Methods and systems for hierarchical de-aliasing time-of-flight (tof) systems

METHODS AND SYSTEMS FOR HIERARCHICAL DE-ALIASING TIME-OF-

FLIGHT (TOF) SYSTEMS

FIELD OF THE INVENTION

[oooi] The invention relates generally to depth imaging systems, especially time- of-flight (TOF) imaging system that acquires depth images at distances (Z) by comparing phase shift (Θ) between emitted optical energy and reflected detected optical energy. Changes in Z produce change in phase shift Θ, but eventually phase shift begins to repeat, e.g., Θ = θ + 2·π, etc. Thus, distance Z is known modulo 2^»TT^»C/2^»CJO = C/2^»f, where f is the modulation frequency. The present invention is directed to disambiguating (or dealiasing) the inherent ambiguity between detected values of phase shift Θ and distance Z, especially in the presence of noise.

BACKGROUND OF THE INVENTION

[0002] Modern time-of-f light (TOF) systems can ascertain depth distances (Z) to a target object by emitting modulated optical energy of a known phase (φ), and examining phase-shift in the optical signal reflected from the target object back to the TOF system. Exemplary such phase-type TOF systems are described in several U.S. patents received by Canesta, Inc., and now assigned to Microsoft, Inc. Such patents include US patent numbers 6,515,740 "Methods for CMOS- Compatible Three-Dimensional Imaging Sensing Using Quantum Efficiency

Modulation", 6,906,793 entitled Methods and Devices for Charge Management for Three Dimensional Sensing, 6,678,039 "Method and System to Enhance Dynamic Range Conversion Useable With CMOS Three-Dimensional Imaging", 6,587,186 "CMOS-Compatible Three-Dimensional Image Sensing Using Reduced Peak Energy ", 6,580,496 "Systems for CMOS-Compatible Three-Dimensional Image Sensing Using Quantum Efficiency Modulation".

[0003] As the present invention is used with such prior art phase-type TOF systems, it is useful at this juncture to review their operation. Fig. 1 A is based upon the above-referenced patents, e.g. the Ί 86 patent, and depicts an exemplary phase-type TOF system.

[0004] In Fig. 1A, exemplary phase-shift TOF depth imaging system 100 may be fabricated on an IC 1 10 that includes a two-dimensional array 130 of pixel detectors 140, which pixel detectors may be single-ended or differential in operation. Preferably each of the pixel detectors 140 has dedicated circuitry 150 for processing detection charge output by the associated detector. IC 1 10 preferably also includes a microprocessor or microcontroller unit 160, memory 170 (which preferably includes random access memory or RAM and read-only memory or ROM), a high speed distributable clock 180, and various computing and input/output (I/O) circuitry 190. Among other functions, controller unit 160 may perform distance to object and object velocity calculations.

[0005] Under control of microprocessor 160, optical energy source 120 is periodically energized by an exciter 1 15, and emits modulated optical energy toward an object target 20. Emitter 120 preferably is at least one LED or laser diode(s) emitting low power (e.g., perhaps 1 W) periodic waveform, producing optical energy emissions of known frequency (perhaps a few dozen MHz) for a time period known as the shutter time (perhaps 10 ms). Typically emitter 120 operates in the near IR, with a wavelength of perhaps 800 nm. A lens 125 is commonly used to focus the emitted optical energy.

[0006] Some of the emitted optical energy (denoted S_out) will be reflected (denoted Sin) off the surface of target object 20. This reflected optical energy S,_n will pass through an aperture field stop and lens, collectively 135, and will fall upon two- dimensional array 130 of pixel or photodetectors 140. When reflected optical energy S_in impinges upon photodetectors 140 in array 130, photons within the photodetectors are released, and converted into tiny amounts of detection current. For ease of explanation, incoming optical energy may be modeled as S,_n =

A^»cos(uM + Θ), where A is a brightness or intensity coefficient, uM represents the periodic modulation frequency, and Θ is phase shift. As distance Z changes, phase shift Θ changes, and Figs. 1 B and 1 C depict a phase shift Θ between emitted and detected signals. The phase shift Θ data can be processed to yield desired Z depth information. Within array 130, pixel detection current can be integrated to accumulate a meaningful detection signal, used to form a depth image. In this fashion, TOF system 100 can capture and provide Z depth information at each pixel detector 140 in sensor array 130 for each frame of acquired data.

[0007] Signal detection within phase-type TOF systems such as system 100 is described more fully later herein with respect to Fig. 2B, but in brief, pixel detection infornnation is captured at at least two discrete phases, preferably 0° and 90°, and is processed to yield Z data.

[0008] System 100 yields a phase shift Θ at distance Z due to time-of-flight given by:

θ = 2·ω·Ζ/0 = 2-(2-TT-f)-Z/C (1 )

[0009] where C is the speed of light, 300,000 Km/sec. From equation (1 ) above it follows that distance Z is given by:

Z = θ ·0/2·ω = θ ·0/(2·2·ί·π) (2) And when θ = 2·π, the aliasing interval range associated with modulation frequency f is given as:

ZAIR = C/(2-f) (3)

[ooio] In practice, changes in Z produce change in phase shift Θ but eventually the phase shift begins to repeat, e.g., Θ = θ + 2·π, etc. Thus, distance Z is known modulo 2·π·0/2·ω) = C/2^»f, where f is the modulation frequency. Thus there can be inherent ambiguity between detected values of phase shift Θ and distance Z, and multi-frequency methods are employed to disambiguate or dealias the phase shift data. Thus, if system 100 reports a distance Zi, in reality the actual distance may be any of Z_N = Zi + N^»C/2f, where N is an integer. The nature of this ambiguity may be better understood with reference to Figs. 1 D and 1 E.

[ooii] Fig. 1 D is a mapping of detected phase Θ versus distance Z for system 100. Assume that system 100 determines a phase angle θ' for target object 20, where this phase information was acquired with a modulation frequency fi of say 50 MHz. As shown by Fig. 1 D, there are several distances, e.g.,∑i, z₂, z₄, z₅, etc. that could be represented by this particular phase angle ... but which is the correct distance? In Fig. 1 D, Z_AIRI represents the Z distance aliasing interval range associated with z data acquired at frequency fi, and is the distance from zi to z₂, or z₂ to z₄, or z₄ to z₅, etc. These various zi, z₂, z₄, z₅, distances are ambiguous and require disambiguation or dealiasing to identify the correct distance value. The terms disambiguate and dealias, or disambiguation and dealiasing, may be used interchangeably herein.

[0012] It is desired to dealias the z data by increasing magnitude of the aliasing interval range Z_AIRI - One prior art approach does this by increasing the ratio C/2f, which is to say, by decreasing the modulation frequency f, see equation (3). Fig. 1 D also shows, in bold line, phase data acquired for a lower modulation frequency f₂. In Fig. 1 D, f₂ is perhaps 20 MHz, in that the slope d0 /dz for the f₂ waveform is less than about half the slope for the fi waveform, where the slope d0 /dz is proportional to modulation frequency f_m. Fig. 1 E is a polar representation in which a vector, depicted as a line rotating counter-clockwise, rotates with velocity ω = d0 /dt = 2 πί. In prior art system 100, data is captured from pixel detectors at at least two discrete phases, e.g., 0° and 180°.

