This invention relates to forming images with light, where the image is determined computationally from time-resolved measurements. The spatial resolution of the computed image is finer than the spatial support of the illuminations and finer than the spatial support of the sensitivities of the light meters. In an embodiment representing a limiting case, images are formed with a single illumination extending across the entire scene and each light meter sensitive to the entire scene.
Imaging is distinguished from other forms of sensing by having the goal of producing an image—a representation in one-to-one spatial correspondence with an object or scene. An image could appear on a continuous substrate such as film or paper and thus require uncountably—many numbers to be described. This invention relates instead to digital images—images described by finitely—many entries, where each entry is finitely—many numbers in discrete or continuous sets. The spatial correspondence of a digital image relates an entry in a digital image to a point, line, patch, or volume in the scene. Lines, patches, or volumes may overlap. Each entry in a digital image can represent brightness, reflectance, or other radiometric properties of a scene known to those of skill in the art. An entry in a digital image comprising a plurality of values may represent radiometric properties distinguished by wavelength or other distinctions known to those of skill in the art.
Traditional imaging involves illumination of the scene with a light source whose intensity does not vary appreciably with time. For example, the reflectance pattern on scene surfaces is acquired by using a lens to focus the incident light onto a plurality of light meters. Those of skill in the art will recognize that light meter can refer to any method of measuring light intensity, including but not limited to chemical and electronic methods, such as a PIN photodiode. Each light meter is sensitive to light originating at some subset of the scene, possibly with some variation of sensitivity with spatial location. The light meters are often in a regular two-dimensional (2D) array, but those skilled in the art will recognize that other arrangements of light meters are possible. Each light meter measures light intensity without time resolution. This means that while each light meter may have a sensitivity to incident light that varies with time—for example, using a shutter or any other method known to those skilled in the art—the light meter produces a single measurement, so variation of the incident light intensity with time cannot be inferred. Each entry of the digital image corresponds to one light meter. The number of light meters equals the number of pixels of the digital image; the number of pixels is also called the size of the digital image.
Many methods of imaging that use light sources and light meters in non-traditional ways are known to those skilled in the art. For example, dual photography [5] uses directed illumination and a single light meter. By Helmholtz reciprocity, directing light at one scene point, line, patch, or volume and measuring at one light meter is approximately equivalent to illuminating the entire scene and focusing reflected light only from the one scene point, line, patch, or volume on a light meter. Thus, by directing the illumination at a plurality of scene points, lines, patches, or volumes, one at a time, the effect of using a plurality of light meters is simulated. The number of distinct illumination points, lines, patches, or volumes equals the size of the digital image. Compressive dual photography [10], structured illumination [11], patterned light collection [12], and computational ghost imaging [13] methods are variations on traditional or dual photography that use simultaneous illumination of and measurement of incident light from a plurality of scene points, lines, patches, or volumes. Digital images are formed through computational means with spatial resolution that is not finer than the smaller of the spatial support of the illumination and the spatial support of the sensitivity of the light meters. These methods do not use time-resolved measurement by the light meters.
The use of time-resolved measurement by light meters, especially very high-speed sensing in photography, is associated with effectively stopping motion [2]. An alternative is to exploit the finite speed of light to associate travel time of light with distances. This is achieved in time-of-flight range measurement systems [3,4], which map time shifts in intensity-modulated light signals to distances. These systems and a generalization that employs multiply-reflected light [6] do not infer properties other than distances, such as brightness or reflectance.
A method is known in which a digital image of a surface not in the line of sight of the illumination source or light meters is produced wife light reflected from an approximately Lambertian reflector surface that is in the line of sight of the illumination source, the surface to be imaged, and a set of light meters [1]. The reflector surface is illuminated with very short impulses of light using a 50-femtosecond Ti:Sapphire laser operating at 75 MHz and with a wavelength of 798 nm. Light that scatters from the reflector to the surface to be imaged, back to the reflector and then to a set of light meters that are optically focused on the reflector are measured with very fine time resolution of 20 picoseconds using a streak camera. Measurements that are obtained with six illumination target points chosen on the reflector are computationally processed to obtain 50-by-50 pixel digital images. This prior art is limited by the use of very short pulses of light (approximately 50 femtoseconds) since fast variation of illumination intensity is more difficult to achieve than slower variation. Furthermore, it is limited by the use of indirect illumination and assumptions of omnidirectional reflections from the reflector surface.
