PATENT ABSTRACT
A distance measurement method includes measuring a plurality of integrated signals at a plurality of modulation phase offsets; estimating at least one integrated signal for at least one of the plurality of modulation phase offsets, respectively, to adjust its reception time relative to an integrated signal for another of the plurality of modulation phase offsets; and determining a distance between the target and receiver in accordance with the estimated at least one signal.

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims foreign priority under 35 U.S.C. §119 to Korean Patent Application No. 10-2009-0082150, filed on Sep. 1, 2009, in the Korean Intellectual Property Office, the disclosure of which is incorporated by reference herein in its entirety. 
     BACKGROUND OF THE INVENTION 
     The present disclosure generally relates to contactless three-dimensional (3D) shape measurements. More particularly, the present disclosure relates to contactless 3D shape measurement methods and devices using delay compensation in modulated optical time-of-flight phase estimation. 
     Contactless 3D shape measurements may use pressure wave or electromagnetic wave emissions. Pressure waves may be ultrasonic signals, for example. Electromagnetic wave emissions may be microwaves or light waves (e.g., l=0.5-1.0 um; f=300-600 THz), for example. For light wave emissions, 3D sensing methods include triangulation, interferometry, and time-of-flight (TOF). Triangulation performs depth detection using geometric angle measurements. Interferometry performs depth detection using optical coherent time-of-flight measurements. TOF may perform depth detection using either pulsed or modulated continuous-wave (CW) optical incoherent time-of-flight measurements. 
     Pulsed type TOF features range sensing by measuring the turn-around time, reduced influence of background illumination, high signal-to-noise ratio (SNR) with low average power for eye safety, low repetition rate (e.g., 10 kHz) of the laser diode (LD) and a low frame rate. Unfortunately, it can be difficult to form pulses with sufficiently short rise and fall times, and dispersion and attenuation can become issues. 
     Modulated CW type TOF features range sensing by measuring phase differences. Modulated TOF can use a wide variety of light sources, such as sinusoidal, square wave, and the like. 
     SUMMARY OF THE INVENTION 
     The present disclosure teaches contactless three-dimensional shape measurement using delay compensation in modulated optical time-of-flight phase estimation. Exemplary embodiments are provided. 
     An exemplary embodiment distance measurement method includes measuring a plurality of integrated signals at a plurality of modulation phase offsets; estimating at least one integrated signal for at least one of the plurality of modulation phase offsets, respectively, to adjust its reception time relative to an integrated signal for another of the plurality of modulation phase offsets; and determining a distance between the target and receiver in accordance with the estimated at least one signal. 
     A further exemplary embodiment includes emitting narrow band electromagnetic energy as a modulated continuous wave; and receiving and integrating signals indicative of electromagnetic energy reflected from a target for the plurality of modulation phase offsets. 
     A further exemplary embodiment includes estimating comprising interpolating the at least one integrated signal at a first time with the at least one integrated signal at a second time, wherein the first time is before the reception time of the integrated signal for the other of the plurality of modulation phase offsets, and the second time is after the reception time of the integrated signal for the other of the plurality of modulation phase offsets. 
     A further exemplary embodiment includes estimating comprising extrapolating the at least one integrated signal at a current time from the at least one integrated signal at a plurality of previous times, wherein the current time is the reception time of the integrated signal for the other of the plurality of modulation phase offsets. 
     A further exemplary embodiment has estimating comprising splitting a time difference between a first integrated signal and a second integrated signal to obtain a median time, interpolating the first integrated signal at a first time with the first integrated signal at a third time to obtain an estimated first integrated signal at the median time, wherein the first time is before the median time and the third time is after the median time, interpolating the second integrated signal at a second time with the second integrated signal at a fourth time to obtain an estimated second integrated signal at the median time, wherein the second time is before the median time and the fourth time is after the median time. 
     A further exemplary embodiment has the narrow band electromagnetic energy has a wavelength between about 850 and about 950 nanometers. 