[0013] Thus in Fig. 1 D, when the lower modulation frequency f₂ is employed, the candidate distance values represented by phase θ' are z₃, z₆, etc. As seen in Fig. 1 D, the aliasing interval range Z_AIR₂ has advantageously increased from a short range Z_AIRI (associated with faster modulation frequency fi) to a greater range ZAIR2- The ratio of the aliasing interval range increase will be the ratio f₂/f i . But acquiring phase data with lower modulation frequency f₂ yields a Z value with less precision or resolution than if acquired with higher modulation frequency fi . This imprecision occurs because the slope of the curve for frequency f₂ is about half the slope for modulation frequency fi . Thus errors in the measurement of phase acquired at f₂ translate to greater errors in Z than errors in phase acquired at fi . For the same signal/noise ratio, errors in phases acquired at f1 and at f2 will be the same, but the corresponding uncertainty errors in Z use phase acquired at the lower f₂ modulation frequency will be about twice as large for the representation of Fig. 1 D. Thus, all things being equal, lowering the modulation frequency undesirably results in lower resolution (greater uncertainty) in accurately determining Z.

[0014] Thus while increasing the aliasing range interval is desired, doing so by decreasing the modulation frequency f is not desirable. This modulation frequency decrease approach to dealiasing is wasteful since lower modulation frequency means lower pixel sensor 140 accuracy per watt of illumination power from emitter 120 (see Fig. 1A). For example, a reduction of modulation frequency by a factor of 2.5, say from f = 50 MHz to f = 20 MHz, will advantageously increase the aliasing interval by the same factor, e.g., from 3 m to 7.5 m, but the penalty is a substantial (2.5)·(2.5) = 6.25X increase in operating power to achieve similar uncertainty performance, assuming effects of ambient sunlight can be ignored. By way of further example, if modulation frequencies of 50 MHz and 10 MHz were used, the dealiasing range would increase from 3 m to 30 m, but at a 25X increase in operating power for the same level of uncertainty. Thus, in practice dealiasing a TOF system simply by lowering the modulation frequency is accompanied by a very substantial performance penalty.

[0015] Ideally, dealiasing a TOF system would achieve the unambiguous range for a low modulation frequency, while providing the distance resolution for a high modulation frequency. But dealiasing in the presence of noise can be

challenging, especially for high modulation frequencies, or for large unambiguous ranges.

[0016] What is needed for a phase-type TOF system is a method of dealiasing or disambiguating Z data results, especially in the presence of noise. Preferably dealiasing should function well in the presence of noise, even with high

modulation frequencies or for large unambiguous ranges.

[0017] The present invention provides a method and system for dealiasing in phase-type TOF systems.

SUMMARY OF THE PRESENT INVENTION

[0018] Embodiments of the present invention hierarchically dealias the depth distance Z acquired by a phase-type time of flight (TOF) system. Depth data is acquired by the TOF system using preferably at least three modulation

frequencies f₁,f₂, - ,f_m separately, wherein if said TOF system were operated solely at one of said modulation frequencies, aliasing interval ranges of

Ζ_1;Ζ₂, ... , Z_m would result. Next, n intermediate frequencies f_DE1,f_DE2, - _OEU are generated sorted by order of f_DE1 < f_DE2 < ... < f_DEn and their corresponding phases are computed from the data acquired separately using three or more modulation frequencies f₁,f₂, - , f_m- Finally there occurs hierarchically dealiasing of the data acquired for the three or more modulation frequencies, separately, using the generated intermediate frequencies. Hierarchical dealiasing may be carried out one step at a time, if desired.

[0019] The result is that the TOF system provides an effective aliasing interval range Z_D > Z_k for k = 1 ... n as if said TOF system operated at a very low modulation frequency f_D, while simultaneously providing depth resolution certainty as if said TOF system operated at a high modulation frequency f_E. Preferably high modulation frequency f_E is a function of all modulation frequencies fi> f2> - > fm > and the function may be, without limitation, an arithmetic mean of fi, f₂, - , fm, or a weighted average of fc,f₂, ... , f_m.

[0020] In one embodiment, the selection of the intermediate frequencies †DEI_>†DE2> - > fDEn can be based on a ratio between two not necessarily

consecutive but shown here for simplicity frequencies ¹⁾, which is limited by a <img file="WO2012012607A2_D0001.tif" />

ratio determined by an uncertainty requirement.

[0021] In another embodiment, hierarchical dealiasing may be carried out in a stepped sequence that includes first de-aliasing phase data of f_DE1 using phase of fjj , and then dealiasing phase of f_DE2 using phase of f_DE1. If desired, at each step, the phase of f_DE(k+i₎ is dealiased using the phase of f_DEk until a last step, at which phase f_E is dealiased using the phase of f_DEn to yield an unwrapped phase of f_E . In one embodiment, the unwrapped phase of f_E is computable by using at a last step of hierarchical de-aliasing, the dealiasing phase of each modulation frequency f₁, f₂, - , f_m using f_DEn to get unwrapped phase of each modulation frequency, and then computing unwrapped phase of f_E using a function between f_E and /i, /₂, ... , f_m .

[0022] In one embodiment, at each step of dealiasing the phase of f_DE(_{k+ 1}) using the phase of f_DEk , a corrected phase cycle of f_DEk within the [0, 2π] cycle

°f fDE _k-i f^{rom a} previous de-aliasing step is used to find a correct phase cycle of oi_?(fe+i) within a cycle of f_DEk . This then enables the correct cycle of /_D£(fe+1) within a total unambiguous range Z_D, or the cycle of f_D, to be found.

[0023] The present invention recognizes that the difficulty of successfully dealiasing data increases as the ratio f_E/fo increases because noise in Θ_Ε is amplified by this ratio when 0_Eused to computer 0_Ds- Consequently for high modulation frequency, which corresponds to a large effective frequency , or for a very large unambiguous range, which corresponds to small difference frequency fo, such amplified noise would produce erroneous results in estimating a dealiasing interval K. The result could cause aliasing the middle of an

unambiguous range.

[0024] Embodiments of the present invention may be implemented in hardware and/or software, which hardware and/or software may be part of the TOF system with which the invention is practiced. Hierarchical dealiasing according to the embodiments of the present invention is preferably relatively low loss.

[0025] Other features and advantages of the invention will appear from the following description in which the preferred embodiments have been set forth in detail, in conjunction with their accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0026] FIG. 1 A is a block diagram depicting a phase-type time-of-flight three- dimensional imaging system, according to the prior art;

[0027] FIGS. 1 B and 1 C depict emitted and reflected optical energy waveforms associated with the imaging system of Fig. 1 A, according to the prior art;

[0028] FIG. 1 D depicts acquired phase-vs-distance Z for two modulation frequencies and demonstrates aliasing intervals, and distance ambiguity, according to the prior art;

[0029] FIG. 1 E is a polar representation of acquired phase data as a function of modulation frequency, according to the prior art;

[0030] FIG. 2A depicts a phase-type time-of-flight three-dimensional imaging system with dealiasing, according to embodiments of the 12/459,033 application;

[0031] FIG. 2B depicts details of exemplary phase capture at 0° and 180°, 90° and

270°, according to embodiments of the present invention;

[0032] FIG 3 acquired phase-vs-distance Z for two close modulation frequencies, and for virtual frequencies fo, , and fos, and resultant large aliasing interval Z_AIR, according to embodiments of the 12/459,033 application;

[0033] FIG. 4A depicts a sequence of eight different captures from one pixel detector, as well as resultant phase shift and frame construction, according to an embodiment of the 12/459,033 application;

[0034] FIG. 4B depicts a sequence in which four adjacent pixel detectors acquire four phases in a single capture, as well as resultant phase shift and frame construction, according to an embodiment of the 12/459,033 application;

[0035] FIG. 4C depicts a sequence in which captures for each phase are offset- cancelled using data from the same pixel detector, as well as resultant phase shift and frame construction, according to an embodiment of the 12/459,033

application; [0036] FIG. 4D depicts a sequence for which direct computation of ΘΕ is made, as well as depicting resultant phase shift and frame construction, according to an embodiment of the 12/459,033 application;

[0037] FIGS. 5A and 5B depict target object locations, real and phantom, determined using two modulation frequencies, according to a least common multiple dealiasing embodiment of 12/459,033 application._.

