Tomographic scanning imager

Harald Hovland

doi:10.1364/OE.17.011371

1. Introduction

Visible and infrared (IR) radiation emitted from, or reflected off objects can be used to detect, track, classify and identify these. This is done using sensors ranging from simple hot-spot trackers giving object location, to multispectral imagers that provide information about the object itself, the surrounding environment or even the climate through structural, textural, shape, radiometric, polarization and spectral information. This, together with a priori information can be used to infer indirect information such as surface temperature, emissivity or reflectivity, and even object fabrics, plant health and if or how an object is used or handled.

Unfortunately, both imaging IR sensors and multispectral imaging sensors can be costly, making their inclusion in low-cost solutions difficult. This work presents a potentially low-cost imager, bridging the gap between simple hot-spot trackers and full-fledged multispectral imagers. With a simple rotational scan, the single pixel TOSCA imager has better noise performance than a conventional single pixel scanning system, and a circular array TOSCA imager performs better than a linear array imager. The potential for remote sensing makes this a useful tool for unmanned aerial vehicles (UAV) capable of surveying wide areas. Other possibilities include sensors operating at wavelengths where focal plane arrays may be difficult or expensive to obtain, such as sensors for detecting THz or ionizing radiation.

1.1 Background

Detection, tracking and identification, and later imaging have to a large extent been driven by military needs. A practical solution for autonomous tracking for missile seekers first appeared with the first successful deployment of the Air Intercept Missile (AIM-9 “Sidewinder”) in 1953 [1]. For missiles, a low-cost solution is desirable. The earliest models employed spin-scan or conical-scan (con-scan) reticle seekers that are simple hot spot trackers, found suitable in relatively simple air target scenarios. Later models employing spin-scan and con-scan optics, such as the circular line scan and crossbar seekers, have addressed certain shortfalls of the former seeker models and also improved their signal to noise ratio. Nevertheless, the elegancy, simplicity and effectiveness of the reticle seeker keep the technology widespread. Figure 1 shows the con-scan reticle seeker principle together with a possible realization.

Although hot spot sensors are suitable for simple air target tracking, imagers are much more versatile, due to the capability of spatially characterizing a target or an environment through shape, texture, orientation and/or structure. Imagers therefore have a wider application range, but their relatively high cost has restricted their use in some wavebands.

Several types of imaging or quasi-imaging systems have been proposed and deployed, including the single pixel rosette scan sensor, the linearly swept linear array sensor, and staring or step-stare focal plane array systems. Several proposals have also been made where spin-scan and con-scan reticle systems are combined with multi-element detectors to provide imaging capability. Driggers et al. [2] and Hong et al. [3] proposed spin-scan FM-reticle systems with relatively simple multi-element focal plane detectors. Several authors have proposed imaging sensors using reticles [4–7], but using an additional scanning mirror or spinning reticle. Hzu, Kopriva and Peršin proposed a detection scheme where a spectral beam-splitter and two detectors received the transmitted signal through the reticle in spin-scan [8, 9] and con-scan [10] optics. Using independent component analysis techniques [11] as well as blind source separation analysis, they were able to discriminate between several point sources.

These solutions have drawbacks requiring multiple pixel arrays, complex scanning or a limitation to very simple scenes. In previous work [12–14] however, I showed that the output signal from a con-scan reticle system with a single-element detector contain imaging information that can be extracted using tomography techniques, and gave an overview of how to reconstruct the image. This paper describes the mathematical details of this procedure, discusses problems that can occur in traditional con-scan reticle systems and shows how to avoid them. Starting with a short introduction to tomography, a mathematical description is then made of the signal extraction from an original image with a con-scan detection system, as well as the tomographic reconstruction process, illustrated by simulations. System limitations and optimization are addressed, and a comparison with conventional systems is presented.

2. Theory

2.1 Tomography

Tomographic imaging starts with the creation of several lower dimensional projections of a scene, which are then processed and combined to reconstruct the scene. In the simplest form, a 2-dimensional scene or object is projected orthogonally onto N different 1-dimensional lines. The image can then be reconstructed using the Radon transform, named after Johann Radon who laid the theoretical foundation of tomography with his 1917 paper [15]. Work on tomography was given a boost after Hounsfield invented the CAT scanner [16]. Tomography is now used in a broad range of applications and fields, including synthetic aperture radars and sonars, astronomy, seismology, medical imaging and imaging through scattering media. Highly readable introductory books on tomography have been written by Kak and Slaney [17], and Deans [18]. The latter contains a translation from German of Radon’s original paper. Natterer’s book has a more stringent mathematical approach [19].

Fig. 1. Schematics and possible realization of con-scan based sensors. (a) Schematics of classical con-scan sensor. (b) Schematics of TOSCA reticle based sensor. (c) Possible realization of classical con-scan sensor. (d) Possible realization of TOSCA reticle based sensor.