     A further exemplary embodiment has the plurality of modulation phase offsets comprises four equally spaced offsets. Another exemplary embodiment has first, second, third and fourth signals for zero degree, 90 degree, 180 degree and 270 degree phase offsets, respectively, are received and integrated by at least one photo sensor. Yet another exemplary embodiment has the at least one photo sensor comprises color pixels and distance pixels, the color pixels disposed on a first integrated circuit and the distance pixels disposed on a second integrated circuit. A further exemplary embodiment has the at least one photo sensor comprises color pixels and distance pixels on a single integrated circuit. 
     An exemplary embodiment distance measurement system includes a narrow band source for emitting electromagnetic energy as a modulated continuous wave; a photo sensor for receiving and integrating signals indicative of electromagnetic energy reflected from a target for a plurality of modulation phase offsets; and a control unit for measuring a plurality of integrated signals at the plurality of modulation phase offsets, estimating at least one integrated signal for at least one of the plurality of modulation phase offsets, respectively, to adjust its reception time relative to an integrated signal for another of the plurality of modulation phase offsets, and determining a distance between the target and receiver in accordance with the compensated at least one signal. 
     A further exemplary embodiment has the source emits narrow band electromagnetic energy with a wavelength between about 850 and about 950 nanometers. An alternate exemplary embodiment has the plurality of modulation phase offsets comprises four equally spaced offsets. A further exemplary embodiment has first and third signals for zero degree and 180 degree phase offsets, respectively, are received and integrated by the photo sensor, and second and fourth signals for 90 degree and 270 degree phase offsets, respectively, are received and integrated by a second photo sensor. An alternate exemplary embodiment has first, second, third and fourth signals for zero degree, 90 degree, 180 degree and 270 degree phase offsets, respectively, received and integrated by the photo sensor. A further exemplary embodiment has the photo sensor comprising color pixels and distance pixels, the color pixels disposed on a first integrated circuit and the distance pixels disposed on a second integrated circuit. A further exemplary embodiment has the one photo sensor comprising color pixels and distance pixels on a single integrated circuit. 
     A further exemplary embodiment has the control unit comprising an estimation unit for interpolating the at least one integrated signal at a first time with the at least one integrated signal at a second time, wherein the first time is before the reception time of the integrated signal for the other of the plurality of modulation phase offsets, and the second time is after the reception time of the integrated signal for the other of the plurality of modulation phase offsets. 
     A further exemplary embodiment has the control unit comprising a estimation unit for extrapolating the at least one integrated signal at a current time from the at least one integrated signal at a plurality of previous times, wherein the current time is the reception time of the integrated signal for the other of the plurality of modulation phase offsets. 
     An exemplary embodiment distance sensor includes a photo sensing array for receiving and integrating signals indicative of electromagnetic energy reflected from a target for a plurality of modulation phase offsets; and a control unit for measuring a plurality of integrated signals at the plurality of modulation phase offsets, estimating at least one integrated signal for at least one of the plurality of modulation phase offsets, respectively, to adjust its reception time relative to an integrated signal for another of the plurality of modulation phase offsets, and determining a distance between the target and receiver in accordance with the estimated at least one signal. 
     A further exemplary embodiment has the photo sensing array comprising a plurality of pixels for sequentially acquiring signal samples at a plurality of modulation phase offsets. An alternate exemplary embodiment has the photo sensing array comprising: a first sensor for sequentially acquiring signal samples at a plurality of first modulation phase offsets; and a second sensor for sequentially acquiring signal samples at a plurality of second modulation phase offsets, wherein the first and second modulation phase offsets alternate in sequence. An alternate exemplary embodiment has the plurality of modulation phase offsets comprising four equally spaced offsets. A further exemplary embodiment has first, second, third and fourth signals for zero degree, 90 degree, 180 degree and 270 degree phase offsets, respectively, being received and integrated by first, second, third and fourth photo sensing arrays, respectively. An alternate exemplary embodiment has first and third signals for zero degree and 180 degree phase offsets, respectively, are received and integrated by the photo sensing array, and second and fourth signals for 90 degree and 270 degree phase offsets, respectively, are received and integrated by a second photo sensing array. Another alternate exemplary embodiment has first, second, third and fourth signals for zero degree, 90 degree, 180 degree and 270 degree phase offsets, respectively, being received and integrated by the photo sensing array. A further exemplary embodiment has the photo sensing array comprising color pixels and distance pixels, the color pixels disposed on a first integrated circuit and the distance pixels disposed on a second integrated circuit. An alternate further exemplary embodiment has the one photo sensing array comprises color pixels and distance pixels on a single integrated circuit. 
