Coded apertures for efficient pyroelectric motion tracking

Coded apertures may be designed to modulate the visibility between source and measurement spaces such that the position of a source amongN resolution cells may be discriminated using logarithm of N measurements. We use coded apertures as reference structures in a pyroelectric motion tracking system. This sensor system is capable of detecting source motion in one of the 15 cells uniformly distributed over a 1.6m× 1.6m domain using 4 pyroelectric detectors. © 2003 Optical Society of America OCIS codes: (110.3080) Infrared Imaging Systems; (040.3060) Infrared Detectors References and links 1. A. Moini, A. Bouzerdoum, K. Eshraghian, A. Yakovleff, X. T. Guyen, A. Blanskby, R. Beare, D. Abbott and R. E. Bogner, “An insect vision-based motion detection chip,” IEEE J. Solid-State Circuits 32, 279-284, (1997). 2. J. R. Baldwin, “Cross-over field-of-view composite Fresnel lens for an infrared detection system,” Hubbell Inc., US Patent 5,442,178, (1995). 3. J. R. Baldwin, “Composite Fresnel lens having array of lens segments providing long narrow detection range,” Hubbell Inc., US Patent 5,877,499, (1999). 4. H. L. Berman, “Infrared intrusion alarm system with temperature responsive threshold level,” Optical Coating Laboratory, Inc., US Patent 4,195,234, (1980). 5. S. D. Feller, E. Cull, D. Kowalski, K. Farlow, J. Burchett, J. Adleman, C. Lin and D. J. Brady, ”Tracking and imaging humans on heterogeneous infrared sensor array for tactical applications,” SPIE Aerosense 2002. 6. W. T. Cathey, E. R. Dowski, “New Paradigm for Imaging Systems,” Appl. Opt. 41, 6080-6092 (2002). 7. D. J. Brady and Z. U. Rahman, “Integrated analysis and design of analog and digital processing in imaging systems: introduction to the feature issue,” Appl. Opt. 41, 6049-6049 (2002). 8. D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164-2166 (1999). 9. T. M. Cannon and E. E. Fenimore, “Coded Aperture Imaging Many holes make light work,” Opt. Eng. 19, 283-289 (1980). 10. G. K. Skinner, “Imaging with Coded-Aperture Masks,” Nucl. Instrum. Methods Phys. Res. A 221, 33-40 (1984). 11. A. J. Bird and M. R. Merrifield, “X-ray all-sky monitoring and transient detection using a coded sphere telescope,” Aston. Astrophys. Suppl. Ser. 117, 131-136 (1996). 12. E. E. Fenimore, “Coded aperture imaging: predicted performance of uniform redundant arrays,” Appl. Opt. 17, 3562-3569 (1978). 13. P. Potuluri, U. Gopinathan, J. R. Adleman, and D. J. Brady, “Lensless sensor system using a reference structure,” Opt. Express11, 965-974 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-965 . 14. J. Fraden, AIP handbook of modern sensors (American Institute of Physics, 1993), Chap. 6. (C) 2003 OSA 8 September 2003 / Vol. 11, No. 18 / OPTICS EXPRESS 2142 #2623 $15.00 US Received June 23, 2003; Revised August 04, 2003


Introduction
Sensors for motion tracking applications have been an active area of research and development in the recent past [1,2,3,4,5].Motion tracking systems may use image sensors [1] or infrared point detectors [2,3,4,5].Video systems are capable of advanced positioning and control but are also associated with high dataloads and computational costs.Tracking systems based on infrared motion sensors on the other hand achieve low dataloads but in general have poor spatial resolution.Design for these motion sensors has focused on shaping the sensor receiver pattern using Fresnel lens arrays [2,3] and conical mirrors [4].
In applications like motion tracking, the goal of the sensor is to extract features of the source by estimating certain source parameters.The features of interest include source position, source classification in a number of discrete states and source shape or size.The source state is a vector of source parameters.The source state estimation is done by preprocessing the source field using digital analysis.In computational imaging systems [6,7,8], the preprocessing is done using non-conventional optical elements like coded apertures [9,10,11,12] and diffractive elements.Recently, a motion tracking system is implemented using an optical element based on a Hadamard mapping [13].In previous motion sensor designs, efficiency of sensing in terms of the number of measurements performed to discriminate a given number of source states has not been a major design criterion.
In the present work, we outline an efficient sensing method and implement a motion sensor based on this method.We define the sensor efficiency as where N is the number of distinguishable source states and m is the number of measurements required to estimate the source state.We use an optical element called a reference structure between the source space and measurement space.The mapping is chosen to reveal the source state so that the source state reconstruction is not computationally intensive.This design approach could discriminate 2 m − 1 source states using m measurements.The motion sensing system we implemented could sense and localize the presence or absence of the source to one of the 15 cells as a single source moves in an area 1.6m × 1.6m using four detector measurements.