[0038] FIG. 6 depicts a phase-type time-of-flight three-dimensional imaging system with hierarchical dealiasing, according to embodiments of the present invention; and

[0039] FIG. 7A-7C depict phase-vs-distance Z for various steps in hierarchical dealiasing, according to embodiments of the present invention.

DESCRIPTION OF THE PRESENT INVENTION

[0040] Embodiments of the present invention implement methods and systems to hierarchically dealias time-of-flight (TOF) phase using at least three frequencies, and will be described with reference to Figs. 6-7D. Conventional de-aliasing relies upon use of two frequencies. However using three or more frequencies can advantageously substantially increase the unambiguous distance of the TOF system, without significantly amplifying noise. Aspects of the present invention can address hierarchical dealiasing, including aspects of probability distribution of phase, and elliptical correction.

[0041] However a greater appreciation and understanding of hierarchical three- frequency dealiasing is obtained by first reviewing two-frequency dealiasing according to the 12/459,033 application. Dealiasing according to the 12/459,033 application is lossless, and could advantageously provides a relatively large aliasing interval range commensurate with a low modulation frequency. This was achieved while also providing high precision certainty with respect to a given Z value, commensurate with a modulation frequency close to the highest modulation frequency f_m. A description of the 12/459,033 application will first be made with reference to Figs. 2A-5B.

[0042] Fig. 2A depicts a phase-type TOF system 200 similar to that described with respect to Figs. 1 A-1 C, except that additional components 210 and software 220 are included to implement embodiments of the 12/459,033 application. As such, software 220 and 210 may be considered as a dealiasing subsystem for TOF system 200. Output DATA' may include information provided as DATA by TOF system 100 in Fig. 1 A. A TOF system such as system 100 (Fig. 1A) or 200 (Fig. 2A) normally is operable at some maximum modulation frequency f_m, perhaps 100 MHz. This is not to say that the system cannot be operated at a modulation frequency greater than f_m, but it is realized that at such higher modulation frequencies, system performance ultimately degrades. Thus, it is understood that f_m denotes the highest modulation frequency at which the system is normally operable, and not the highest modulation frequency at which it can be operated.

[0043] Before describing dealiasing operation of system 200, it is useful to briefly describe multiphase detection with reference to Fig. 2B, which shows two of the many photodetectors (PD) in array 130, namely photodetectors 140-1 , and 140-N, as well as some of their associated electronics, namely 150-1 , 150-N. This particular embodiment employs quantum efficiency (QE) modulated differential photodetectors or pixels 140, whose pixel detection information is captured at at least two discrete phases 0° and 90°, and more preferably four discrete phases 0° and 180°, and 90° and 270°. These discrete phase operations of the pixel detectors are not to be confused with the phase shift data Θ that is sought to be detected. These discrete phases represent shift between modulator 1 15 and optical energy emitter 120, whereas the phase shift data Θ that is sought to be detected is shift between emitted optical energy S_out from emitter 120, and pixel detectors 140 in array 130 (see Fig. 2A).

[0044] The detection phase data that is captured at the discrete phases is denoted herein as captures C° and C¹⁸⁰, C⁹⁰ and C²⁷⁰ and is processed to implement dealiasing according to the 12/459,033 application. Acquisition using four phases is preferred so as to remove so-called fixed pattern offset. The C° acquisition yields data but may include an error offset from zero, e.g., the value C° may not be zero when there is no signal to be detected. By the same token, the C¹⁸⁰ acquisition should have the same, but inverted, data and will have the same fixed pattern offset. Advantageously by subtracting (C°-C¹⁸⁰) and preferably also subtracting (C⁹⁰-C²⁷⁰), phase and Z data information is preserved but the fixed pattern offset is canceled out. However embodiments of the 12/459,033 application may be used to dealias multiphase TOF data that is obtained from single-ended phase detectors, as well as from detection systems that do not employ QE modulation detection.

[0045] Phase angle Θ can be computed from captures C°, C⁹⁰, C¹⁸⁰, C²⁷⁰, as follows:

n o/ ^&lt;90 ^270 ^0 ^18( . . . 6/ = a tan (C - C , C - C ) (4) where atan2(X,Y) is the trigonometric function corresponding to atan(Y/X)

[0046] The configuration and operation of what is shown in Fig. 2B is similar to what was described with respect to a fixed phase delay embodiment (Fig. 10) in earlier-referenced U.S. patent nos. 6,580,496 and 7,906,793. In Fig. 2B, detection-generated photocurrent from each QE-modulated differential pixel detector, e.g., 140-1 , is differentially detected (DIF. DETECT) and differentially amplified (AMP) to yield signals B^»cos(0), B^»sin(0), where B is a brightness coefficient, A fixed discrete 0° or 90° phase shift delay (DELAY), and more preferably a fixed 0° or 180 or 90° or 270° phase shift is switchably insertable responsive to a phase select control signal (PHASE SELECT) that can be commanded by clock unit 180'. Phase data, e.g., C° and C¹⁸⁰, C⁹⁰ and C²⁷⁰, is acquired or captured from the pixel detectors at these discrete phases. A more detailed description of such phase detection may be found in the above- referenced patents, but without limitation, the fixed discrete phase shifts may be inserted by clock unit 180', whose signal is used to drive detector array 130, or may be introduced by modulator 1 15; see Fig. 2B.

[0047] According to the 12/459,033 application, preferably close together modulation frequencies fi and h, each typically less than the system maximum modulation frequency (f_m) are combined. The resultant phase data acquired from each, denoted θ i and Θ ₂, is used to provide two goals: (1 ) obtaining the equivalent of a high modulation frequency measurement from system 200 that yields a low level of Z resolution uncertainty, and (2) obtaining the equivalent of a low modulation frequency measurement from system 200 that provides a long aliasing interval Z range. In this fashion the approximate Z range is determined from the long aliasing interval, while more precise determination of the Z value is determined from the high modulation frequency measurement.

