due to Fees

• Current Assignee: Inter Science GmbH
• Original Assignee: Inter Science GmbH
• Inventors: Beckstead, Jeffrey A., Barton, George G.
• Primary Examiner(s): Teska, Kevin J.
• Assistant Examiner(s): FERRIS III, FRED O
• Application Number:
  US09/511,712
• Time in Patent Office: 1,581 Days
• Field of Search: 703/2, 703/7, 359/725, 359/201, 348/36
• US Class Current: 703/2
• CPC Class Codes: G02B 13/06 Panoramic objectives; So-called "sky lenses" including panoramic objectives having reflecting surfaces

Abstract

An improved conical panoramic mirror element design is disclosed such that the panoramic vertical field of view is not fixed and the image covers at least 90% of the toroidal image pixels of an imaging device. The data required to prescribe the panoramic conical element includes the position of the detector device, the most negative vertical scene angle, the most positive vertical scene angle, the panoramic cone'"'"'s base diameter, the cone'"'"'s apex to base ratio. These are utilized according to a mathematical prescription that optimizes the mirror element'"'"'s design.

Claims

1. A method for designing a fabrication profile for a panoramic mirrored cone element, said cone element comprising a base and an apex, using calculating means for performing mathematical calculations, comprising the steps, starting with a plurality of predetermined input parameters for said cone element, of;

calculating a parabolic profile for said cone element;

calculating a spherical profile for said cone element, based on said parabolic profile;

calculating a composite profile for said cone element, based on said parabolic profile and said spherical profile; and

calculating an elliptical profile for said cone element, based on said composite profile;

wherein

said fabrication profile comprises said elliptical profile.

2. The method of claim 1, said predetermined input parameters comprising:

a position of a detector device in a proposed application of said cone element;

a most positive vertical scene angle Ø

1;

a most negative vertical scene angle Ø

2;

an initial diameter of said base of said cone element;

an initial aspect ratio of said cone element, defined as a ratio of said apex of said cone to said base of said cone.

3. The method of claim 2, said step of calculating said parabolic profile for said cone element further comprising the steps of:

A) for a first approximation;

1;

solving for y1, x1;

y1=Kp/Tan²(θ

1)

x1=2(Kp*y1)^1/2

2;

solving for y2, x2;

y2=Kp/Tan²(θ

2)

x2=2(Kp*y2)^1/2,

where θ

1=45−

Ø

1/2 and θ

2=45−

Ø

2/2, and Kp=Db/2, where Db is a diameter of said base of said cone element, and

3;

determining a correction factor dX for Kp;

dX=(Rc*(x1−

x2))/(Rc−

1);

where Rc denotes an aspect ratio of said cone element apex to said cone element base;

B) for a second approximation;

4;

setting Kp=(Db/2)²/(X2−

dX),

recalculating said equations 1, 2, and 3 using said value of Kp determined by said equation 4, and calculating the factors;

KpX=dX, recalculated with said Kp of equation B)4,

KpY=y1, recalculated with said Kp of equation B)4,

Ym=y2−

y1, where Ym is a height of said cone element;

C) generating data files containing angular and associated radial positions of all points of said parabolic profile, and y and x coordinates of all points of said parabolic profile and associated tangents to an optical axis and a value of Ø

for each value of said y coordinates; and

D) using said data files generated in said step C), solving for x, Tan (θ

)=f(y), the parabolic profile equations;

4. The method of claim 3, said step of calculating said spherical profile for said cone element further comprising the steps of:

E) determining spherical constants Kc, KcY, KcX by calculating;

1;

a length of a chord to a face arc;

RO=(dy²+dx²)^1/2

2;

a spherical radius;

Kc=RO*Sin((θ

2−

θ

1)/2)/2

3;

spherical constants;

KcY=Kc*Sin(θ

1)

KcX=Kc*Cos(θ

1),

where said height of said cone element dy=y2−

y1, a width of a face arc dx=x2−

x1, and tangents to a surface which are θ

2 at y2 and θ

1 at y1, using said y1, y2, x1, and x2 generated from said step of calculating said parabolic profile; and

F) solving the spherical profile equations;

5. The method of claim 4, said step of calculating said composite profile for said cone element further comprising the steps of:

G) using the parameters and data files generated in said steps of calculating said parabolic and spherical profiles, calculating;

1;

Dr=(x2−

x1)/(PxR−

PxA), radial displacement per pixel,

2;

DØ

=(Ø

1−

Ø

2)/(PxR−

PxA), Ø

angle displacement per pixel,

3;