     A further exemplary embodiment has the control unit comprising an estimation unit for interpolating the at least one integrated signal at a first time with the at least one integrated signal at a second time, wherein the first time is before the reception time of the integrated signal for the other of the plurality of modulation phase offsets, and the second time is after the reception time of the integrated signal for the other of the plurality of modulation phase offsets. 
     A further exemplary embodiment has the control unit comprising an estimation unit for extrapolating the at least one integrated signal at a current time from the at least one integrated signal at a plurality of previous times, wherein the current time is the reception time of the integrated signal for the other of the plurality of modulation phase offsets. 
     A further exemplary embodiment has the photo sensing array comprising: a first integrated circuit for acquiring signal samples for color pixels; and a second integrated circuit for acquiring signal samples for distance pixels. 
     A further exemplary embodiment has the photo sensing array comprising: a first integrated circuit for acquiring signal samples for color and distance pixels. 
     The present disclosure may be further understood from the following description of exemplary embodiments, which is to be read in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure provides contactless three-dimensional shape measurement using delay compensation in modulated optical time-of-flight phase estimation in accordance with the following exemplary figures, in which: 
         FIG. 1  is a schematic diagram of a contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 2  is a schematic diagram of another contactless 3D shape measurement device in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 3  is a graphical diagram of a signal plot in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 4  is a graphical diagram of another signal plot in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 5  is a schematic diagram of a contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 6  is a schematic diagram of images from a contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 7  is a schematic diagram of a 2-tap contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 8  is a schematic diagram of a 2-tap pixel unit in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 9  is a circuit diagram of a pixel sensor circuit in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 10  is a graphical diagram of a 2-tap signal plot and timing diagram of IR signals and gate signals in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 11  is a graphical diagram of a 2-tap sampling point plot in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 12  is a graphical diagram of another 2-tap sampling point plot in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 13  is a schematic diagram of a 2-tap timing diagram illustrating the estimation process for a digital pixel signal according to the operation of the depth sensor at estimation time in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 14  is a graphical diagram of simulation results for conventional versus a 2-tap embodiment in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 15  is a graphical diagram of a 2-tap comparative simulation graph in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 16  is a graphical diagram of a 2-tap comparative simulation graphs in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 17  is a schematic diagram of a 1-tap contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 18  is a schematic diagram of a 1-tap pixel unit in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 19  is a graphical diagram of a 1-tap sampling point plot in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 20  is a schematic diagram of a 1-tap timing diagram illustrating the estimation process for a digital pixel signal according to the operation of the depth sensor at estimation time in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 21  is a graphical diagram of plots of 1-tap simulation results in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 22  is a graphical diagram of a 1-tap comparative simulation graph in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 23  is a graphical diagram of a 1-tap comparative simulation graphs in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 24  is a flow diagram of a method of depth estimation in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 25  is a circuit diagram of an exemplary three-transistor (3T) APS structure in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 26  is a circuit diagram of an exemplary four-transistor (4T) APS structure in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 27  is a circuit diagram of a first exemplary five-transistor (5T) APS structure in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 28  is a circuit diagram of a second exemplary 5T APS structure in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 29  is a schematic diagram of a contactless 3D shape measurement system using a two-chip solution in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 30  is a schematic diagram of a contactless 3D shape measurement system using a single-chip solution in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 31  is a schematic diagram of a contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 32  is a schematic diagram of a contactless 3D shape measurement system in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 33  is a schematic diagram of a contactless 3D shape measurement partial circuit and schematic signal diagram in accordance with an exemplary embodiment of the present disclosure; 
         FIG. 34  is a schematic diagram of a contactless 3D shape measurement partial circuit and schematic signal diagram in accordance with an exemplary embodiment of the present disclosure; and 
         FIG. 35  is a circuit diagram of APS structures in accordance with exemplary embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Preferred embodiment contactless three-dimensional (3D) shape measurement methods and devices perform depth detection using modulated continuous-wave (CW) optical incoherent time-of-flight (TOF) measurements, and feature delay compensation in TOF phase estimation. 