[0048] Referring back to Fig. 2A, software 220 when executed by processor 160 can alter normal operation of clock circuit 180' by virtue of components 210. System 200 is thus caused to operate using at least first and second frequencies fi , f2, where fi &gt;f2, fi is close to f_m, preferably within about 40% or less of f_m, and f2 is within about 35% of fi and preferably closer. Thus fi may be said to lie in a range of about 60% to 100% of the frequency of f_m such that if f_m were say 100 MHz, then fi would have a frequency of about 60 MHz to as high as 100 MHz. Similarly the range of f₂ is about 65% to about 99% the frequency such that if fi was say 90 MHz, then f₂ would have a frequency of about 58.5 MHz to about 89.1 MHz.

[0049] According to the 12/459,033 application, using modulation frequencies fi and f₂ cause system 200 to behave with respect to dealiasing interval as though phase data were collected while system 200 was being operated at a very slow modulation frequency h that preferably is proportional to (fi - f₂). For example, assume that operation of system 200 at modulation frequency fi provides an aliasing interval range ZAIRI and that operation of system 200 at modulation frequency f₂ provides an aliasing interval range Z_MR2,. Embodiments of the

12/459,033 application process data acquired at modulation frequency fi and at modulation frequency f₂ to provide an effective aliasing interval range Z_AIRD &gt; Z_AIR₂ ZAIRI■

loose] Further, such use of preferably close together modulation frequencies fi and f₂ advantageously produced good resolution precision of Z measurements as though system 200 were collecting phase data operating with a high effective modulation frequency close to f_m, perhaps (fi + f₂)/2 or other combination of fi and f₂. It is noted that the frequencies (fi - f₂), (fi + f₂)/2 and other combinations of these modulation frequencies are really mathematical constructs or virtual frequencies, and system 200 does not physically operate at those virtual frequencies. Of course it is understood that more than two modulation

frequencies fi , f₂ may be used, which different modulation frequencies preferably are close in frequency to each other and to f_m. Using multiple modulation frequencies also results in Z resolution certainty or precision better than would be achieved if the TOF system processed data acquired solely while operating at modulation frequency fi and disregarded data acquired at modulation frequency f₂, or operated at modulation frequency f₂ and disregarded data acquired at modulation frequency fi . [0051] As will be appreciated from what is shown in Fig. 3, in contrast to prior art dealiasing wherein f_m is decreased with loss of system efficiency, embodiments of the 12/459,033 application operated system 200 at high frequency, preferably close to f_m, which maintains high operating system efficiency. Further, because modulation frequencies fi and h preferably are close to each other and to f_m, phase information captured by pixel detectors 140 at one modulation frequency may be shared with data frames captured at the other modulation frequency (or frequencies, if more than two modulation frequencies are employed).

[0052] Fig. 3 is a plot of phase versus distance Z, according to the 12/459,033 application. A waveform is shown for phase data acquired using first modulation frequency fi , which data is denoted θι . For purposes of explanation, assume fi is about 50 MHz. Fig. 3 also shows a plot of phase data acquired using a slightly lower second modulation frequency h, data for which is denoted Θ2. For purposes of explanation, assume h is about 31 MHz, and according its slope is less than that of the fi waveform, and its period (or aliasing interval range) is longer. For these exemplary values of fi and h, a typical value of f_m for system 200 might be 70 MHz. The h waveform is drawn with a heavier line than the fi waveform to promote clarity in the figure. As was the case with the plot described in Fig. 1 D, at phase multiples of 2π the data folds-over or wraps around. The wrap-around somewhat complicates the calculations of ΘΕ and 0DS as noted later herein.

[0053] As noted, one aspect or goal of the 12/459,033 application was to provide a large aliasing interval range Z_AIR by making it appear as though system 200 acquire phased phase data using a relatively low modulation frequency. The following description will demonstrate that defining a difference frequency h that preferably is a function of fi and h and defining a phase difference Θ D = (Θ1- Θ 2) will achieve the goal of providing a low modulation frequency suitable for a large aliasing interval range. Without limitation, an exemplary function for f_D may be a difference function (a-f b-f2), where a and b may be weighting factors.

[0054] Mathematically, phase delay Θ may be expressed in terms of absolute target distance Z as:

Θ = 2π(^ +Ν) (5) [0055] Differentiating equation (5) yields: ^{Se =}— - (6)

loose] Therefore the absolute (dealiased) distance Z_ABS is given by:

^{MS ~} π δ ⁽⁷⁾

[0057] In this instance, the differential in equation (7) can be replaced with small differences without loss of precision, to yield:

^ C ΑΘ

Z = (8)

4π Af '

[0058] Equation (8) shows that Z can be determined from Δ Θ, which is 0D or (or Θ ₁- Θ ₂) and from Δ1 Note that this is the same equation one would use to

determine Z from a modulation frequency Δί = f_D = fi-f2 and phase ΔΘ = 0D = Θ1-Θ2. Thus with knowledge of ^ ,f₂, Θ₁, 0₂, one can compute a measurement for distance Z that is akin to a calculation for Z where the actual modulation frequency was physically (f ₁ -f ₂) - This represents the combination of modulation frequencies fi and f₂ that will yield a large aliasing interval range, as though system 200 were acquiring phase data while operating at a very low modulation frequency. Of course many other combinations of fi and f₂ could be used to achieve this goal, and one could employ more than two modulation frequencies, e.g., fi , f₂, h,■■■■ Preferably the highest of these modulation frequencies fi would be close to the system highest maximum modulation frequency f_m and at least within 40% of f_m. Preferably at least the closest together of the various modulation frequencies would be within about ±35% of each other, and preferably closer together than that.

[0059] Preferably maximum modulation frequency f_m is close to the optimal operating point for TOF system 200. This means that f_m is near a maximum of operating efficiency, and therefore curve of system operating efficiency is relatively flat in the vicinity of f_m. Generally this maximum is quite flat and therefore operating frequencies fi and f₂ will also be close to the optimal operating conditions for system 200.

[0060] Design consideration in implementing a high effective (e.g., virtual) modulation frequency commensurate with a desired low level of Z resolution uncertainty will now be presented, according to the 12/459,033 application. Phase data obtained from system 200 (see Fig. 2A) operating at modulation frequencies fi and f₂ are combined to yield an effective (virtual) frequency measurement at frequency f_E. Rather than use data obtained from system 200 operating at modulation frequency fi alone, or at modulation frequency f₂ alone, embodiments of the 12/459,033 application advantageously combines data acquired at both modulation frequencies to further lower uncertainty in the Z data. (It is understood that if additional modulation frequencies are used, e.g., f₃, f , ... phase data acquired while system 200 was operating at these modulation frequencies would also preferably be used).

[0061] Combining phase data from all modulation frequencies used (e.g., fi and f₂ in the example at hand) averages noise in the phase data measurements, and advantageously results in lower noise than would be obtained using data acquired solely from fi or f₂ alone. Furthermore, because modulation frequencies fi and f₂ are preferably relatively close to maximum modulation frequency f_m, each measurement is obtained with relatively high precision and lowered noise. In general for close together frequencies fi and f₂, system 200 performance will be somewhat similar. Advantageously, uncertainty obtained after combining data from frequency fi and f₂ will be about 0.7 times the uncertainty when phase data acquired using modulation frequency fi or modulation frequency f₂ was used alone_.

[0062] Thus this second aspect of the 12/459,033 application relates to combining phase data acquired by system 200 operating at different (but preferably close to each other) modulation frequencies so as to emulate system operation at a high modulation frequency that yields a low Z resolution uncertainty. While some specific methods of combining data will now be described, it is understood that many other combinations could also be used.