Dpx=(Ø

n−

Ø

1)/DØ

−

(xn−

x1)/Dr, pixel error, for all records in said data files, and if abs (Dpx)>

a prior Dpx, storing the larger magnitude Dpx,

4;

1−

Dpx/(PxR−

PxA), linearity, where;

PxR=number of pixels per radius in detector,

PxA=PxR/Rc=number of radial pixels subtended by said apex of said cone element, and

PxR−

PxA=number of active pixels;

H) determining proportionality constants Js for said spherical profile and Jp for said parabolic profile, as;

Js=Dpp/(Dpp+Dps)

Jp=1−

Js,

where Dpp=Parabolic profile Dpx and Dps=Spherical profile Dpx; and

I) solving the composite profile equations;

6. The method of claim 5, said step of calculating said elliptical profile for said cone element further comprising the steps of:

J) approximating an eccentricity e from the data produced by said steps of calculating said parabolic, spherical, and composite profiles for said cone element, and by iterative procedure, determining a value of e for a least pixel error;

K) calculating elliptical constants Ke, KeY, KeX using parameters b=a(1−

e²)^1/2 and Ym=Parabolic y2−

y1, with initial conditions Km=(1−

e²)^1/2 and Ke=an initial, user-defined value, according to;

1;

y1=Ke*Tan(θ

1)/(Km²+Tan²(θ

1))^1/2

2;

y2=Ke*Tan(θ

2)/(Km²+Tan²(θ

2))^1/2

3;

Ke=Ke*Ym/(y1−

y2)

4;

Recalculate equations K) 1, 2, 3

5;

x1=Km*(Ke²−

y1²)^1/2

6;

x2=Km*(Ke²−

y2²)^1/2

7;

KeX=(x2−

(Rc*x1))/(1−

Rc)

8;

Ke=Ke*Db/2/(x2−

Kex)

9;

Recalculate equations K) 1, 2, 5, 6, 7

10;

KeY=y1

11;

Ya=KeY−

y, y being a controlling variable; and

L) solving the elliptical profile equations;

 

X=Km*(Ke²−

Ya²)^1/2

7. The method of claim 1, further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

8. The method of claim 6, further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

9. The method of claim 7, further comprising the step of fabricating a central cylinder running longitudinally through said panoramic mirrored cone element.

10. A panoramic mirrored cone element product, produced by a process for designing a fabrication profile for said cone element, said cone element comprising a base and an apex, said process comprising the steps, starting with a plurality of predetermined input parameters for said cone element, of:

calculating a parabolic profile for said cone element;

calculating a spherical profile for said cone element, based on said parabolic profile;

calculating a composite profile for said cone element, based on said parabolic profile and said spherical profile; and

calculating an elliptical profile for said cone element, based on said composite profile;

wherein

said fabrication profile comprises said elliptical profile.

11. The product of claim 10, said predetermined input parameters comprising:

a position of a detector device in a proposed application of said cone element;

a most positive vertical scene angle Ø

1;

a most negative vertical scene angle Ø

2;

an initial diameter of said base of said cone element;

an initial aspect ratio of said cone element, defined as a ratio of said apex of said cone to said base of said cone.

12. The product of claim 11, said step of calculating said parabolic profile for said cone element further comprising the steps of:

A) for a first approximation;

1;

solving for y1, x1;

y1=Kp/Tan²(θ

1)

x1=2(Kp*y1)^1/2

2;

solving for y2, x2;

y2=Kp/Tan²(θ

2)

x2=2(Kp*y2)^1/2,

where θ

1=45−

Ø

1/2 and θ

2=45−

Ø

2/2, and Kp=Db/2, where Db is a diameter of said base of said cone element, and

3;

determining a correction factor dX for Kp;

dX=(Rc*(x1−

x2))/(Rc−

1);

where Rc denotes an aspect ratio of said cone element apex to said cone element base;

B) for a second approximation;

4;

setting Kp=(Db/2)²/(X2−

dX),

recalculating said equations 1, 2, and 3 using said value of Kp determined by said equation 4, and

calculating the factors;

KpX=dX, recalculated with said Kp of equation B)4,

KpY=y1, recalculated with said Kp of equation B)4,

Ym=y2−

y1, where Ym is a height of said cone element;

C) generating data files containing angular and associated radial positions of all points of said parabolic profile, and y and x coordinates of all points of said parabolic profile and associated tangents to an optical axis and a value of Ø

for each value of said y coordinates; and

D) using said data files generated in said step C), solving for x, Tan (θ

)=f(y), the parabolic profile equations;