     Referring to Table A, Equations A1, A2, A3 and A4 represent return signal amplitudes between an emissive sensor and an object. Here, the amplitudes for each of four phase offsets is comprised of a background noise component, alpha, and a reflected signal component, beta. The reflected signal component, beta, indicates an intensity that may vary with distance and reflectivity of the target, for example. 
     Equation A5 defines the time-of-flight as a function of the distance to and from the target divided by the sped of light. Equation A6 defines an integration or summation for each amplitude by intervals T-int. Equation A7 defines the phase delay estimation. Equation A8 defines the depth estimation. 
     Referring now to Table B, Equation B1 defines a proportionality for the intensity of an emitted signal from an infra-red source. Equation B2 defines the intensity of an emitted signal from an optical source. Equation B3 defines the intensity of the received signal. Equation B4 defines an angle as a function of four phase-offset sampling points. Equation B5 defines the luminosity as a function of the angle. Equation B6 defines a change in luminosity as proportional to a function of brightness and amplitude. 
     In operation, three-dimensional (3D) time-of-flight (TOF) imaging may use an invisible light source to illuminate a subject. A sensor chip can measure the distance that the light has traveled to reach each pixel within the chip. Embedded imaging software uses a depth map to perceive and identify objects in real time. An associated end-user device can react appropriately to the resulting object identification. 
     For example, one type of sensor might use a modulation frequency of 44 MHz with a wavelength of 3.4 m, and have a measurement range of about 15 cm to about 3 m with a depth resolution of under 3 cm. Another type of sensor might have a larger pixel array, use a modulation frequency of 20 MHz with a wavelength of 7.5 m, and have a measurement range of around 15 m, with a depth resolution of about 1.3 cm. 
     Distance or Z-measuring using a sinusoidal waveform can use a relatively inexpensive light source rather than a laser, and a lower bandwidth amplifier. The measurement is typically repeated to get a ‘Z’ value by accumulated charge, where Z may be on the order of a wavelength of the emitted light. 
     A Z-measuring optical system can use a nonlinear Geiger-mode to detect small numbers of photons. Here, the optical system is composed to effectively transfer the reflected light from the object to a detector. 
     It can be difficult to detect depth information by differential signaling caused by pixel saturation due to background light that is noise. Thus, elimination of a common mode signal by the background signal can help. To prevent pixel saturation, either resetting the CA voltage when it is saturated and/or periodically resetting the voltage may be performed. 
     Undesired photons generated by background light may decrease the signal-to-noise ratio (SNR). To increase the SNR, the voltage may be calculated by background light during the time that the modulated light is off, and the calculated voltage for the background light may be subtracted from the measured voltage during the time that the modulated light is on or emitting. 
     High and low sensitivities may be obtained by long and short detection periods, respectively. For increased SNR, a distance value for each image element in the range image is calculated based on each electric charge picked up in synchronization with the specific detection period, and then the range image is constructed. 
     A delayed phase signal may include two components, namely pixel-wise delay and frame-wise delay. Of these, the pixel-wise delay is more significant. Pixel-wise delay is the delay between the 0, 90, 180, and 270 degree components. 
     Referring to Table C, a 2-tap structure has two sampling times to obtain the four measurements, where the second sampling time is defined by Equation C1. Here, the phase angle is defined by Equation C2. A 4-tap structure has four sampling times to obtain the four measurements, each offset by a time interval. Here, the phase angle is defined by Equation C3. Frame-wise delay is the delay between the first pixel and the last pixel captured with a rolling shutter. 