The present invention overcomes the aforementioned drawbacks of previous imaging systems and methods by generating a digital image with spatial resolution finer than the smaller of the spatial support of the illumination intensity characteristic and the spatial support of the light meter sensitivity characteristic using light directed at a scene to be imaged that has time-varying intensity. In a preferred embodiment of the invention, a plurality of time-resolved light meters is provided for receiving light reflected from the scene to generate time-resolved samples of the intensity of light incident at each light meter; systems and methods involving only one light meter are also described. The temporal variation in the intensity of light incident at each sensor is related to radiometric properties of the scene, and a computer means processes site discrete-time samples to form a digital image. For any fixed desired image accuracy, suitable illumination bandwidth, sampling frequency, number of illumination spatial intensity characteristics, number of light meter spatial sensitivity characteristics, spatial extent of the light meters, distance between imaging device and scene, and image sure scale together. In one example, 16 light meters in a regular array of size 4 m by 4 m provide a sharp 50 pixel by 50 pixel image of a scene 10 m from the imaging device with illumination bandwidth approximately 375 MHz and sampling frequency approximately 750 MHz. Those skilled in the art will recognize that many other combinations of parameters may be used. This opens up possibilities for forming digital images without leases and mirrors and enables practical implementations of the imaging of occluded scenes. Need for a plurality of light meters can be removed by moving a single light meter to multiple locations or using a plurality of illuminations with differing spatial intensity characteristics. Such a plurality of illuminations could be produced with a plurality of light sources or a plurality of configurations of light sources and optical devices such as mirrors, lenses, and spatial light modulators.
As in a time-of-flight range measurement system, the present invention exploits that the speed of light is finite. Differences in optical path lengths combined with temporal variation of illumination give intensity variations with both space and time in the light emitted by the scene. The generation of a digital image is achieved through an unmixing of the radiometric information that is combined at each light meter because distinct optical paths may have equal path length.
a and 2b are plots illustrating that time delay from source to scene to light meter is a continuous function of the position in the scene.
a and 3b are plots showing a single measurement function according to m embodiment of the invention herein.
a is a graph of amplitude against time showing responses to impulsive illumination in
b is another graph of amplitude against time showing continuous time responses.
The invention is described through one exemplary configuration. Those skilled in the art can generate many other configurations.
Consider the imaging scenario depicted in
We assume that the position, orientation, and dimensions (L-by-L) of the planar surface 10 are known. Many methods for estimating these geometric parameters are known to those skilled in the art; a method using diffuse illumination and time-resolved sensing was recently demonstrated [6].
Formation of art ideal gray scale image is the recovery of the reflectance pattern on the surface, which can be modeled as a 2D function ƒ:[0,L]2→[0,1]. Those skilled in the art can replace the gray scale reflectance with other radiometric scene properties, including but not limited to a set of 2D functions representing reflectance at various wavelengths. We assume the surface to be Lambertian so that its perceived brightness is invariant to the angle of observation [7]; those skilled in the art may incorporate any bidirectional reflectance distribution function (BRDF), including but not limited to the Phong model, the Blinn-Phong model, the Torrance-Sparrow model, the Cook-Torrance model, the Oren-Nayar model, the Ashikhmin-Shirley model, the He-Torrance-Sillion-Greenberg model, the Lebedev model, the fitted Lafortune model, or Ward's anisotropic model.