13. The product of claim 12, said step of calculating said spherical profile for said cone element further comprising the steps of:

E) determining spherical constants Kc, KcY, KcX by calculating;

1;

a length of a chord to a face arc;

RO=(dy²+dx²)^1/2

2;

a spherical radius;

Kc=RO*Sin((θ

2−

θ

1)/2)/2

3;

spherical constants;

KcY=Kc*Sin(θ

1)

KcX=Kc*Cos(θ

1),

where said height of said cone element dy=y2−

y1, a width of a face arc dx=x2−

x1, and tangents to a surface which are θ

2 at y2 and θ

1 at y1, using said y1, y2, x1, and x2 generated from said step of calculating said parabolic profile; and

F) solving the spherical profile equations;

14. The product of claim 13, said step of calculating said composite profile for said cone element further comprising the steps of:

G) using the parameters and data files generated in said steps of calculating said parabolic and spherical profiles, calculating;

1;

Dr=(x2−

x1)/(PxR−

PxA), radial displacement per pixel,

2;

DØ

=(Ø

1−

Ø

2)/(PxR−

PxA), Ø

angle displacement per pixel,

3;

Dpx=(Ø

n−

Ø

1)/DØ

−

(xn−

x1)/Dr, pixel error, for all records in said data files, and if abs (Dpx)>

a prior Dpx, storing the larger magnitude Dpx,

4;

1−

Dpx/(PxR−

PxA), linearity, where;

PxR=number of pixels per radius in detector,

PxA=PxR/Rc=number of radial pixels subtended by said apex of said cone element, and

PxR−

PxA=number of active pixels;

H) determining proportionality constants Js for said spherical profile and Jp for said parabolic profile, as;

Js=Dpp/(Dpp+Dps)

Jp=1−

Js,

where Dpp=Parabolic profile Dpx and Dps=Spherical profile Dpx; and

I) solving the composite profile equations;

15. The product of claim 14, said step of calculating said elliptical profile for said cone element further comprising the steps of:

J) approximating an eccentricity e from the data produced by said steps of calculating said parabolic, spherical, and composite profiles for said cone element, and by iterative procedure, determining a value of e for a least pixel error;

K) calculating elliptical constants Ke, KeY, KeX using parameters b=a(1−

e²)^1/2 and Ym=Parabolic y2−

y1, with initial conditions Km=(1−

e²)^1/2 and Ke=an initial, user-defined value, according to;

1;

y1=Ke*Tan(θ

1)/(Km²+Tan²(θ

1))^1/2

2;

y2=Ke*Tan(θ

2)/(Km²+Tan²(θ

2))^1/2

3;

Ke=Ke*Ym/(y1−

y2)

4;

Recalculate equations K) 1, 2, 3

5;

x1=Km*(Ke²−

y1²)^1/2

6;

x2=Km*(Ke²−

y2²)^1/2

7;

KeX=(x2−

(Rc*x1))/(1−

Rc)

8;

Ke=Ke*Db/2/(x2−

Kex)

9;

Recalculate equations K) 1, 2, 5, 6, 7

10;

KeY=y1

11;

Ya=KeY−

y, y being a controlling variable; and

L) solving the elliptical profile equations;

16. The product of claim 10, said process further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

17. The product of claim 15, said process further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

18. The product of claim 10, further comprising a central cylinder running therethrough.

19. A computerized device used for designing a fabrication profile for a panoramic mirrored cone element, said cone element comprising a base and an apex, said computerized device comprising means, starting with a plurality of predetermined input parameters for said cone element, for:

calculating a parabolic profile for said cone element;

calculating a spherical profile for said cone element, based on said parabolic profile;

calculating a composite profile for said cone element, based on said parabolic profile and said spherical profile; and

calculating an elliptical profile for said cone element, based on said composite profile;

wherein

said fabrication profile comprises said elliptical profile.

20. The computerized device of claim 19, said predetermined input parameters comprising:

a position of a detector device in a proposed application of said cone element;

a most positive vertical scene angle Ø

1;

a most negative vertical scene angle Ø

2;

an initial diameter of said base of said cone element;

an initial aspect ratio of said cone element, defined as a ratio of said apex of said cone to said base of said cone.