     Thus, the numbers of photons captured at actual sampling times may be interpolated or extrapolated to a compensated sampling time by using sensors activated by photogates for a desired phase offset range of a cycle or period, where the desired phase offset range is summed or integrated over many cycles to capture sufficient numbers of photons. In addition, a single correlated double sampling may be performed after such integration over many cycles. Although sensors configured for a typical 2×2 Bayer or like mosaic pattern are sufficient, striped and/or other patterns such as 2×4 and 3×3 may be implemented in alternate embodiments. Similarly, complement colors such as yellow, cyan and magenta may be substituted for the green, blue and red of the Bayer pattern. An emitted wavelength of about 850 to about 950 nm is preferred for outdoor sensing, since the sun has a lower output at 850 nm, yet such wavelengths are just outside of normal human vision in the near infrared band. 
     As shown in  FIG. 1 , a contactless 3D shape measurement system is indicated generally by the reference numeral  100 . The contactless 3D shape measurement device  100  uses depth compensation for modulated optical time-of-flight phase estimation, and includes an emitter  110 , an object or reflector  112  some distance from the emitter, a receiver  114  some distance from the object, and a phase measurement unit  116  in signal communication with the receiver, where the receiver provides a reference and a signal to the phase measurement unit. 
     Turning to  FIG. 2 , another contactless 3D shape measurement device is indicated generally by the reference numeral  200 . Here, a combined emitter/receiver  210  is disposed a distance from an object  212 . The time of flight is the time taken for the light to travel from the emitter to the reflector to the receiver. 
     Turning now to  FIG. 3 , a signal plot is indicated generally by the reference numeral  300 . The plot  300  includes an emitted light signal  310 , a reflected or received light signal  312 , a 0-degree phase-offset signal  314 , and a 180-degree phase offset signal  316 . Here, the 0-degree phase-offset signal  314  includes a 0 th  quadrant n th  cycle integral  318  and a 0 th  quadrant (n+1) th  cycle integral  319 . Similarly, the 180-degree phase-offset signal  316  includes a 2 nd  quadrant n th  cycle integral  320  and a 2 nd  quadrant (n+1) th  cycle integral. 
     As shown in  FIG. 4 , another signal plot is indicated generally by the reference numeral  400 . The plot  400  includes an emitted light signal  410 , and a detected signal  420 . Here, a phase delay angle between the emitted and detected signals may be used to provide distance information; an alternating-current (AC) or time-varying amplitude of the detected signal may be used to provide accuracy information; and a direct-current (DC) or constant amplitude of the detected signal may be used to provide brightness information. 
     Turning to  FIG. 5 , a contactless 3D shape measurement system is indicated generally by the reference numeral  500 . The system  500  includes an invisible light source  510  that illuminates a subject  511 , a sensor chip  512  that measures the distance light travels to each pixel within the chip from the subject and the source, an embedded imaging unit  513  that comprises a depth map to perceive and identify the subject in real time, and an end-user device  514  in signal communication with the imaging unit for responding to the perceived subject. 
     Turning now to  FIG. 6 , images from a contactless 3D shape measurement system are indicated generally by the reference numeral  600 . Here, an imaging unit such as the imaging unit  513  of  FIG. 5  perceives a scene  610  as the depth map  612 , and perceives the scene  620  as the depth map  622 . In this case, the depth map  622  sufficiently matches a reference map  624 , indicating that a subject is holding a recognized object, such as a coffee cup. 
     As shown in  FIG. 7 , a 2-tap contactless 3D shape measurement system is indicated generally by the reference numeral  700 . The system  700  includes a device  710  and an object  711 . The device  710  includes an infra-red (IR) emitter  712 , a sensor array  714  including a plurality of sensor pixels  716 , which each receive reflected light from the object through an IR pass filter  717 , a correlated double sampling analog-to-digital converter (CDS/ADC) unit  718  that receives amplitudes or photon counts from the array, and a timing and control signal from a timing and control (T/C) unit  720 . The T/C unit is also in signal communication with an X-decoder  722  for the array  714 , and the IR emitter  712 . The CDS/ADC unit  718  passes sampled amplitudes or photon counts to a memory  724 , which, in turn, provides the sampled amplitudes or photon counts to a depth estimator  726 . The depth estimator  726  provides signals indicative of object depth or distance from the emitter and sensor pixels of the device  710 . 