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2.2 The con-scan reticle sensor

The classical con-scan reticle sensor in Fig. 1(a) (schematics) and 1(c) (possible realization) mainly consists of three elements: Scanning focusing optics, a fixed reticle and a detector. The optics focuses the target scene image onto and scans it across the fixed reticle in a circular movement. The fixed reticle is a patterned mask with transmitting fields, through which the incoming light passes onto the detector to produce a signal. In the basic con-scan seeker, the target image, generally reduced to a “hot spot”, is scanned across radially distributed reticle sectors. Assuming a constant scene, the signal produced from a central target is a fixed-frequency pulse train, with a stationary fundamental (carrier) frequency. Offset targets leads to carrier frequency modulation, the modulation depth and phase giving the target position.

2.3 The TOSCA principle

The con-scan reticle system’s imaging properties arises as follows: The target image movement across the reticle follows a circular yet translational path, as the target image orientation remains fixed. The knife edge transitions between reticle regions occur at well defined orientations, as shown in Fig.2. The con-scan then becomes a sequence of knife-edge scans of the scene with a well defined angular distribution. Tomographic reconstruction of knife-edge scans was demonstrated by Quabis et al. in another imaging application [20].

Fig. 2. TOSCA scan principle. The target orientation remains constant, whereas the knife edges have a regular angular distribution. The scan follows a circular path.

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A circular aperture is added to the con-scan sensor [13]. The aperture overlaps the static reticle, but rotates with the optical axis and has an eccentricity given by it, making the reticle appear static relative to the scene image. It also limits the field of view such that only one knife edge is seen at a time, hiding the knife edge ends. The resulting system is shown in Fig. 1(b) (schematics) and 1(d) (potential realization).

2.4 Reconstruction in the continuous case

We now assume a con-scan reticle system with a circular scanning aperture. We furthermore assume ideal conditions with negligible noise, ideal optics and detector, and both the aperture and the reticle to be in the image plane. We choose a coordinate system moving with the image. If the focusing optics produces an image I(r,t), the “apertured” image becomes:

I_{A} (r, t) = I (r, t) A (r), A (r) = {\begin{array}{c} 1, & ∥ r ∥ \leq r_{0} \\ 0, & ∥ r ∥ > r_{0} \end{array}

Here r is a image plane vector and A(r) is the transmission of the aperture with radius r₀. Due to the aperture, the reticle transmission can be modeled as a sequence of unit step functions:

K (r, t, i) = u (k_{i} ∙ (r - r_{i} (t))), u (x) = {\begin{array}{c} 1, & x > 0 \\ 0, & x < 0 \end{array}

Here, i denotes the knife-edge index, k _i=(cos θ_i, sin θ _i) denotes a unit vector normal to the knife-edge and oriented towards the transmitting region, r i(t) is a point on the knife-edge, and u(x) is a unit step function. The signal as the scene is scanned across the knife-edges becomes:

S (t, i) = \int \int_{r \in P} I_{A} (r, t) K (r, t, i) d r = \int \int_{r \in P} I_{A} (r, t) u (k_{i} ∙ (r - r_{i} (t))) d r

Here, P is the image plane. The first order time derivative of this signal is given by:

\frac{d}{dt} S (t, i) = \int \int_{r \in P} (\frac{d}{dt} I_{A} (r, t)) u (k_{i} ∙ (r - r_{i} (t))) d r

The first term of equation (4) is due to a scene variation, and the second part results from the knife-edge movement, as indicated in Fig. 3(a). We will here assume the scene to be time invariant and return to this assumption in the discussion. We then get:

\frac{d}{dt} S (t, i) = [k_{i} ∙ (- \frac{d}{dt} r_{i} ((t)))] \int \int_{r \in P} I_{A} (r, t) δ (k_{i} ∙ (r - r_{i} (t))) d r

Introducing Cartesian coordinates, we define the knife-edge line L_i as:

L_{i} = {r_{i} = (x_{i}, y_{i}), x_{i} \cos θ_{i} + y_{i} \sin θ_{i} = \int_{τ_{i}}^{t} V_{C, i} (t) dt}

Here, time is denoted by t, θ_i is the angle of the i ^th knife-edge normal, and τ_i is the time at which the knife-edge crosses the origin. V_C,i is defined as the instantaneous scan speed:

V_{C, i} (t) = k_{i} ∙ (- \frac{d}{dt} r_{i} (t))

Combining equations (5–7), we get:

\frac{d}{dt} S (t, i) = V_{C, i} (t) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I_{A} (x, y) δ (x \cos θ_{i} + y \sin θ_{i} - \int_{τ_{i}}^{t} V_{C, i} (τ) d τ) dx dy

This corresponds to the expression of a line scan. If the scan speed component V_C,i(t) does not change its sign during the knife-edge scan, we can define an effective scan distance T_i:

T_{i} (t) = \int_{τ_{i}}^{t} V_{C, i} (τ) d τ

Here, τ_i is the time at which the knife edge crosses the origin. The time derivative of T_i is:

\frac{d T_{i}}{d t} = V_{C, i} (t)

We now define the Radon transform of I_A(x,y) as:

P_{θ_{i}} (T) = ℜ [I_{A} (x, y)] = \int_{- \infty}^{\infty} \int_{\infty}^{\infty} I_{A} (x, y) δ (x \cos θ_{i} + y \sin θ_{i} - T) dx dy

Combining equations (8–11), we find an important relationship between the scan distance derivative of the signal and the Radon transform of the apertured image:

P_{θ_{i}} (T_{i}) = \frac{1}{d T_{i}} (\frac{d T_{i}}{d t}) P_{θ_{i}} (T_{i}) dt = \frac{d S (t, i)}{d T_{i}}

This relationship between the extracted signal and the Radon transform of the apertured field is crucial, enabling the use of tomographic reconstruction methods from parallel beam projection. The appendix describes the image reconstruction in the continuous case.