The light incident at Light Meter k is a combination of the time-delayed reflections from all points on the planar surface. For any point x=(x1,x2)ε[0,L]2, let d(1)(x) denote the distance from illumination source to x, and let dk(2)(x) denote the distance from x to Light Meter k. Then dk(x)=d(1)(x)+dk(2)(x) is the total distance traveled by the contribution from x. This contribution is attenuated by the reflectance ƒ(x), square-law radial fall-off, and cos(θ(x)) to account for foreshortening of the surface with respect to the illumination, where θ(x) is the angle between the surface normal at x and a vector from x to the illumination source. Thus, when the intensity of the omnidirectional illumination is abstracted as a unit impulse at time 0, denoted s(t)=δ(t), the contribution from point x is the light intensity signal αk(x)ƒ(x)ε(t−dk(x)), where we have normalized to unit speed of light and
is the light meter spatial sensitivity characteristic for Light Meter k. Examples of distance functions and light meter spatial sensitivity characteristics are shown in
Combining contributions over the plane, the total light incident at Light Meter k is
g
k(t)=∫0L∫0Lαk(x)ƒ(x)δ(t−dk(x))dx1dx2. (2)
Thus, evaluating gk(t) at a fixed time t amounts to integrating over xε[0,L]2 with dk(x)=t. Define the isochronal curve Ckt=[x:dk(x)=t]. Then
g
k(t)=∫αk(x(k,u))ƒ(x(k,u))du (3)
where x(k,u) is a parameterization of Ckt∩[0,L]2 with unit speed. The intensity gk(t) thus contains the contour integrals over Ckt's of the desired function ƒ. Each Cktis a level curve of dk(x); as illustrated in
A digital system can use only samples of gk(t) rather than the continuous-time function itself. We now see how uniform sampling of gk(t) with a linear time-invariant (LTI) prefilter relates to linear functional measurements of ƒ. This establishes the foundations of a Hilbert space view of diffuse imaging. Those skilled in the art can incorporate effects of non-LTI device characteristics.
Suppose discrete samples am obtained at Light Meter k with sampling prefilter hk(t) and sampling interval Tk:
y
k
[n]=(gk(t)*hk(t))|, n=1,2, . . . , N. (4)
A sample yk[n] can be seen as a standard L2(R) inner product between gk and a time-reversed and shifted hk [8]:
y
k
[n]=<g
k(t),hk(nTk−t)>. (5)
Using (2), we can express (5) in terms of ƒ using the standard L2([0,L]2) inner product:
yk[n]=<ƒ,φk,n> where (6a)
φk,n(x)=ak(x)hk(nTk−dk(x)). (6b)
Over a set of sensors and sample times, [φk,n] will span a subspace of L2([0,L]2), and a sensible goal is to form a good approximation of ƒ in that subspace.
For ease of illustration and interpretation, let Tk=T, meaning all light meters have the same time resolution, and
for all k, which corresponds to “integrate and dump” sampling. Now since hk(t) is nonzero only for tε[0,T], by (4) or (5), the sample yk[n] is the integral of gk(t) over tε[(n−1)T, nT]. Thus, by (3), yk[n] is an α-weighted integral of ƒ between the contours Ck(n-1)T and CknT. To interpret this as an inner product with ƒ as in (5), we see that φk,n(x) is αk(x) between Ck(n-1)T and CknT and zero otherwise. for a single light meter have disjoint supports because of (7); their partitioning of the domain [0,L]2 is illustrated in
To express an estimate {circumflex over (ƒ)} of the reflectance ƒ, it is convenient to fix an orthonormal basis for a subspace of L2([0,L]2) and estimate the expansion coefficients in that basis. For an M-by-M pixel representation, let
so that {circumflex over (ƒ)}=ΣMΣ
Mci,jψi,j in the span of (ψi,j) is constant on Δ-by-Δ patches, where Δ=L/M. Those skilled in the art can generalise the reconstruction space to other finite-dimensional manifolds in L2([0,L]2).
For {circumflex over (ƒ)} to be consistent with the value measured by Sensor k at time n, we must have
yk[n]=<{circumflex over (ƒ)},φk,n>=ΣMΣ
Mci,j<ψi,j,φk,n>. (9)
Note that the inner products [<ψi,j,φk,n>] exclusively depend on Δ, the positions of illumination and sensors, the plane geometry, the sampling prefilters [hk]K, and the sampling intervals [Tk]
K—not on the unknown reflectance of interest ƒ. Hence, we have a system of linear equations to solve for the coefficients [ci,j]. (In the case of basis (8), the coefficients are the pixel values multiplied by Δ.)