     In operation of this exemplary embodiment 2-tap structure, the depth sensor  710 , object  711 , one or more IR emitters  712 , 2-tap depth sensor array  714 , sensing pixels  716 , IR pass filter  717 , CDS/ADC unit  718 , timing controller  720 , memory  724  and depth estimator  726  form an effective system. 
     Referring to Table E, Equation 1, t-delta is the time difference between the emitted light (EL) and reflected light (RL) where d is depth information, distance between sensor and objects, and c is the speed of light. The RL can pass through the additional lens or lens module located in front of the IR pass filter  717 . The IR emitter  712  may emit modulated IR toward the outside and can be configured with light emitting diodes (LEDs), organic light emitting diodes (OLEDs), or laser diodes (LDs), for example. 
     Each depth sensing pixel  716  having this 2-tap pixel structure can measure pixel signals A 0 ′/A 2 ′ and A 1 ′/A 3 ′ in accordance with gate signals Ga and Gb, respectively, which have a 180-degree phase difference. 
     Thus, a plurality of sensing pixels  716  accumulate RL comprising photo-generated electrons introduced by reflected IR incident through the IR pass filter  717  for a predetermined time period, such as a pre-defined integration time, and output pixel signals A 0 ′/A 2 ′ and A 1 ′/A 3 ′ generated by the accumulation. 
     In Table E, Equation E2 represents pixel signals A 0 ′/A 2 ′ and A 1 ′/A 3 ′ generated by each pixel  716 . Ak′ is derived from the phase difference of the gate signal. When it is 0 degrees, this yields k 0 ; 90 degrees yields k 1 ; 180 degrees yields k 2 ; 270 degrees yields k 3 , where ak,n is the number of photo-generated electrons in the depth sensor  716  while inputting the nth gate signal with a phase difference according to the ‘k’, and N=fm*Tint, where fm is the modulated IR or EL, and Tint is the integration time. 
     Turning to  FIG. 8 , a 2-tap pixel unit is indicated generally by the reference numeral  800 . The pixel unit includes a pixel  816 , such as one of the pixels  716  of  FIG. 7 . The pixel includes a first region  821  and a second region  822 . The first region includes a first tap  823 , and the second region includes a second tap  824 . 
     Turning now to  FIG. 9 , a pixel sensor circuit is indicated generally by the reference numeral  900 . The circuit includes a photo-sensor device (PSD)  910  for receiving photons, a gate transistor  912  connected to the PSD, a reset transistor  914  connected between the gate transistor and a source voltage, a second transistor  916  gated by the gate transistor and connected between the source voltage and a select transistor  918 , and a load transistor  920  connected between the select transistor and ground. Thus, the pixel sensor circuit includes transistors and photo-electric converting devices in active areas  821  and  822  of  FIG. 8 . 
     As shown in  FIG. 10 , a 2-tap signal plot and timing diagram of IR signals and gate signals is indicated generally by the reference numeral  1000 . The plot  1000  includes an emitted light signal  1010 , a reflected or received light signal  1012 , a O-degree phase-offset signal  1014 , and a 180-degree phase offset signal  1016 . Here, the 0-degree phase-offset signal  1014  includes a 0 th  quadrant n th  cycle integral  1018  and a 0 th  quadrant (n+1) th  cycle integral  1019 . Similarly, the 180-degree phase-offset signal  1016  includes a 2 nd  quadrant n th  cycle integral  1020  and a 2 nd  quadrant (n+1) th  cycle integral. 
     In operation of the 2-tap sensor of  FIGS. 8 ,  9  and  10  uses gate signals Ga and Gb, which have a phase difference of 180 degrees, and which are supplied for each photo-electric converting device including photogates  823  and  824  of the depth sensing pixel  816  of  FIG. 8 . 