System Model
A schematic of the system model is shown in Fig. 1.Let s(r,t) describe the source as a function of position r and time t.In the present work, r is assumed to lie in a two dimensional plane.In the following discussion, source space refers to the two dimensional plane in which the source moves and measurement space refers to the discrete set of points at which measurement is done.A reference structure modulates the visibility of source space.The visibility of a point r in source space to a point r i in measurement space is defined by a visibility function v (r i , r) denoted as v i (r).The visibility function is binary valued with value 1 if r i is visible to r and 0 otherwise and it is implemented by the reference structure.The reference structure maps the source motion through the spatio-temporal grid (r,t) to a temporal signal at a point r i in measurement space The response m i (t) of the detector at the point r i to the stimulus s i (t), assuming the detector is linear and time-invariant is given by where h(t) is the impulse response of the detector.A discrete number of measurements is done in the measurement space.In this work, we implement an efficient sensing method to discriminate 2 m − 1 source states by performing m measurements.For this case the measurement space consists of m discrete measurement points.Every source point r is associated to a binary signature vector χ (r) ∈ {0, 1} m .The i th element of this vector is v i (r).The reference structure is so designed that the source space has 2 m − 1 regions henceforth referred to as a cell.A cell is the collection of all points with the same non-zero signature vector.Let χ j denote the non-zero signature vector of the points in the j th cell.The source parameter s(r,t) at a given time instant t is assumed to have a non-zero value only in one of the 2 m − 1 cells.If s(r,t) = 1 for r ∈ A j , then s(r,t) = 0 for r ∈ A \ A j .Here, A denotes the set of all points in source space, A j denotes the set of all points in the j th cell and index j ranges from 1 to 2 m − 1.The source state can be represented as a vector s.The j th element of this vector denoted by s j is 1 for the source presence in the cell A j .The measured signal m i (t) in Eq. ( 3) is transformed to an event signal mi (t) by a transformation f The index i ranges from 1 to m.At a given time instant t, one can define a binary state vector M (t) ∈ {0, 1} m , the i th element being the mi (t).If we assume the above mentioned source constraint, we can define a one-to-one mapping between the 2 m − 1 state vectors and 2 m − 1 source states.At a given time instant t, one can estimate the source state vector s as follows In the above equation, χ j denotes the signature vector of the points in the j th cell.Many encodings (mappings of space cells to unique signal signatures) are possible, however, self revealing encodings are preferred so that no expensive search is required to identify the index of a space cell from a given signal signature.For instance, a binary encoding would only  require a logarithmic search complexity, equivalent to determining a decimal number from a sequence of binary bits.Alternative assignments of equivalent complexity can be based on appropriate hash functions.The assignment of signatures to space cells can also be dictated by preferences regarding the continuity of the space that each detector monitors.One may also want to compromise and trade sensing efficiency for accuracy by utilizing additional sensor measurements as error detecting and correcting codes.The motion tracking system discussed in this paper, localizes the source motion to one of 2 m − 1 cells within a time interval δt, as the source moves through the spatio-temporal grid (r,t).The spatial resolution is determined by the size of the cell which in turn is determined by the reference structure.The temporal resolution δt is limited by the response time of the sensor used for the measurement.

System design and implementation
Based on the above model, we implemented a sensor system which can localize source motion to one of 15 cells using four detector measurements.In the coming sections, we discuss in greater detail the design aspects of such a system.

Design of Reference Structure
The reference structure implements a visibility mapping between the source space and measurement space.The measurement space consists of four points that lie in a 2D plane.A sensing element is located at each of these points.The reference structure is designed so that the source space is divided into 15 equally sized cells.The visibility mapping is as shown in Fig. 2. The reference structure consists of four two dimensional binary masks with regions opaque and transparent to source radiation.Each cell (a dotted square in Fig. 2) has two sets of points.One set of points has a zero vector as the signature vector.All other sets of points (colored area in Fig. 2) have a unique signature vector which identifies each cell.The signature vector associated with each point has four elements, each element being the visibility of the point to the measurement point, as discussed in the previous section.The map showing the different signature vectors is shown in Table 1.

System implementation
The visibility mapping of Fig. 2 is implemented using four binary masks.Each of four detectors see the source space through only one of the masks.The masks have regions transparent and opaque to heat radiation at 10-12µm.The masks are built using CAD software and automaticallly generated code from our simulator.The masks are printed in a 3D stereo-lithography printer.The transparent regions are squares of dimension 2mm × 2mm and are positioned to implement the map shown in Table 1.The distance between the detector plane and binary mask is 2cm, and between the detector plane and source plane is 200cm.Each square region in the mask maps to a region of visibility 1 (colored region in Fig. 2).The four detectors, each with a sensing area 2mm × 2mm, are positioned on the vertices of a square of side 2cm.The visibility mapping covers an area of 1.6m × 1.6m in the source plane.Each cell has an area 40cm × 40cm.The colored square within a cell is of size 20cm × 20cm.

Pyroelectric detector as sensing element
A single element pyroelectric detector is used as the sensing element in this work.Pyroelectric detectors are heat flow sensors and generate an electrical signal proportional to the change in heat flux falling on it [14].The source is described by the distribution of the flux radiated by the source φ (r,t) (given by the Stefan-Boltzman law) as it moves through a spatio-temporal grid (r,t).The response of a single element pyroelectric detector at the point r i to an impulse at a point (r,t) in spatio-temporal grid can be described as follows The impulse response of the detector is weighted by a factor f (θ ) where θ is the angle between the unit vector r−r i |r−r i | and the normal to the sensor.f (θ ) restricts the source space that can be seen by the detector thereby modulating the visibility of the source space.f (θ ) represents the angular response pattern of the detector and sets the field of view of the detector.f (θ ) for the detector used in this work is plotted in Fig. 3.The response of the sensor to a step input can be approximated by S = So 1 − e −t τe e −t τ t (7) where So is the steady state response.The time constant τ e is the electrical time constant and τ t is the thermal time constant.The frequency response of the detector has a bandpass characteristic.The upper cutoff frequency is related to the electrical time constant τ e and controls how fast the sensor can respond to an input stimulus.The lower cutoff frequency is related to thermal time constant τ t and controls the lowest frequency of the input stimulus to which the detector can respond.The impulse response g(t) is obtained by differentiating the step response in time.

Fig. 2 .
Fig. 2. Visibility map for each of the four points in the measurement space.The colored area has a visibility 1 and the other regions have a visibility 0

Table 1 .
The map showing signature vectors of different cells