When we specialize to the box sensor impulse response (7) and basis (8), many inner products <ψi,j,φkn> are zero so the linear system is sparse. The inner product <ψi,j,φk,n> is nonzero when reflection from the (i, j) pixel affects the light intensity at Sensor k within time interval [(n−1)T,nT]. Thus, for a nonzero inner product the (i, j) pixel must intersect the elliptical annulus between Ck(n−1)T and CknT. With reference to
To express (9) with a matrix multiplication, replace double indexes with single indexes (i.e., vectorize, or reshape) as
y=Ac (10)
where yεRKN contains the data samples (yk[n]), the first N from Light Meter 1, the next N from Light Meter 2, etc.; and cεRM
Assuming that A has a left inverse (i.e., rank(A)=M2), one can form an image by solving (10). The portion of A from one sensor cannot have full column rank because of the collapse of information along elliptical annuli depicted in
Those skilled in the art will appreciate that the rank condition is not necessary for producing a digital image. Any method of approximate solution of (10) could be employed. In the case that non-LTI effects are included, the analogue of (10) may be nonlinear, in which case rank would not apply. The invention applies to any method of approximate solution of the resulting system of equations.
Those skilled in the art will recognize that the discretized linear system (10) is not the only way to process the measurements in (6) to form a digital image. For example, the measurements are a one-dimensional projection of an elliptical Radon transform, and methods such as those in [14,15,16] can be employed.
A Dirac impulse illumination is an abstraction that cannot be realized in practice. One can use expensive, ultrafast optical lasers that achieve Terahertz bandwidth as an approximation to impulsive illumination, as in [1]. The present invention allows practical, non-impulsive illuminations to improve upon impulsive illumination for typical scenes and sensors.
Light transport is linear and time invariant. Hence, the effect of a general illumination intensity waveform s(t) is the superposition of effects of constant illuminations over infinitesimal intervals. This superposition changes the light incident at Light Meter k from gk(t) in (2) to gk(t)*s(t). Thus, the block diagram in
A typical natural scene ƒ has a good bandlimited approximation. Integration over elliptical contours Ckl further smooths the signal. Plotted in
In [1], high-bandwidth illumination and sampling were used under the assumption that these would lead to the highest reconstruction quality. However, impulsive illumination severely limits the illumination energy, leading to poor SNR, especially due to the radial fall-off attenuations in (1).
Here we compare impulsive and lowpass illumination. All image reconstructions are obtained with (10) regularised by the l2 norm of the discrete Laplacian, with regularization parameter optimised for l2 error. This conventional technique for backprojection [9] mildly promotes smoothness of the reconstruction; additional prior information, such as sparsity with suitable [ψi,j], is beneficial but would obscure the novelty of the invention. Results are for M=50 and several values of sampling period T and sensor array extent W.
In direct analogy with [1], we simulated short-pulsed, high-bandwidth illumination using a square-wave source with unit amplitude and time width equal to one-fifth of T.
Using non-impulsive, low-bandwidth illumination, we show that high SNR can be achieved while improving the reconstruction resolution. We chose s(t) to be the truncated impulse response of a third-order Butterworth filter, with again a unit peak amplitude. As illustrated in
The proof-of-concept experiments in [1] show that diffuse imaging can succeed in forming image reconstructions. In this patent application we have used signal processing abstractions to show that using lowpass time-varying illumination instead of impulsive sources improves the SNR and reconstruction quality.
Assigning dimensions to our simulation enables the specification of required device capabilities for one instantiation of the exemplary configuration. Suppose the 50-by-50 pixel image reconstruction with W=4 and T=0.1 shown in
The numbers in brackets refer to the references listed herein. The contents of all of these references are incorporated herein by reference as is the provisional application to which this application claims priority.
It is recognised that modifications and variations of the present invention will be apparent to those of skill in the art—including but not limited to the variations mentioned in the description of the exemplary configuration—and it is intended that all such modifications and variations be included within the scope of the appended claims.
This application claims priority to provisional application No. 61/524795 filed on Aug. 18, 2011, the contents of which are incorporated herein by reference.
This invention was made with government support under Grant No. CCF-0643836 awarded by the National Science Foundation. The government has certain rights in this invention.
Number | Date | Country | |
---|---|---|---|
61524795 | Aug 2011 | US |