     Therefore, each photogate transfers the photo-generated electrons generated by the reflected light (RL) to the floating diffusion (FD) region through the transfer gate  912  of  FIG. 9  during the high times of Ga and Gb. Each pixel signal A 0 ′/A 2 ′ and A 1 ′/A 3 ′ corresponding to photo-generated electrons is generated by respective photo-electric converting devices  823  and  824  through the source follower transistor  916  and the select transistor  918 . The reset transistor  914  resets FD to Vdd in accordance with a reset signal (RST). 
     The load transistor  920  is connected between the output node of the depth sensing pixel and ground, and operates by following a load signal VLOAD. Upon the signal of the timing controller  720  of  FIG. 7 , the digital CDS/ADC circuit  718  executes correlated double sampling and ADC for each pixel signal A 0 ′/A 2 ′ and A 1 ′/A 3 ′, and outputs each digital pixel signal A 0 /A 2  and A 1 /A 3 . 
     Turning to  FIG. 11 , a 2-tap sampling point plot is indicated generally by the reference numeral  1100 . The plot  1100  shows two sampling points, t 0  and t 1 . At the first sampling point t 0 , the zeroth and second phase-offset quadrant photon counts are sampled. At the second sampling point t 1 , the first and third phase-offset quadrant photon counts are sampled. 
     In operation, the memory  724  of  FIG. 7 , configured by buffers, receives and stores each digital pixel signal A 0 /A 2  and A 1 /A 3  outputted from the CDS/ADC circuit  718 . The depth estimator  726  estimates phase differences based on each digital pixel signal A 0 /A 2  and A 1 /A 3 . The phase difference estimated by the depth estimator  726  is defined by Table E, Equation E3. 
     The depth estimator  726  estimates depth information based on estimated phase differences in accordance with equation E4, where c is the velocity of light, and fm is the modulated frequency of reflected light (RL), and outputs depth information. 
     The timing diagram illustrates the time difference generated in depth sensing pixels having the 2-tap pixel structure, as set forth in Equation E5. If gate signals Ga and Gb, which each has a phase offset of about 0 degrees and 180 degrees, respectively, are input at t 0 , then depth sensing pixels  716  having the 2-tap pixel structure output simultaneously measured pixel signals A 0 ′ and A 2 ′. In addition, if the gate signals, which each have a phase offset of about 90 degrees and 270 degrees, respectively, are input at time t 1 , then depth sensing pixels  716  having the 2-tap pixel structure output simultaneously measured pixel signals A 1 ′ and A 3 ′. Thus, the depth sensing pixels  716  each measure pixel signals twice in each time interval Tint, since the depth sensing pixels  716  do not measure each pixel signal A 1 ′, A 2 ′, A 3 ′ or A 4 ′ at the same time. 
     The depth estimator  726  estimates the phase difference based on each digital pixel signal A 0 , A 1 , A 2 , and A 3  according to Equation E5. The depth estimator  726  estimates depth information based on estimated phase differences and outputs depth information, d-hat. 
     Turning now to  FIG. 12 , another 2-tap sampling point plot is indicated generally by the reference numeral  1200 . The plot  1200  shows two actual sampling points t 0  and t 2 , and two interpolated sampling points t 1  and t 3 . 
     This conceptual diagram is provided for explaining estimation of digital pixel signals. In the exemplary embodiment of  FIG. 7 , alternate embodiment depth estimators  726  may be used for compensating the phase error according to the time difference, Tint. For example, the estimator  726  can estimate respective digital signals at estimation time t 2  using a plurality of digital pixel signals already measured at the same time t 2 . Here, the depth sensor  710 , including depth sensing pixel  716  having the 2-tap pixel structure, measures respective pixel signals A′(k−1), A 3 ′(k−1) at time t 1  and A 0 ′(k), A 2 ′(k) at time t 2 , and A 1 ′(k), A 3 ′(k) at time t 3 . 
     Each pixel signal A′(k−1), A 3 ′(k−1), A 0 ′(k), A 2 ′(k), A 1 ′(k), and A 3 ′(k) is converted to a digital signal A(k−1), A 3 ( k− 1), A 0 ( k ), A 2 ( k ), A 1 ( k ), and A 3 ( k ), and stored in memory  724 . Thus, the depth estimator  726  estimates two estimated values according to Equation E6 at time t 2 . 