[0063] With respect to this second aspect or goal, one approach is to combine that raw data readings from pixel detectors 140 in array 130 (see Figs. 2A and 2B) from captures (C¹o, Ci⁹⁰, Ci¹⁸⁰, Ci²⁷⁰) obtained at modulation frequency fi and add this raw data to data readings (C₂°, C₂°, C₂¹⁸⁰, C₂¹⁸⁰) obtained at a frequency f₂ to produce (Ci⁰ +C₂°, Ci⁹⁰ +C₂°, Ci ¹⁸⁰ +C₂¹⁸⁰, Ci²⁷⁰ +C₂¹⁸⁰). This exemplary combination is equivalent to adding the respective phase vectors together in a polar representation and obtaining the phase angle for the sum vector. [0064] Consider now a polar representation of the phase vector for a target object 20 at a given distance Z, acquired by system 200 operating at modulation frequency fi or at modulation frequency f₂. (Fig. 1 E may be regarded as a polar representation for a single phase vector.) The phase vectors may be represented in polar form as Vi = (p, θι) and V₂ = (p, θ₂) for captures at modulation frequencies fi , f₂ respectively. For ease of explanation it will be assumed that the small change in modulation frequency does not affect modulation contrast, and hence both Vi and V₂ have the same magnitude p. (The term "modulation contrast" denotes a measure of collection efficiency within pixel detectors 140, e.g., how well incoming photon energy from signal S_in is captured and converted into a detection signal. A high modulation contrast is desirable.)

[0065] Adding phase vectors Vi = (p, θι) and V₂ = (p, θ₂) yields:

V! + V₂ = (2p. sin((0₁-0₂)/2),(0₁+ θ₂)/2) (9)

[0066] Thus if (θι-θ₂) is not a multiple of π, the phase of \Λ +V₂ that would make the vector null will be the same phase (θι+ θ₂)/2 as the phase vector for target object 20 measured at modulation frequency (fi + f₂)/2. In practice, however, modulation contrast varies with frequency, which complicates implementation of this addition method. A further complication is variation in modulation contrast ratio with temperature, which makes the mathematics somewhat unwieldy.

[0067] In an alternate embodiment, actual phase angles from the two

measurements θι and 0₂ at modulation frequencies fi and f₂ are themselves combined to produce an effective phase angle Θ_Ε. Typically the average

(θι + 0₂)/2 of the phase angles is used, although other combinations of the actual phase angles are possible.

[0068] As noted above, values for ΘΕ and 0D may be determined in a variety of way. Regardless of the specific method used to obtain these values, preferably 0_E and 0D are both used to determine distance Z, as shown by Fig. 3. Fig. 3 depicts phase angles θι, θ₂, 0D = 6 0₂ and Θ_Ε = (θι+θ₂)/2 associated, respectively, with frequencies fi , f₂, h = (fi-f2&gt;, and†E = (fi +f₂)/2. It is noted that θι and 0₂ are normalized so as to be 0 for Z= 0. Fig. 3 also depicts 0DS_, which may be derived from 0_D as 0DS = 0D* i - Angle 0DS corresponds to the same frequency as 0_E and thus has the same slope as 0_E as shown in Fig. 3. However because angle 0_Ds is mathematically derived from 0D , it advantageously has the same large aliasing interval as 0D, denoted Z_AIR.

[0069] Note that both 0_D and Θ_Ε are adjusted to take account the foldover or wrap around repetitions of θι and θ₂. For example, before the end of the first aliasing interval of Θ_Ε, θι wraps around. Simply computing Θ_Ε = (θι+θ₂)/2 would not suffice because Θ_Ε would decrease by π when θι wraps around. However when θι &lt; θ₂ , it is known that Θ1 must have wrapped around, and hence π must be added to Θ_Ε . Such trigonometric corrections are well known in the art and are assumed to have been performed on the data.

[0070] Having thus appropriately trigonometrically corrected 0_E and 0D, since 0DS and 0_E have the same slope and differ only by their aliasing interval, in the absence of noise it follows that 0_Ds = ΘΕ ⁺ 2π. K is an integer that represents the index of the aliasing interval of 0_E. In the presence of noise, 0DS≡ 0E +

Κ2π. Finding the correct aliasing interval involves selecting K so as to minimize the absolute value of (0_Ds - ΘΕ + 2π). Θ_Ε + Κ2π then represents an accurate measure of Z, but with unknown aliasing interval. The expression Θ_Ε +

Κ2π represents the dealiased value of Θ_Ε. It is of course assumed that the target object is at a distance less than the aliasing interval for f_D. It is understood that other equivalent mathematical approaches to determine suitable values for K may also be employed.

[0071] It is important to select an appropriate difference frequency fo. If fp is too big, the corresponding dealiasing interval for f_D may be too small. Conversely, if f_D is too small, then resolution certainty in Z measurements at f_D can become too large. Thus, difference frequency fo should be as small as possible, subject to the noise constraints affecting resolution uncertainty. The aliasing interval index K should be determined with high certainty to ensure the proper dealiasing interval is selected.

[0072] If it assumed that f_D « , then error(0_Ds) » error(0_E) and one can generally neglect error from Θ_Ε. Let K_s = ((0_Ds - θ_Ε)/2π. Then K is the closest integer to K_s. To find the correct K, the error on K_s must be «0.5, which means the error on Q_DS /2n = Q_{D E}/h/2n must also be substantially less than 0.5. [0073] The following section will now describe exemplary approaches to

identifying good capture sequences for modulation frequencies fi and h, according to the 12/459,033 application. Many different capture sequences may be used to generate dealiased frames. A frame represents a complete Z image acquired by sensor array 130, which is to say that each pixel detector 140 in the array is associated with a corresponding Z value from target object 20. Assume first that pixel detectors 140 in system 200 can only achieve one capture at a time. Under this assumption, some exemplary sequences are as follows.

[0074] A first case is depicted in Fig. 4A, where it is assumed that the same pixel detector is sequenced though eight different captures to acquire the desired phase shift data θι , θ₂. Thus as shown in Fig. 4A, with system 200 operating at modulation frequency fi a sequence of captures Ci°, Ci⁹⁰, Ci ¹⁸⁰, Ci²⁷⁰ is acquired over time. Next, with system 200 operating at modulation frequency f₂, a sequence of captures C2⁰, C2⁹⁰, C2¹⁸⁰, C2²⁷⁰ is acquired, after which the system is returned to operating at modulation frequency fi and a sequence of captures Ci°, C1⁹⁰, Ci¹⁸⁰, C1²⁷⁰ is made, and so on. Thus in Fig. 4A (as well as in Figs. 4B-4D), the phase information displayed beneath the relevant sequence of captures is the phase data acquired by system 200 operating at the associated modulation frequency. Thus it is understood that subscript 1 denotes captures associated with modulation frequency fi and subscript 2 denotes captures associated with modulation frequency f₂. The frame information displayed beneath the phase information shows how frames are preferably constructed, according to the sequence shown. Once Θ1 and Θ2 have been determined, θ E can be calculated from (θι+θ₂)/2.