2.5 Narrow slit reticle

Physically, equation (12) is intuitive, given that the change in the signal corresponds to the signal integrated along a line at the knife-edge, and the Radon component is just such a line integral. Mathematically, the double integral in equation (5) is a rewritten line integral of the apertured image along the knife-edge. This means we can reconstruct the image scanning the scene across narrow reticle slits, the reticle transmission becoming a sequence of delta functions. Equation (2) becomes:

K (r, t, i) = δ (K_{i} ∙ (r - r_{i} (t)))

Equation (3) of the instantaneous signal is replaced by:

S (t, i) = \int \int_{r \in P} I_{A} (r, t) K (r, t, i) d r = \int \int_{r \in P} I_{A} (r, t) δ (k_{i} ∙ (r - r_{i} (t))) d r

Comparing equation (14) and the development in equations (5–12), we find that:

P_{θ_{i}} (T_{i}) = S (t, i)

2.6 Reconstruction algorithm

We finally get the following reconstruction processing scheme in the continuous case:

1. For the knife-edge configuration, determine the scan length derivative of each knife-edge scan signal. For the narrow slit configuration, use the original signal.

2. Obtain the Fourier transform of the signal obtained in 1.

3. Multiply the frequency components in 2 by the absolute value of their frequencies.

4. Take the inverse Fourier transform of the filtered frequency components found in 3.

5. “Back project” the signal obtained in 4 along the scan lines in the image space.

6. Integrate the result in 5 for all scan angles.

Fig. 3. Geometrical considerations. (a) Scene masked by circular aperture and knife-edge. The latter is defined by the line L_i and the normal unit vector k _i. Also indicated is r _i, a point along L_i. The indicated scan velocity gives a positive scan speed and an increasing signal. The terms in equation (4) are due to scene variations and the moving knife-edge. (b) When reconstructing an image using discrete samples, the frequencies of a shaded area are represented by the value of its centre point. The sampled value is therefore multiplied by the area it represents in Fourier space. The two darkest patches are represented by the origin in the given scan. The angles are in the range [0,π[, and the frequency values include both positive and negative values.

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2.7 Reconstruction in the discrete case

A practical system will have a limited number of knife-edges or narrow slits, and a limited number of sample points for each angular scan. A discrete reconstruction is therefore more appropriate for a realistic approach. We must now account for several aspects:

• The Fourier transform components of the resulting samples maps onto a polar grid, which generally do not coincide with a rectangular grid. The replacement of frequency component that falls in between the points on the rectangular grid by corresponding elements on the grid leads to a coupling to all the grid points, making this transition either inaccurate or computationally complex and time consuming.

• The sample point density in the Fourier domain is higher near the origin, but do not tend towards infinity as in the continuous case. The multiplication by the absolute value of the frequencies in the continuous case must therefore be modified at the origin, where the coefficient should equal ¼ of the frequency step length. This can be seen by considering the area of the elements surrounding each point in Fig. 3(b).

• In the continuous case, the time differentiated signal is Fourier transformed, multiplied with the absolute value of the frequency |U| and then inverse Fourier transformed. This corresponds to an aperiodic convolution. In the finite discrete case, the implementation of the fast Fourier transform (FFT) is periodic. A simple replacement of the aperiodic convolution with a periodic convolution leads to a “dishing” artefact due to an inter-period interference [17]. Zero-padding the time differentiated signal for each knife-edge scan with at least the number of original elements can remove this artefact. By making the total number of elements of each scan a power of 2, an efficient FFT algorithm can be used.

• A limited number of scan angles and low sampling density creates aliasing artefacts. The number of independent scan line angles and sample points along each scan line should therefore satisfy the Nyquist sampling criterion. The higher tangential point density close to the origin in a polar grid means the lack of angular resolution is more pronounced at high frequencies. With a too coarse sampling, typical artefacts appear as regularly distributed rays around hot spots, or lozenge patterns in more complex images. Filtering the higher frequencies reduces these aliasing artefacts. Hsieh [23] gives examples and analyses of errors present when the Nyquist criteria are not met.

We still assume mathematically perfect components (scan mechanism, optics, detector, and sampling) in the system. The process is identical to that of the continuous case until sampling of the detected signal occurs. For a knife-edge configuration, equation (4) will be replaced by:

\frac{Δ S (t_{j}, i)}{Δt} = \int \int_{r \in P} \frac{Δ I_{A} (r, t_{j})}{Δ t} u (k_{i} ∙ (r - r_{i} (t_{j}))) d r

Here, Δt is the sampling time interval, and the other increments, annotated with a Δ, correspond to the value differences between samples. The time t is subscripted with a j to indicate the sampling times, whereas the subscript i denotes the i ^th knife-edge scan. Spatial information is still continuous, as it corresponds to the physical integration.

