Method for determining the complex amplitude of the electromagnetic field associated with a scene
A method for determining the complex amplitude of the electromagnetic field associated with a scene, comprising a) capturing a plurality of images of the scene by means of a photographic camera, the images being focused in planes of focus arranged at different distances, wherein the camera comprises a lens of focal length F and a sensor arranged at a certain distance from the lens in its image space, taking at least one image pair from the plurality of images and determining the accumulated wavefront to the conjugate plane in the object space corresponding to the intermediate plane with respect to the planes of focus of the two images of the pair.
1. A method for determining the complex amplitude of the electromagnetic field associated with a scene, comprising:
a) capturing a plurality of images of the scene by a photographic camera, the images being focused in planes of focus arranged at different distances, wherein the camera comprises a lens or system of lenses of focal length F and a sensor arranged at a certain distance from the lens in the image space,
b) taking at least one image pair from the plurality of images and determining the accumulated wavefront to a conjugate plane in an object space corresponding to an intermediate plane with respect to the planes of focus of the two images of the pair, determining the wavefront W(x,y) as:
W
(
x
,
y
)
=
∑
p
=
0
N
-
1
d
p
Z
p
(
x
,
y
)
where {Z p (x,y)} is a predetermined set of polynomials and N is the number of polynomials used in the expansion,
wherein the coefficients d j are determined by means of solving the system of equations:
u
2
X
(
j
)
-
u
1
X
(
j
)
2
z
=
(
∑
p
=
0
N
-
1
d
p
∂
∂
x
Z
p
(
x
,
y
)
)
x
=
u
1
X
(
j
)
+
u
2
X
(
j
)
2
,
y
=
u
1
Y
(
k
)
+
u
2
Y
(
k
)
2
u
2
Y
(
k
)
-
u
1
Y
(
k
)
2
z
=
(
∑
p
=
0
N
-
1
d
p
∂
∂
x
Z
p
(
x
,
y
)
)
x
=
u
1
X
(
j
)
+
u
2
X
(
j
)
2
,
y
=
u
1
Y
(
k
)
+
u
2
Y
(
k
)
2
where 2z is the distance between the planes of focus of the two images of the pair, where {(u 1X (j),u 1Y (k)),j,k=1 . . . T} are points belonging to the first image of the pair, and {(u 2X (j),u 2Y (k)),j,k=1 . . . T} are points belonging to the second image of the pair, such that for each 1≤j, k≤T, the following is verified
∫ −∞ u 1Y (k) ∫ −∞ u 1X (j) ƒ 1XY ( x,y ) dxdy=s ( j ) s ( k )
and
∫ −∞ u 2Y (k) ∫ −∞ u 2X (j) ƒ 2XY ( x,y ) dxdy=s ( j ) s ( k )
where s(j) is a sequence of real numbers of values between 0 and 1, monotonically increasing for each 1≤j≤T,
where ƒ XY is the two-dimensional density function which takes into account the probability of occurrence of a photon and is determined in each case by the normalized intensity I(x,y) of the corresponding image of the pair, i.e.:
∫ −∞ u 1Y (k) ∫ −∞ u 1X (j) I 1 ( x,y ) dxdy=s ( j ) s ( k )
∫ −∞ u 2Y (k) ∫ −∞ u 2X (j) I 2 ( x,y ) dxdy=s ( j ) s ( k ).
2. The method according to claim 1 , wherein the wavefront is determined by the expression:
W
(
x
,
y
)
=
∑
p
=
0
N
-
1
∑
q
=
0
N
-
1
d
pq
Z
pq
(
x
,
y
)
where
Z
pq
(
x
,
y
)
=
1
N
e
2
π
i
N
(
px
+
qy
)
for each 0≤p, q≤N−1.
3. The method according to claim 1 , further comprising determining the accumulated wavefront for a plurality of image pairs.
4. The method according to claim 3 , further comprising determining a phase shift between two planes of the object space as the subtraction of the accumulated wavefronts to said planes.