[0075] Fig. 4B depicts another sequencing example, in which four adjacent pixel detectors, shown as a block, in the pixel array are used to acquire all four phases in a single capture. Whereas the eight capture sequence of Fig. 4A is susceptible to motion blur due to the length of time to acquire all eight captures, the sequence shown in Fig. 4B should inherently exhibit lower motion blur. But in Fig. 4B, the C°-C¹⁸⁰ offset cancellation is done with data from different pixels and performance can suffer in that offsets are not fully cancelled. As before, θ E is calculated from (θι+θ₂)/2. [0076] Fig. 4C depicts yet another sequencing example, one in which captures for each phase are cancelled with respect to C°-C¹⁸⁰ offset with the same pixel detector, shown as a block. In this configuration, performance is quite good, and motion blur is acceptable. As before, θ _E is calculated from (θι+θ₂)/2.

[0077] Fig. 4D depicts an alternative embodiment in which phase θι and phase Θ₂ are not C°-C¹⁸⁰ offset corrected. As a result, data quality is somewhat poor, but advantageously Θ_Ε is computed directly from the captures Ci° - C₂¹⁸⁰ and Ci⁹⁰ - C₂²⁷⁰. The method is as described previously, where C₂¹⁸⁰ is substituted for C₂° and where C₂²⁷⁰ is substituted for C₂⁹⁰. Advantageously, data quality for Θ_Ε is good but 0D = Θ₁-Θ₂ is of poor quality and as a result the dealiasing interval decision quality suffers.

[0078] An alternative embodiment, not depicted explicitly, is similar to that of Fig. 4D, but wherein Θ_Ε is calculated from (θι+θ₂)/2. As noted earlier, acquired phase angles preferably are normalized to start at zero for Z=0. In the alternative embodiment at hand, if pre-normalization differences between Θ₁ and θ₂ are small, then offset induced errors in Θ₁ and θ₂ will be opposite of each other and will advantageously partially cancel out in Θ_Ε = (θι+θ₂)/2. If O is the assumed small offset, then from equation (4), Θ₁ = atan2(Ci° + O, Ci⁹⁰+ O), and θ₂ = atan2(-C₂¹⁸⁰+ O, -C₂²⁷⁰+ O). If modulation contrast is similar for both modulation frequencies fi and f₂, and θι ^« θ₂, then Ci° ^« C₂¹⁸⁰ and C₁^{90 « "} C₂²⁷⁰. Thus, Θ₁ * atan2(Ci° + O, Ci⁹⁰+ O) and θ₂ = atan2(Ci°- O, Ci⁹⁰- O) + π. As such, phases Θ₁ and θ₂ move in opposite directions in roughly equal amounts with respect to offset O. Thus, advantageously the effect of offset O will be partially cancelled in the sum Θ_Ε = (θι+θ₂)/2. In some advanced TOF systems, the time of flight pixels can acquire more than one phase or modulation frequency at each capture. For example such multiphase-capture pixels effectively function as though they simultaneously capture Ci° and C₁⁹⁰. Such effective functionality may be implanted, for example, by rapidly time-slicing or time-multiplexing between different modulation

frequencies and/or phases, or by using multiple detectors operating at different modulation frequencies and/or phases. Using such pixels, the time to assemble sequences for a dealiased frame is considerably shortened. The basic

methodology described earlier herein remains the same. [0079] Thus far, the various described dealiasing embodiments have been lossless, e.g., dealiasing was carried out with a very small TOF system

performance penalty. A lossy so-called least common multiple (LCM) dealiasing embodiment will now be described reference to Figs. 5A and 5B. In Figs. 5A and 5B, two distance measurements are made using two modulation frequencies fi and f2, for which frequencies Z_AIRI and Z_AIR2 represent the associated maximum unambiguous aliasing interval ranges. In Fig. 5A, ^ is 25 MHz, and Z_AIRI is 3 M, while in Fig. 5B, f₂ is a lower 18.75 MHz, and Z_AIR2 is a longer 4 M. Assume a target object is present 7 m from the TOF sensor array. If a single modulation frequency fi = of 25 MHz is used to acquire data, one can infer that the Z distance of the object is either 1 m, 4 m, 7 m, or 12 m as suggested by the cross-hatched rectangular regions in Fig. 5A. On the other hand, if a single modulation frequency h = 18.75 MHz is used, one can infer the target object is at 3 m, 7 m, or 1 1 m, as suggested by the cross-hatched rectangular regions in Fig. 5B. .

loose] According to the LCM embodiment, the above-two results are combined, leading to the conclusion that the target object must be at a distance of 7 m. The target object is drawn in Figs. 5A and 5B with heavy bold lines to distinguish from phantom locations, which are drawn in phantom. The true location of the target object could still be confused with another location at 19 M, but compared to the one-modulation frequency case, the use of two modulation frequencies has extended the unambiguous interval range substantially. According to the

12/459,033 application, if two modulation frequencies are used, the effective unambiguous interval range is increased according to the LCM least common multiple of Z_AIRI and Z_MR2. One strategy in selecting modulation frequencies is to maximize the LCM of the corresponding unambiguous interval ranges by choosing two frequencies close to each other.

[0081] Another embodiment of the 12/459,033 application provides a lossy so- called branch and bound hierarchical approach to dealiasing. According to this embodiment, a relatively very low modulation frequency may be used to determine whether a target object is within a first or a second Z range or bin, e.g., perhaps between a 0 m to 50 m range, or perhaps between a 50 m to 100 m range. If TOF system 200 determines that the target object is within say 0 m to 50 m, then software 200 can cause clock 180' to double the modulation frequency to determine whether the target object lies within a 0 m to 25 m sub-range (or narrower bin), or within a 25 m to 50 m sub-range (or narrow bin). A similar analysis and modulation frequency change is performed should the object initially have been found to lie with a 50 m to 100 m range. This method of estimating distance range for the target object and then changing modulation frequency to better ascertain the distance range is repeated preferably until the range is determined with a desired range granularity. Again this method is preferably carried out automatically under command of software 220. Of course while this example assumed the various alternate ranges or bins were 2X apart, other ratios may be used.

[0082] During a branch and bound hierarchical dealiasing process it is not necessary to obtain full certainty (e.g., best granularity) at each modulation frequency. For lower modulation frequencies it can suffice to obtain lower resolution. For example to decide between the 0 m to 50 m, and 50 m to 100 m ranges or bins, a fairly coarse level of resolution is sufficient. For example, if a highest quality depth image acquired by system 200 uses say ten bits, a fairly coarse level of resolution may only require three to four bits of resolution.

Advantageously therefore captures at low modulation frequency may have short shutter times to reflect these relaxed requirements. In general for a hierarchical sweep of modulation frequencies, e.g., 1 MHz, 10 MHz, 100 MHz, the shutter will be short for all but the highest swept frequency, here 100 MHz. So doing advantageously reduces the amount of integration time the sensor array is operated at low modulation frequencies, which further boosts effective

performance.

[0083] Understandably, using branch and bound hierarchical dealiasing, it may be desirable to minimize the number of bins for each relatively low modulation frequency. In turn, the hierarchical modulation frequency sweep process tends to maximize the effective system TOF modulation frequency by keeping the resolution requirement at each lower modulation frequency as small as feasible. If desired, a branch and bound embodiment may be combined with lossless dealiasing embodiment at one or more steps in the hierarchical process.