We define discrete samples T_i,j of the scan distance T_i. The Radon transform is defined as before, but the sampled values T_i,j replace the continuous values. Equation (12) then becomes:

P_{θ_{i}} (T_{i, j}) = \frac{Δ S (t_{j}, i)}{Δ T_{i, j}}

For the narrow slit configuration we retain equation (15), but replace T_i and t by T_i,j and t_j.

We have not made any assumptions about the sampled values of T_i,j. By resampling the signals obtained in this known geometry, it is possible to find the values for constant scan distances. We may then follow the method proposed in for example [17] to do the reconstruction. This gives us the following processing scheme:

1. If necessary, resample the signal to obtain signals at equidistant scan length intervals.

2. With the knife-edge configuration, find the scan length derivative of each knife-edge scan signal. With a narrow slit configuration, use the signal obtained in 1.

3. Zero-pad the signal obtained in 2 and find the FFT of the result.

4. Create a discrete ramp filter with a length equal to the signal length in 3. The filter may be modified with a window function described in detail below.

5. Multiply the signal obtained in 4 value by value by the filter obtained in 4, and obtain the inverse FFT of the result

6. “Back project” the signal obtained in 5 along the scan lines in image space using interpolation, and sum up the result for all scan angles.

The filter applied to the signal in process steps 3–5 contains two parts:

a) The modified ramp (or RamLak) filter is given by [17,21,22]:

h (U) = {\begin{array}{c} ∣ U ∣, & ∣ U ∣ > 0 \\ \frac{Δ U}{4}, & U = 0 \end{array}

b) A windowing filter modifying the ramp filter. One alternative to equation (18) is:

h (U) = {\begin{array}{c} ∣ U ∣, & U_{C} \geq ∣ U ∣ > 0 \\ \frac{Δ U}{4}, & U = 0 \\ 0, & ∣ U ∣ > U_{C} \end{array}

This modification, termed a hard windowing filter, imposes a frequency limit on the system. The choice of frequency limit could be dictated by the matrix size, but could also be a lower limit. This can be used in for example over-sampling techniques.

A soft windowing filter can replace the hard windowing filter to yield better performance in noisy environments. A Hamming filter is proposed as a general purpose filter by Slaney and Kak [17], but the filter choice generally depends on the specific application.

3. TOSCA optimization and improvements

3.1 Reticle shapes

If a circular moving aperture is used together with a reticle containing knife-edges, the maximum number of useful samples per scan angle is roughly the ratio between the optical spot size and the moving circular aperture diameter. The distance between consecutive knife edges measured along the scan circle has a lower limit determined by the aperture radius r_Aperture and the scan circle radius R_{Scan circle}. The maximum number of non-overlapping knife-edge scans is:

N_{Max} \leq \frac{π}{\arcsin (r_{Aperture} ⁄ R_{Scan circle})}

The knife-edge orientations should be evenly distributed as all angles are assumed to be equally important. The radial sector distribution in Fig. 4(a) may seem logical, but is in fact not the best solution. The number of transparent and non-transparent fields is equal, making the number of knife-edge transitions between them even. Combined with the regular angular distribution, this means any knife-edge has a correspondent located diametrically opposite, the half of them thus being redundant. The number of independent knife-edge scans is thus:

N_{Eff, 1} = \frac{N_{Max}}{2} \leq \frac{π}{2 \arcsin (r_{Aperture} ⁄ R_{Scan circle})}

A compact alternative is to use a configuration with knife-edges defined as follows:

1. The knife-edges are angularly evenly distributed, but the first and last are parallel. The angles between the N subsequent (non-parallel) knife-edges is 2π/(N-1).

2. The knife-edges are separated such that the moving circular aperture located with its centre on the scan circle can be placed between two consecutive knife-edges.

We here use the term compact to indicate that this solution minimizes redundancy; hence the space wasted is minimized, leading to the smallest possible scan circle. Figure 4(b) shows a compact knife-edge reticle. Only one knife-edge pair represents a redundancy, but their crossing with the scan circle are separated slightly more than in the radial case, and we get:

N_{Eff, 2} \leq N_{Max} - 1

Equation (22) is only an upper bound, and not the limiting function itself. With a circular scan with constant rotational speed, the effective scan speed V_C,i varies as:

V_{C, i} = R_{Scan circle} ω \cos (ω (t - τ_{i}) + φ_{i})

Here ω is the angular scan speed, and φ_i is the angle between the scan line normal and the tangent of the scan circle where it crosses the scan line. The scan speed variation for the radial configuration is identical for all knife edges, symmetrical around each scan midpoint and approximately parabolic. The ratio between the lowest and the highest scan speed is given by:

V_{C, i, min} ⁄ V_{C, i, max} = \sqrt{1 - {(r_{Aperture} / R_{Scan circle})}^{2}}

The scan speed variation of an optimum compact knife-edge reticle is non-symmetrical and differs for each scan, the minimum ratio between the lowest and the highest scan speed being:

V_{C, i, min} / V_{C, i, max} = \sqrt{1 - {(2 r_{Aperture} / R_{Scan circle})}^{2}}

The compact reticle version thus gives almost twice the angular resolution compared to a radial reticle for a given spot size and field of view at the price of a 4 times bigger scan speed variation. This variation, however, will be small and can be compensated for in the linear interpolation look-up tables, or the sample rate can be varied.