21. The computerized device of claim 20, said means for calculating said parabolic profile for said cone element further comprising means for:

A) for a first approximation;

1;

solving for y1, x1;

y1=Kp/Tan²(θ

1)

x1=2(Kp*y1)^1/2

2;

solving for y2, x2;

y2=Kp/Tan²(θ

2)

x2=2(Kp*y2)^1/2,

where θ

1=45−

Ø

1/2 and θ

2=45−

Ø

2/2, and Kp=Db/2, where Db is a diameter of said base of said cone element, and

3;

determining a correction factor dX for Kp;

dX=(Rc*(x1−

x2))/(Rc−

1);

where Rc denotes an aspect ratio of said cone element apex to said cone element base;

B) for a second approximation;

4;

setting Kp=(Db/2)²/(X2−

dX),

recalculating said equations 1, 2, and 3 using said value of Kp determined by said equation 4, and

calculating the factors;

KpX=dX, recalculated with said Kp of equation B)4,

KpY=y1, recalculated with said Kp of equation B)4,

Ym=y2−

y1, where Ym is a height of said cone element;

C) generating data files containing angular and associated radial positions of all points of said parabolic profile, and y and x coordinates of all points of said parabolic profile and associated tangents to an optical axis and a value of Ø

for each value of said y coordinates; and

D) using said data files generated in said step C), solving for x, Tan (θ

)=f(y), the parabolic profile equations;

22. The computerized device of claim 21, said means for calculating said spherical profile for said cone element further comprising means for:

E) determining spherical constants Kc, KcY, KcX by calculating;

1;

a length of a chord to a face arc;

RO=(dy²+dx²)^1/2

2;

a spherical radius;

Kc=RO*Sin((θ

2−

θ

1)/2)/2

3;

spherical constants;

KcY=Kc*Sin(θ

1)

KcX=Kc*Cos(θ

1),

where said height of said cone element dy=y2−

y1, a width of a face arc dx=x2−

x1, and tangents to a surface which are θ

2 at y2 and θ

1 at y1, using said y1, y2, x1, and x2 generated from said means for calculating said parabolic profile; and

F) solving the spherical profile equations;

23. The computerized device of claim 22, said means for calculating said composite profile for said cone element further comprising means for:

G) using the parameters and data files generated in said steps of calculating said parabolic and spherical profiles, calculating;

1;

Dr=(x2−

x1)/(PxR−

PxA), radial displacement per pixel,

2;

DØ

=(Ø

1−

Ø

2)/(PxR−

PxA), Ø

angle displacement per pixel,

3;

Dpx=(Ø

n−

Ø

1)/DØ

−

(xn−

x1)/Dr, pixel error, for all records in said data files, and if abs (Dpx)>

a prior Dpx, storing the larger magnitude Dpx,

4;

1−

Dpx/(PxR−

PxA), linearity, where;

PxR=number of pixels per radius in detector,

PxA=PxR/Rc=number of radial pixels subtended by said apex of said cone element, and

PxR−

PxA=number of active pixels;

H) determining proportionality constants Js for said spherical profile and Jp for said parabolic profile, as;

24. The computerized device of claim 23, said means for calculating said elliptical profile for said cone element further comprising means for:

J) approximating an eccentricity e from the data produced by said steps of calculating said parabolic, spherical, and composite profiles for said cone element, and by iterative procedure, determining a value of e for a least pixel error;

K) calculating elliptical constants Ke, KeY, KeX using parameters b=a(1−

e²)^1/2 and Ym=Parabolic y2−

y1, with initial conditions Km=(1−

e²)^1/2 and Ke=an initial, user-defined value, according to;

1;

y1=Ke*Tan(θ

1)/(Km²+Tan²(θ

1))^1/2

2;

y2=Ke*Tan(θ

2)/(Km²+Tan²(θ

2))^1/2

3;

Ke=Ke*Ym/(y1−

y2)

4;

Recalculate equations K) 1, 2, 3

5;

x1=Km*(Ke²−

y1²)^1/2

6;

x2=Km*(Ke²−

y2²)^1/2

7;

KeX=(x2−

(Rc*x1))/(1−

Rc)

8;

Ke=Ke*Db/2/(x2−

Kex)

9;

Recalculate equations K) 1, 2, 5, 6, 7

10;

KeY=y1

11;

Ya=KeY−

y, y being a controlling variable; and

L) solving the elliptical profile equations;

25. The computerized device of claim 19, further comprising means for fabricating said panoramic mirrored cone element using said fabrication profile.

26. The computerized device of claim 24, further comprising means for fabricating said panoramic mirrored cone element using said fabrication profile.

27. The computerized device of claim 25, further comprising means for fabricating a central cylinder running longitudinally through said panoramic mirrored cone element.