[0084] The above description was directed to lossy hierarchical type dealiasing embodiments. A lossless hierarchical dealiasing embodiment of the 12/459,033 application will now be described. Earlier herein it was shown that 0_D * lfo/2n should be substantially less than 0.5. In an embodiment where a very long aliasing interval is desired, use of relatively high modulation frequencies fi and f₂ yield an f_E fD ratio that becomes very large, making it difficult to keep 0D * lfo/2n « 0.5. For such applications, a lossless dealiasing method may also be applied hierarchically. In this fashion, at each step in the hierarchical process, as governed by noise consideration and as controlled by software 220, the ratio /h is kept relatively small.

[0085] Consider the following example, in which modulation frequencies f_a = 100 MHz, f _b = 1 10 MHz, f_c = 1 1 1 MHz are used, and wherein there are defined difference frequencies Di = f_b - f_a = 10 MHz, D₂ = f_c - f_a = 1 1 MHz., and

hierarchically applied difference frequency Ei = D₂ - Di = 1 MHz.

[0086] In a so-called top-down sweep, initially frequency Ei is used to dealias acquired phase data for Di and for D₂. In this initial step, fi = Di , f ₂ = D₂ and fo = Ei . The ratio /h advantageously is not too large, here approximately 10. Thus, in this example, dealiased Z values may be found for an effective frequency θ_Ε=(θι +θ₂)/2.

[0087] Going down one step in the hierarchical process, the above found and dealiased value of Θ_Ε will now be used in the current step as dealiasing phase 0_D for effective frequency 0.33(f_a+fb+f_c) and effective phase ΘΕ = O.33(0_a+0b+6_c). Advantageously, it is noted that the ratio f_E fD remains relatively small, here approximately 10. In this fashion, as the bottom of the hierarchy, 0_E is 0.33(100 MHz+1 10+1 1 1 MHz), i.e., close to 100 MHz, yet is has been dealiased as though its frequency were Ei =1 MHz, with Z_AIR of about 150 M.

[0088] In addition, it is noted that TOF system 200 is not necessarily physically acquiring phase data with a modulation frequency f_c = 1 1 1 MHz. Frequency f_c may be computed mathematically, for example as f_c = (1 10 MHz + 100 MHz/100), e.g., fb+(f_a)/100. Thus only frequencies f_a and % need to be physically measured by the TOF system.

[0089] In summary, dealiasing according to embodiments of the 12/459,033 application is lossless, and provided a relatively large aliasing interval range commensurate with a low modulation frequency. These embodiments also provide high precision certainty with respect to a given Z value, commensurate with a modulation frequency close to the highest modulation frequency f_m. Dealiasing could be carried out relatively losslessly or with loss, depending upon the embodiment.

[0090] Turning now to Fig. 6, a phase-based TOF system 300 implements the present invention. TOF system 300 includes a clock unit 180" that may be modified over clock unit 180 in Fig. 1 A and, or alternatively, includes hardware 310 and/or software 320. Software 320 may but need not be executed by processor 160. This hardware and/or software implement the various

embodiments of the present invention, which will now be described.

[0091] Recall the following relationships from equation (2) and equation (8) presented earlier herein that show unambiguous range Z is inversely proportional to modulation frequency f:

C Θ eC , C ΔΘ

[0092] Z =— x— = and Z

2/ 2π 4nf 4_π f where Af = f_D = - h, phase ΔΘ = 6 = Θ1-Θ2. The undesirable sensitivity to noise of two frequency dealiasing is seen in Fig. 3, where the relationship between ODS and 0D is given by: f H~ f Θ H~ Θ where f_E = -^ ¹, 9_D = θ₁ - θ₂ is phase associated with f_D, Θ_Ε = -^-^ is phase associated with f_E. Phase 9_D was scaled to yield 9_DS, whose frequency slope is given by the above equation. Note that 6_DS = Θ_Ε + 2ηπ.

[0093] Unfortunately in the presence of noise, the difficulty of de-aliasing increases as the ratio increases. This is because noise present in Θ_Ε will be amplified by this ratio when Θ_Ε is used to compute Q_DS. Therefore, for high modulation frequency (corresponding to large effective frequency f_E) or very large unambiguous range (corresponding to small difference frequency f_D), such amplified noise would produce erroneous results in estimating de-aliasing interval K and would cause aliasing in the middle of the unambiguous range. This is apparent from Fig. 3 if one imagines the angle (or slope) of Q_DS increased or decreased, as the ratio ^ increases or decreases, as 6_DS Θ_Ε + 2ηπ.

TD

Thus an increased ratio— represent amplified noise in Θ_Ό and is undesired

†D

because it contributes to a wrong dealiasing decision. What is desired is to simultaneously maintain a small ratio— , while achieving a dealiasing range

†D

associated with small f_D and a precision of depth associated with a large f_E. This is accomplishing using preferably at least three modulation frequencies, as will now be shown. Embodiments of the present invention preferably employ an n- step hierarchical dealiasing approach to avoid aliasing problems caused by very large amplification of the noise in B_E.

[0094] Assume there are m modulation frequencies f₁,f₂, - , f_m (A &gt; f₂ &gt; ■■■&gt; &gt; f_m), and it is desired that TOF system 300 achieve unambiguous range Z_D, which corresponds the frequency f_D as in Z_D = -^— . Z_D = -^— . One embodiment uses f_D to de-alias and achieve the distance resolution of the effective

frequency f_E , where f_E = g₀ (f₁, f₂, - , f_m) is a function of the modulation

frequencies f_x, f₂, - , f_m , preferably an arithmetic mean or a weighted average.

[0095] Instead of using f_D to dealias the phase for effective frequency f_E in one step dealiasing, embodiments of the present invention preferably generate a set of N— 1 intermediate frequencies f_DE1, f_DE2, - , f_DE{N-i) , where f_D &lt; f_DE1 &lt; f_DE2 &lt;

[0096] f_D &lt; f_DE1 &lt; f_DE2 &lt; ... &lt; _D£(_W_D &lt; f_E. Each of the intermediate

frequencies is a function of the modulation frequencies f₁, f₂, - , f_m as in f_DEk = 9_k k_{&gt; &gt;} - , fm) where k = 1,2,3 ... , N - 1.

[0097] f_DEk = 9_k (fi&gt; f &gt; - &gt; fm) where = 1,2,3 ... , TV - 1, and let _DB0 = /D, and<img file="WO2012012607A2_D0002.tif" />

[0098] At each step of the hierarchical de-aliasing, a preferred method de-aliases the phase e_DEk of f_DEk using the phase e_DE{k+1) of f_DE{k+1) (k = 0,1, .. N - 1) by finding the correct de-aliasing interval m_k such that 9_{DEk scaled} 0_D£(fc+1) + m_k2n, w9_DEk __scai_ed = 9_DE(k+1^ + m_fc 277:here 9_{DEk scaled} = 9_DEk.

[0099] The final unwrapped phase for _£ will be given as:

[ooioo] Θ_Ε + τη_Ν_₁2π + m_N_₂ 2π — m ^ ^g(fe+1) 2π — m- 2π

<img file="WO2012012607A2_D0003.tif" />

[ooioi] The number of the intermediate frequencies N— 1 and the ratio ¹⁾<img file="WO2012012607A2_D0004.tif" /> between frequencies in each consecutive pair preferably are determined by the uncertainty of the depth. As a rule of thumb, preferably the uncertainty of the amplifying factor at each step is sufficiently small such that the probability of choosing m_k incorrectly is low.