Fig. 4. Possible reticle and detector array configurations. Dark sectors are transparent regions, light grey circles show the maximum size of the moving circular aperture. Slashed circles show the scan circle. Thick lines in (a) and (b) indicates (some of the) redundancies due to parallel knife edges. (a) Radial spoke knife-edge reticle. (b) Compact knife-edge reticle. (c) Thin slit reticle configuration. (d) Circular detector array configuration, enabling a smaller scan radius.

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We now consider the narrow slit reticle, exemplified in Fig. 4(c). As parallel slits lead to redundancy, an odd number of slits is preferred. These reticles give several benefits:

1. Only the signal from a small part of the scene is integrated, reducing the required dynamic range, enabling higher radiometric resolution and reducing photon noise and the sensitivity to temporal scene variations, compared to the knife-edge configuration. The required dynamic range scales with the maximum number of pixels seen, which is proportional to the ratio between the aperture and spot diameters for the narrow slit reticle, and the ratio between the areas of the aperture and the spot for the knife-edge reticle. The photon noise levels scales with the square root of these figures. Omitting the differentiation used with knife edge scans also reduces sensitivity to high frequency detector and background noise.

2. An odd number of radial slits can be used, eliminating redundancy while maintaining a more regular scan speed. The effective number of line scans and the speed variation are given by equations (20) and (24), respectively.

3.2 Circular detector array configuration

The reticle configurations have certain disadvantages. First, the collecting optics must be able to collect all the light passing through the moving circular aperture, either by bringing the light back to the rotational axis or by letting the detector follow (or even be) the moving aperture. The aperture then determines the detector diameter. Second, only one knife-edge or thin slit can be exposed through the aperture at a time, requiring a large scan radius. The scan circle perimeter is then given by the spot size, the sample number per angular scan and the number of scans. A small spot size requires optics with a small f-number. A short focal length is therefore desired, but the requirement of a large scan circle puts a lower limit to the focal length. Relay optics can be used, but adds weight, cost and complexity to the sensor. A small spot size also requires costly precision mechanics. These hurdles set aside, the required scan circle radius may still be so large that it becomes a system limiting factor. As an example, using a reticle with 129 scan lines, 129 samples per scan line and a 30 µm spot size gives a 79 mm scan circle radius.

Replacing the slit reticle by a circular array of radially oriented line detectors simplifies optics and reduces the required detector size, giving less noise and faster response. If each detector replacing a narrow slit has its own signal processing chain, several line detectors can scan the moving circular aperture simultaneously. The scan circle can then be reduced, enabling a shorter focal length and a simpler and smaller sensor. The scan radius is minimized when the (rectangular) detectors form a dense star shape as shown in Fig. 4(d), giving:

{R'}_{Scan circle} = n s (\frac{1}{2} + \frac{1}{2 π}) = \frac{n s (π + 1)}{2 π}

Here, s is the detector element width, n being the number of elements (of length ns). Compared to the smallest reticle based configuration, this corresponds to a scan circle radius reduction by a factor n/(π+1). With n=129 and s=30 µm, the scan radius is reduced to 2.6 mm. As the scan speed is proportional to the required frame rate and the scan radius, it can be reduced, along with the required detector bandwidth, leading to less detector and background noise. This increases the scan speed variations accordingly, but these variations can be accounted for in the re-sampling and/or filtering calculations. Another advantage is that a larger fill factor can be achieved, leading to a higher sensor responsivity.

The issues encountered are the same as for the narrow slit reticle configuration. An even number of detectors should be avoided due to redundancy. As an example, less information is available from a crossbar detector (four detectors forming a cross) than the corresponding optimum configuration using three detectors.

4. Noise considerations

This section contains a simplified and fairly generic analysis of background and detector noise to indicate how the different TOSCA-configurations compare with other imaging solutions. Conventionally, two-dimensional images can be created in several ways: A single pixel detector scanned in 2-dimensions, linear detector array scanned linearly, and the staring 2-dimensional detector array. We will in the following analysis make several assumptions to enable the comparison to highlight general trends. These assumptions are:

• An image is represented as a matrix containing n×n pixels, and it is assumed that n angular line scans are required to reconstruct the image with sufficient fidelity. The spot size and the frame rate are denoted s and F, respectively.

• The dominating noise contribution is assumed to be either photon noise from the field of view (from now on termed just photon noise) or detector noise. The maturity of material science for detectors and optics and use of low noise electronics, shielding and cooling makes this assumption valid for a large range of systems operating in the ultraviolet, visible and infrared. Both types of noise are assumed to be proportional to the square root of both the active area and bandwidth. The active area for the photon noise is considered to be the transparent region (reticle+circular aperture) or the part of a line detector visible through the aperture.

• The photon harvesting efficiency is the average proportion of the incoming photons collected by the sensor, and is listed in table 1. The detector signal is proportional to this efficiency, and both noise types are proportional to its square root.

