System and method for optical three-dimensional particle localization
Embodiments include methods that may be used to optically obtain the precise three-dimensional location of multiple objects from one or more two dimensional images. An optical point spread function having a transverse shape which varies with axial distance may be implemented to obtain depth information. The transverse variation in the PSF with depth may be produced using a cylindrical lens. The objects may be imaged by a focal plane array detector. One or more 2D images may be used to find the 3D location of the objects using sparse signal reconstruction methods.
1. A method for localizing an object containing an ensemble of light-switchable elements in three-dimensional space comprising the steps of:
causing a first sparse subset of the elements to illuminate;
acquiring a first image b having dimensions m (1/2) ×m (1/2) of said activated first sparse subset through optics having a point spread function (PSF) that varies predictably with axial object distance;
causing said first sparse subset of the elements to extinguish;
repeating the previous three steps for different sparse subsets of elements;
defining a sensing array Φ=m×n×l which represents a discrete set of possible PSF's that any of said elements can generate within a given volume having dimensions n (1/2) ×n (1/2) ×l; and
creating a three dimensional image of said elements by solving the following linear programming problem:
min Σ l ∥x l n ∥ 1 subject to ∥Σ l (Φ l mnl x l n )− b m ∥ 2 ≦ε(Σ m b m m ) 1/2 and x l n ≧0,
where: x=l-many n×1 image vectors comprising said n (1/2) ×n (1/2) ×l three dimensional volume and ε=user set tolerance on convergence and
where: m is the number of pixels in each image b, n is the total number of possible transverse locations in which a light-switchable element may localized and l is the number of planes of images.
2. The method of claim 1 , further comprising processing simultaneously a plurality of said sensing arrays.
3. The method of claim 1 , wherein each of said steps is repeated to create a plurality of results, and further comprising summing said results.
4. A method for localizing an object containing an ensemble of light-switchable elements in three-dimensional space comprising the steps of:
causing a first sparse subset of the elements to illuminate;
acquiring a first image b having dimensions m (1/2) ×m (1/2) of said activated first sparse subset through optics having a point spread function (PSF) that varies predictably with axial object distance;
causing said first sparse subset of the elements to extinguish;
repeating the previous three steps for different sparse subsets of elements;
defining a sensing array Φ=m×n×l which represents a discrete set of possible PSF's that any of said elements can generate within a given volume having dimensions n (1/2) ×n (1/2) ×l; and
creating a three dimensional image of said elements by solving the following linear programming problem:
min
∑
i
c
i
nT
x
i
n
subject
to
∑
i
(
Φ
1
mnl
x
1
n
)
-
b
m
≤
ɛ
(
∑
m
b
m
m
)
1
/
2
and
x
l
n
≥
0
,
where: x=l-many n×1 image vectors comprising said n (1/2) ×n (1/2) ×l three dimensional volume, where: m is the number of pixels in each image b, n is the total number of possible transverse locations in which a light-switchable element may localized and l is the number of planes of images,
ε=user set tolerance on convergence and
weighting factors are defined for n-many transverse locations each of said elements can occupy for each axial plane l, and wherein said weighing factors are represented by the equation
c
l
n
=
1
max
(
[
ϕ
11
l
+
ϕ
21
l
+
…
ϕ
m
1
l
⋮
ϕ
1
n
l
+
ϕ
2
nl
+
…
ϕ
mnl
]
)
[
ϕ
11
l
+
ϕ
21
l
+
…
ϕ
m
1
!
⋮
ϕ
1
nl
+
ϕ
2
nl
+
…
ϕ
mn
!
]
.
5. The method of claim 4 , wherein each of said steps is repeated to create a plurality of results, and further comprising summing said results.
6. A system for creating an image of an object comprising an ensemble of light-switchable elements, the system comprising:
an image sensor;
optics having a point spread function (PSF) that varies predictably with axial object distance;
an energy source; and
a computer; wherein
said computer: controls said energy source to illuminate a first portion of said light-switchable elements; controls said image sensor to acquire a first image of said light-switchable elements; controls said energy source to illuminate a second portion of said light-switchable elements; controls said image sensor to acquire a second image of said light-switchable elements and
defines a sensing array Φ=m×n×l which represents a discrete set of possible PSF's that any of said elements can generate within a given volume having dimensions n (1/2) ×n (1/2) ×l; and
creates a three dimensional image of said elements by solving the following linear programming problem:
min Σ l ∥x l n ∥ 1 subject to ∥Σ l (Φ l mnl x l n )− b m ∥ 2 ≦ε(Σ m b m m ) 1/2 and x l n ≧0,
where: x=l-many n×1 image vectors comprising said n (1/2) ×n (1/2) ×l three dimensional volume and ε=user set tolerance on convergence and where: m is the number of pixels in each image b, n is the total number of possible transverse locations in which a light-switchable element may localized and l is the number of planes of images.
7. The method of claim 5 , wherein said defining of said sensing array, said creating of said three dimensional image and said solving are performed on a field programmable gate array.
8. The method of claim 5 , wherein said optics include a cylindrical lens.