[00102] Embodiments implementing two-step (three-frequency) hierarchical dealiasing will now be described with reference to Fig. 6 and Figs. 7A-7D. For two-step hierarchical de-aliasing, at least N=3 modulation frequencies are needed. The total unambiguous range of the system is determined by f_D, which is a function of the three modulation frequencies.

[00103] As shown in Fig. 3, below, first phase 9_D\s used for f_D to de-alias the phase 9_DE of some intermediate frequency f_DE, and the correct de-aliasing interval m for 9_DE such that, 9_DS 9_DE + πιλπ.

[00104] In the second step of hierarchical de-aliasing, each de-aliasing interval of 9_DE is used to de-alias the effective phase 9_E of the effective frequency f_E by finding the correct de-aliasing interval n such that 9_DS = Θ_Ε + ηλπ.

[00105] Considers now Figs. 7A-7D. Combining the two de-aliasing steps enables determination of the final de-aliased values for 9

<img file="WO2012012607A2_D0005.tif" />

[00106] Note that the amplification ratio for the de-aliasing steps are— and

<img file="WO2012012607A2_D0006.tif" /> respectively. Advantageously, this method achieves the desired large ratio

_ fE_ _ [^DE_ J^E_ wjfhout amplifying the noise in 9_E by such a large ratio. The

<img file="WO2012012607A2_D0007.tif" />

beneficial result occurs because the de-aliasing intervals m and n are determined separately and the noise is amplified at much smaller ratio at each method step.

[00107] Exemplary algorithm details for two-step hierarchical dealiasing will now be described with reference to Figs. 7A-7D. It is understood that these steps are preferably carried out with TOF system 300, e.g., by clock unit 180", and/or modules 310, 320. Embodiments of the present invention preferably use at least three modulation frequencies. Given three modulation frequency f_{l t} f₂ and f₃, it is desired to achieve an effective frequency f_E that is as close as possible to the TOF system maximum modulation frequency f_m. Let f_D = f - f₂ be designated as the slowest frequency associated with the total dealiasing range, and let fDE = fi - h ^De designated as the intermediate frequency. The final effective frequency can be f_E = ^l+^^{2+ 3} or other weighted linear combination of f_{l t} f₂ and [00108] As shown in Fig. 7A, one can first use phase fofor frequency†D to dealias the phase 6DE of some intermediate frequency†DE and the correct dealiasing interval m for phase 6DE such that 0_Ds = ΘΕ + ιτι2ττ. As shown in Fig. 7B one can now use each dealiasing interval of Θ_0Ε to dealias the effective phase θ_Εο1 the effective frequency by finding the correct dealiasing interval n, such that ODES = Θ_Ε + η2π. Figs. 7C and 7D depict how to dealias phase θ, οί individual frequency /j (i = 1 , 2, or 3, for three frequenices), using phase ODES- For example, Fig. 7C shows that 0_DE and 0, are likely to wrap around at different distances. Thus one cannot dealias phase more than the first cycle of 6DE- (TO dealias 6DS as shown in Fig. 7A, one would have to dealias Θ_0Ε for all of the four cycles shown in the figure.) Fig. 7D shows that one can compute the offset-corrected phase 6°^ffset, which will always start at zero phase at the beginning of each cycle of ODES- One can then use this offset-compensated phase to dealias phase θ, ονβΓ the multiple cycles of 9_DEs- [00109] Thus, a first method step is to dealias f_DE = f₁ - f₃ using f_D = fi - f₂. Let θ₁ be the phase of /_1; let θ₂ be the phase of f₂, and let θ₃ to be the phase of ₃. First it is necessary to unwrap θ₁ using θ₂ to get θ₁^unwraP~² Θ₁^unwraP~² = θ₁ + 2π * (θ₁ &lt; θ₂), and unwrap θ₁ using θ₃ to get θ₁^unwraP~³ . One can then obtain phase for f_D = f_x - f₂ a s 9_D = θ₁^unwraP~² - e₂ and one can obtain phase for f_DE = f_x - f₃ as e_DE = θ₁ ^wrap-3 _

[ooiio] One can now rescale θο to the same slope as 6DE and create Q_DS, where 8_DS θ₀. Completing the firsf step, one can next find the correct <img file="WO2012012607A2_D0008.tif" />

dealiasing interval m by minimizing \6_DS - (θ_ΟΕ + τη2π) \ for m = 0,1,2,

[ooiii] Consider now step two of the method, which involves dealiasing f_E = l⁺ ^ ^{+ 3} from f_DE = f - f₃. Although it is desired to dealias f_E, this cannot be done direction because one cannot get the phase corresponding to f_E =

Λ+ 2 +/3 ^_{e correc}t warpping-aroung method. Instead embodiments of the present invention dealias f_x, f₂ and f₃ separately and get the unwrapped phases φ₁ = θ₁ + η 2π, φ₂ = θ₂ + η₂2π, and φ₃ = θ₃ + η₃2π.

[00112] The unwrapped phase for f_E = ^l+^^{2+ 3} can be calculated as φ_Ε = ^Φι+φ^^+φ3_

[00113] Dealiasing f_t (i = 1,2,3) to get the unwrapped phase φι will now be described with reference to Fig. 7C and Fig. 7D. As shown in Fig. 7C, Q_DE and e_t are likely to wrap around at different Z distance because of the frequency differences. This will cause problems in the second step of dealiasing unless additional constraints on the frequencies are imposed, e.g., 0_£ is an multiplier of Θ_ΌΕ. Otherwise, if one directly applied the dealiasing method as the first step to dealias 0_£ using 9_DE, one could only dealias the first cycle of 9_DE because in the remaining cycles, 0_£ and Q_DE will not start at the same position and one cannot satisfy the relationship of Q_DES = 0_£ + η λπ.

[00114] Referring to Fig. 7D, preferably an "offset compensation" method is used to avoid adding additional constraints that limit the choices of frequencies. First one can remove the offset of Θ, caused 0_£ by the wrapping around of ODES and 9_DES and compute the offset-corrected phase e°^ffset. The offset-corrected phase e°^ffset will always start at zero phase at the beginning of each cycle of 6_DES- One can then find the correct de-aliasing interval by minimizing:

DES - (0°^ffset + η_£2π) | for n_£ = 0,1,2, ... where i = 1,2,3

[eons] The unwrapped phase for each frequency ^■ is computed as:

[00116] φι = Q° ^set + η 2π + ηΐ 2π

[00117] The unwrapped phase for f_E = ^l+^^{2+ 3} is then φ_Ε = ^Φι+φ^^+φ3 _

[eons] It will be appreciated from the description of the preferred embodiments, that the present invention provides hierarchical dealiasing for a TOF system that functions even in the presence of noise. Preferably a number N, N &gt; 3, of frequencies that are close to each other are used to create a slow dealiasing frequency and at least one intermediate frequency that can be used to dealias a long distance hierarchically. Advantageously, rather than amplifying noise by a very large ratio as in two-frequency dealiasing, embodiments of the present invention only amplify noise by a small ratio at each dealiasing step.

Modifications and variations may be made to the disclosed embodiments without departing from the subject and spirit of the invention as defined by the following claims.