• As the reticle based TOSCA-configuration detector does not require imaging properties, the use of a concentrator enables a smaller detector. It is assumed (somewhat arbitrarily) that the detector area can be given approximately as ¼n ² s ².

• The TOSCA processing contains an intrinsic noise reduction mechanism due to multiple sampling of each pixel with different angular scans. The total effect is a signal to noise improvement for both noise types by a factor n ^1/2.

Mathematically, the noise reduction can be described as follows: If N_A is the active area normalized by the spot size (i.e. the number of pixels covered by the active area) and 〈φ〉 is the expected sample noise level produced in each pixel, the expected sample noise level 〈N_Sample〉 in the whole active area, where the spatial noise adds incoherently, is:

N_{Sample} = N_{A}^{1 / 2} 〈 φ 〉

We denote the expected signal level from the contribution of one pixel in a sample as 〈S_Sample〉. The total signal contribution 〈S〉 from one pixel in a entire frame is the coherent linear addition from all the line scans, thus we get a total frame pixel signal to be:

〈 S 〉 = n 〈 S_{Sample} 〉

The noise on the other hand adds incoherently, giving an effective total frame pixel noise 〈N〉 to be:

〈 N 〉 = n^{1 / 2} 〈 N_{Sample} 〉

This gives an increase in the signal to noise ratio of the frame relative to that of the sample:

〈 S 〉 / 〈 N 〉 = n^{1 / 2} 〈 S_{sample} 〉 / 〈 N_{sample} 〉

We now consider the bandwidths B required for the various TOSCA configurations. The bandwidth required for the reticle based narrow slit reticle configuration is given by:

B_{Narrow line reticle} = n^{2} F \approx F \frac{2 π R_{Scan circle}}{S}

The maximum bandwidth required for the optimized circular array configuration is a function of the frame rate, scan perimeter and spot size. Using equation (20), we get:

B_{Narrow line \det ector} = F \frac{2 π {R'}_{Scan circle}}{s} = F n (π + 1)

Given the scan speed variation, bandwidth tuning during the scan can enhance performance.

Table 1 shows different factors accounting for the noise and signal levels (accumulated in a single frame) in 6 different configurations, normalized to the 2-dimensional array detector.

Table 1. General parameters governing noise performance of various detector configurations for a single image frame, normalized to the 2-dimensional array detector. A channel is here a pixel in the reconstructed image, and n denotes the array length or diameter, as well as the number of independent angular scans.

View Table

To sum up, in terms of noise performance both TOSCA reticle versions perform better than the single pixel sensor configuration and worse than the linear array configuration. Between them, the thin slit reticle version has better background noise performance than the knife-edge version. The TOSCA circular array configuration performs better than the linear array in terms of noise, while the 2-dimensional array performs even better.

5. Simulations

The algorithms used to generate the simulated detector signals as well as to reconstruct the images were implemented in Matlab® to demonstrate the concept. The source code is freely available upon request.

An original 512 by 512 pixel image “Lena”, see Fig. 5(a) was filtered spatially by a 250 pixel radius circular aperture. The resulting image was then scanned by 51 independent knife-edges with regularly distributed orientations. The reconstruction is seen in Fig. 5(b). Angular artefacts are visible as a lozenge pattern in the image, but the scene is easily recognized despite a compression by 90%. Figure 5(c) shows the reconstructed image using 501 independent knife-edge orientations. Here there are no artefacts. There is no difference between this image and the image obtained using a line scan configuration in this noise-free, ideal environment.

Fig. 5. Monochrome “Lena” (a), reconstructed using 51 (b) and 501 (c) independent scans.

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Fig. 6. Normalized detector signal from knife-edge reticle based TOSCA configuration with 51 independent scans. (a) Signal from integer frame (b) Detail from 4 angular scans.

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Fig. 7. Normalized detector signal from narrow-slit reticle based TOSCA configuration with 51 independent scans. (a) Signal from integer frame (b) Detail from 4 angular scans.

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Figure 6 shows the detector signal from a knife-edge reticle sensor scanning the mono-chrome “Lena” original with 51 independent knife-edges. Figure 7 shows the corresponding signal from a narrow 51-slit reticle sensor scanning the same image. The structural differences between scans are more easily seen here. Besides the sign change in every second scan, this signal equals the scan length derivative of the signal from the knife-edge reticle sensor.

Figure 8(a) shows a colour version of “Lena”. Figure 8(b) shows a reconstruction using 51 independent scan lines. The artefacts do not appear visually to distort the image to a significantly higher degree than was found for the monochrome image. As seen in Fig. 8(c), no artefacts are visible when using 501 line scans in the reconstruction, as it is a linear superposition of three artefact-free monochrome reconstructions. The lack of resolution for 51 scan lines appears to induce localized structural distortions. The feature induced distortions are less significant close to the feature. Localized features on an otherwise homogeneous background are therefore fairly well preserved, even with high compression.

Fig. 8. Colour “Lena” (a), reconstructed using 51 (b) and 501 (c) independent scans.