5. The method according to claim 4 , further comprising determining the phase shift for a plurality of object planes.
6. The method according to claim 1 , further comprising:
determining, from P images selected from the plurality of captured images, a value of a light field (L) focused at distance F at M values other than u, M≤P, as the values of the light field verifying the system of equations:
Σ n=1 M L F ( n +[( x−n )/ a j ], n )=α j 2 F 2 I j ( x ), ∀ j∈{ 1 . . . P}∧∀x∈{ 1 . . . k}
where P is the number of images considered for determining the light field, F is the focal length of the lens, L F is the value of the light field focused at distance F, α j F is the focus distance of the image j and I j (x) is the intensity of the image j, and where [x] denotes the integer closest to x,
obtaining as a result for each image j, with 1≤j≤P, the light field L F (x) evaluated at the value of u j resulting from the fit, i.e., the view of the light field corresponding to the value u j , where x and u are the two-dimensional vectors determining the position in the sensor and in the lens of the camera, respectively.
7. The method according to claim 6 , wherein the value of the light field is determined by solving the system of equations by means of least squares, i.e., minimizing:
∥Σ n=1 M L F ( n +( x−n )/α j ,n )−α j 2 F 2 I j ( x )∥ 2 .
8. The method according to claim 1 , wherein the two images of each selected image pair are taken, respectively, on both sides of the focus.
9. The method according to claim 8 , wherein the two images of each selected image pair are taken from symmetrical distances on both sides of the focus.
10. A device for determining the amplitude of the electromagnetic field associated with a scene, the device comprising:
means for capturing images comprising a lens of focal length F and an image sensor arranged parallel to the lens, at a certain distance from the lens in its image space; and
processing means configured for carrying out step b) of the method according to claim 1 .
11. The device according to claim 10 , wherein the processing means are additionally configured for determining the wavefront by the expression:
W
(
x
,
y
)
=
∑
p
=
0
N
-
1
∑
q
=
0
N
-
1
d
pq
Z
pq
(
x
,
y
)
where
Z
pq
(
x
,
y
)
=
1
N
e
2
π
i
N
(
px
+
qy
)
for each 0≤p, q≤N−1.
12. The device according to claim 10 , wherein the processing means are additionally configured for determining the accumulated wavefront for a plurality of image pairs.
13. The device according to claim 12 , wherein the processing means are additionally configured for determining a phase shift between two planes of the object space as the subtraction of the accumulated wavefronts to said planes.
14. The device according to claim 13 , wherein the processing means are additionally configured for determining the phase shift for a plurality of object planes.
15. The device according to claim 10 , wherein the processing means are additionally configured for:
determining, from P images selected from the plurality of captured images, a value of a light field (L) focused at distance F at M values other than u, M≤P, as the values of the light field verifying the system of equations:
Σ n=1 M L F ( n +[( x−n )/ a j ], n )=α j 2 F 2 I j ( x ), ∀ j∈{ 1 . . . P}∧∀x∈{ 1 . . . k}
where P is the number of images considered for determining the light field, F is the focal length of the lens, L F is the value of the light field focused at distance F, α j F is the focus distance of the image j and I j (x) is the intensity of the image j, and where [x] denotes the integer closest to x, and
obtaining as a result for each image j, with 1≤j≤P, the light field L F (x) evaluated at the value of u j resulting from the fit, i.e., the view of the light field corresponding to the value u j , where x and u are the two-dimensional vectors determining the position in the sensor and in the lens of the camera, respectively.
16. The device according to claim 15 , wherein the processing means are additionally configured for determining the value of the light field by solving the system of equations by means of least squares, i.e., minimizing:
∥Σ n=1 M L F ( n +( x−n )/α j ,n )−α j 2 F 2 I j ( x )∥ 2 .
17. The device according to claim 10 , wherein the processing means are additionally configured for taking the two images of each selected image pair, respectively, on both sides of the focus.
18. The device according to claim 17 , wherein the processing means are additionally configured for taking the two images of each selected image pair from symmetrical distances on both sides of the focus.