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6. Discussion

6.1 Noise performance comparison of TOSCA imagers and alternative techniques

As indicated in table 1, the single pixel TOSCA configuration performs better than a conventional single pixel configuration. The knife-edge reticles do not have any advantages compared to the thin slit reticles, except that it is possible to electronically modify the number of sampling points per scan line. There might also be a manufacturability issue with very narrow slits, especially if air gaps are required.

The circular array TOSCA configuration performs better than a conventional linear array for larger array sizes (n>16), at the cost of a four times larger maximum detector bandwidth. The latter is due to the increased scan distance compared to that of an ideal linear scan. The staring array imager has by far the lowest noise figure according to this simplified model. This noise analysis, however, only includes a simplistic model of the most fundamental errors. Available fill factor is not accounted for, and detector and background noise can also differ significantly from the above mentioned shot noise. As an example, low-frequency (“1/f”) noise is typically a significant issue with staring systems, and can typically be a much bigger problem than normal shot noise. Non-uniformity correction is also an important issue with staring systems, and the possibility to let the system look at reference light sources during parts of the scan can significantly enhance the performance of scanning systems compared to staring arrays. Reconstruction errors can of course modify these comparisons.

Other errors and noise mechanisms can modify the picture. These include errors caused by optics and optomechanics, detectors, signal conditioning, analog to digital conversion and digital effects. Being outside the scope of this text, these issues will not be pursued.

6.2 Multispectral imaging

The TOSCA imaging technique only requires a single pixel detector to create images. An interesting aspect of the system is that the image is created by temporal modulation. If the signal that comes through the reticle is sent through one or several spectral beam-splitters, it is possible to create a multispectral imager. The same can also be achieved using multispectral single-pixel detectors. Among the advantages of the TOSCA reticle configuration is that the detector only needs to collect the light, and there is therefore no need for an extremely precise detector alignment as only the fields of view are aligned and not the individual pixels. Furthermore, the signals in all the spectral bands can be sampled simultaneously and with the exact same geometry, giving the same point spread function in all spectral bands. In spectral imaging, geometrical registration between bands is crucial to avoid unphysical mixture spectra [24]. Any image blurring then creates only linear mixing between image constituents. Finally, adding extra sensor bands is relatively inexpensive.

The circular array configuration has superior noise performance compared to the corresponding reticle based system, but it is difficult to achieve multispectral detection with more than perhaps two or three detector layers (for example with the detectors stacked on top of each other), in which case the thin slit reticle configuration would be the first choice. Comparing the TOSCA system with spinning filter wheel systems is somewhat complicated. The initially high resolution and low noise gives the spinning wheel system an initial advantage. Increasing the number of bands reduces the photon harvesting efficiency reduction of the spinning wheel configuration, improving the relative TOSCA configuration performance. An increase in the number of bands might also limit the upper frame rate limit of the spinning wheel configuration.

6.3 Other issues

Introducing mechanical scan mechanisms can induce vibrations, leading to wear and potential malfunction. On the other hand, the scan mechanism can, if well designed, be used to gyroscopically stabilize and even steer the sensor. The single pixel configuration features the most complex scan mechanism. The linear array detector and the TOSCA system will have a similar complexity. 2-dimensional staring arrays require no moving parts and this, together with the good noise performance currently makes the staring array the configuration of choice for most imaging purposes. Certain aspects, however, might give an advantage to the TOSCA configurations:

• If the detector type generates too much heat in large array structures. As an example, Semiconductor material that cannot be doped to become both p- and n-type in a stable manner can only be operated in photoconduction mode. This can be the case for certain high-bandgap II–VI type and extremely low-bandgap semiconductors.

• If the radiation, such as ionizing radiation, can harm unshielded read-out electronics.

• If the detection process requires significant absorption lengths compared to detector structures or the focal depth in the image zone. This can be the case for detection with semiconductors at photon energies near or below the band gap energy, detection of very long wave infrared and THz radiation, and when using scintillation detectors.

• If the detector material supports neither embedded electronics nor flip-chip read-out.

• If feature sizes are too small for available readout electronics structures. An example of this is near-field microscopy with super-resolution using a super-lens [25–27].

A potential obstacle for the TOSCA configurations is to find sufficiently fast detector types. This is not only because of the high bandwidths required, but also the large detectors required.

7. Conclusion

This paper presents the tomographic scanning (TOSCA) imaging principle in detail. TOSCA is based on the extraction of imaging properties from a modified con-scan hot-spot tracker by means of computational tomography. Three versions of the TOSCA imaging configurations are presented. The configuration with the best noise performance consists of a simple nutating focusing optics which images a scene through an eccentrically rotating circular aperture. The light is then absorbed by a circular array of detectors. In the two other versions the circular detector array is replaced by a fixed reticle, collecting optics and a single element detector. The reticle contains transmitting and non-transmitting fields, either in the form of sectors or narrow slits. The light passing through the reticle is then collected by a detector.

A mathematical proof of the imaging capabilities of the system is provided, and the relationship between these imaging systems and the parallel ray tomography is demonstrated. Simulation results visualize the imaging capabilities of the system, and the consequences of limiting the angular resolution. It is shown that a system using narrow slits improves the number of useful samples for a given optical spot size and a given scan circle diameter, has better performance in terms of background noise and dynamics, and relaxes the requirements of scene stability compared to a system using knife edges to scan the scene. In terms of noise performance, a single element TOSCA sensor should be better than a single pixel imager using conventional scanning, and a TOSCA sensor with a circular detector array should be better than a linear array sensor using conventional scanning and the same number of scanning elements. The concurrent sampling and exact geometric match of all wavebands also makes the system an interesting candidate for multispectral imaging.

Appendix: Development of the continuous case reconstruction

We take the Fourier transform of the Radon transform of I_A(x,y) in equation (11):

D_{i} (U) = FT [P_{θ_{i}} (T)] = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} P_{θ_{i}} (T) e^{- j 2 π U T} d T

The 2-dimensional Fourier transform of I_A(x,y) can be defined as:

F (u, v) = FT (I_{A} (x, y)) = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I_{A} (x, y) e^{- j 2 π (ux + vy)} dx dy

From equation (A2), the expression in equation (A1) is proportional to Fourier components along the knife-edge normal:

D_{i} (U) = \sqrt{2 π F} (U \cos θ_{i}, U \sin θ_{i})

Scans in opposite directions as well as scans in the same direction but with opposite knife-edge orientations give equivalent information. Switching knife-edge orientations corresponds to a change in knife-edge normal angles θ_i by θ_i’=θ_i+π, and using (A3), we have:

D_{i'} (U) = D_{i} (- U)

Scans in opposite directions correspond to a time reversal of the initial signal, and means that the integral limits in the first line of equation (A1) should be switched. We then get:

D_{i'} (U) = D_{i} (- U)

The signal extracted from each knife-edge scan determines the Fourier component values of the apertured image along the knife-edge normal. These lines cross the origin in Fourier space. An infinite number of knife-edge scans maps the entire Fourier space, enabling a perfect reconstruction of the image using the inverse Fourier transform. Practical systems, though, use a limited number of angular scans and samples per scan. Calculating the frequency spectrum through interpolation, followed by an inverse Fourier transformation is either inaccurate or time-consuming. A better and the most popular reconstruction technique by far is the filtered back projection technique invented by Bracewell and Riddle [22]. A modified version is presented in the following. The standard transition between polar (U, θ_i) and Cartesian (X,Y) coordinates is found using the Jacobian, and we can rewrite (A3) to get:

D_{i} (U) U d θ_{i} d U = \sqrt{2 π F} (U \cos θ_{i}, U \sin θ_{i}) U d θ_{i} d U = \sqrt{2} π F (X, Y) d X d Y

We now define g(k) as an inverse Fourier transform of D_i(U)|U|:

g (k) = {FT}^{- 1} (D_{i} (U) ∣ U ∣) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} D_{i} (U) ∣ U ∣ e^{j 2 π Uk} d U

Using (A6) and the inverse of (A2), the reconstruction of the image is obtained as follows:

I_{A} (x, y) = {FT}^{- 1} (F (X, Y))

We then make a variable change in the last line, and use (A4) and (A7) to get:

I_{A} (x, y) = {(2 π)}^{- 3 / 2} \int_{0}^{π} \int_{0}^{\infty} D_{i} (U) {Ue}^{j 2 π U (x \cos (θ_{i}) + y \sin (θ_{i}))} d U d θ_{i}

Acknowledgements

The author is very grateful to Prof. Svein Erik Hamran and Dr. Torbjørn Skauli for all the helpful discussions and able guidance they have provided.

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Configuration	2-D det. array	Linear det. array	Single pixel det.	TOSCA knife-edge reticle	TOSCA slit reticle	TOSCA circular array
Photon harvesting efficiency	1	n^-1	n^-2	n^-1	n-1	(π+1) ^-1
Required bandwidth	1	n	n²	n²	n²	n(π+1)
Detector area	1	1	1	¼n²	¼n²	n
(Average) active focal plane area/channel	1	1	1	½n²	n	n
TOSCA noise reduction factor	1	1	1	n^-1/2	n^-1/2	n^-1/2
Detector noise/channel	1	1	1	½n	½n	n^1/2
Photon noise/channel	1	1	1	2^-1/2n	n^1/2	n^1/2
S/N factor (detector noise limited)	1	n^-1	n^-2	2n^-2	2n^-2	(π+1)^-1n^-1/2
S/N factor (photon noise limited)	1	n^-1	n^-2	2^1/2n^-2	n^-3/2	(π+1)^-1n^-1/2

Tomographic scanning imager

Abstract

1. Introduction

1.1 Background

2. Theory

2.1 Tomography

2.2 The con-scan reticle sensor

2.3 The TOSCA principle

2.4 Reconstruction in the continuous case

2.5 Narrow slit reticle

2.6 Reconstruction algorithm

2.7 Reconstruction in the discrete case

3. TOSCA optimization and improvements

3.1 Reticle shapes

3.2 Circular detector array configuration

4. Noise considerations

5. Simulations

6. Discussion

6.1 Noise performance comparison of TOSCA imagers and alternative techniques

6.2 Multispectral imaging

6.3 Other issues

7. Conclusion

Appendix: Development of the continuous case reconstruction

Acknowledgements

References and links

Cited By

Figures (8)

Tables (1)

Equations (54)

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