     Here, the background noise may be assumed constant while the object is moving as a function of time. Using the current and past values of the captured signals, A 1  and A 3 , the compensated signals at which A 0  and A 2  are measured can be estimated. For example, a simple interpolation algorithm may be used. Alternatively, extrapolation may be used in an alternate embodiment. 
     Referring now to Table D, Equation D1 defines an interpolated measurement as a function of the actual measurements at the two actual sampling points. Equation D2 shows the exemplary simple interpolation used here. Equation D3 calculates the phase angle theta from the interpolated and actual measurements. 
     If the depth estimator  726  uses a linear interpolation, for example, then it can estimate values using Equation E7. The depth estimator  726  also estimates the remaining estimated values using the same equation. The depth estimator  726  then calculates the phase difference at time t 2  using A 0 ( k )/A 2 ( k ) and Equation E8. 
     As shown in  FIG. 13 , a 2-tap timing diagram illustrating the estimation process for a digital pixel signal according to the operation of the depth sensor at estimation time is indicated generally by the reference numeral  1300 . Equation E8 is re-written as Equation E9, where Tint is the integration time, Tread is the readout time from pixel out(Ak′) to digital out(Ak), and Tcal is the time that was taken by the depth estimator  726  to calculate or estimate digital pixel signals. The digital pixel signals are provided by the depth estimator  726  according to Equation E10. 
     Turning to  FIG. 14 , plots of simulation results are indicated generally by the reference numeral  1400 . Here, a simulation graph that represents phase difference calculation error in a case where the object  711  moves at the speed of 1 mm/s˜1 m/s is shown in plot  1410 , and a simulation graph that represents phase difference calculation error in a case where the object  711  moves at the speed of 1 mm/s˜1 m/s, but corrected by linear interpolation according to a preferred embodiment of the present disclosure is shown in plot  1420 . 
     Before compensation, in the case of using a conventional algorithm, the phase difference error that is estimation error increases as the integration time increases and/or the speed of a moving object  711  increases. After compensation, in the case of using an algorithm according to an exemplary embodiment method of the present disclosure, the phase difference calculation error is significantly decreased even if the integration time and/or speed of the object increase. 
     Turning now to  FIG. 15 , a 2-tap comparative simulation graph is indicated generally by the reference numeral  1500 . Here, the plot  1510  represents the phase difference calculation error of a conventional method, while the plot  1520  represents the phase difference calculation error of an exemplary embodiment of the present disclosure. 
     As shown in  FIG. 16 , 2-tap comparative simulation graphs are indicated generally by the reference numeral  1600 . Here, the plot  1610  shows the phase difference calculation error without compensation. The plot  1620  shows the phase difference calculation error with conventional compensation. The plot  1630  shows the phase difference calculation error with compensation in accordance with an exemplary embodiment of the present disclosure. The plot  1640  shows the mean squared error for the conventional method  1642  versus an exemplary embodiment of the present disclosure  1644 . 
     Turning to  FIG. 17 , a 1-tap contactless 3D shape measurement system is indicated generally by the reference numeral  1700 . The system  1700  includes a device  1710  and an object  1711 . The device  1710  includes an infra-red (IR) emitter  1712 , a sensor array  1714  including a plurality of sensor pixels  1732 , which each receive reflected light from the object through an IR pass filter  1717 , a correlated double sampling analog-to-digital converter (CDS/ADC) unit  1718  that receives amplitudes or photon counts from the array, and a timing and control signal from a timing and control (T/C) unit  1720 . The T/C unit is also in signal communication with an X-decoder  1722  for the array  1714 , and the IR emitter  1712 . The CDS/ADC unit  1718  passes sampled amplitudes or photon counts to a memory  1724 , which, in turn, provides the sampled amplitudes or photon counts to a depth estimator  1726 . The depth estimator  1726  provides signals indicative of object depth or distance from the emitter and sensor pixels of the device  1710 . 
     In operation, the depth sensor  1710 , object  1711 , IR emitters  1712 , depth sensor array  1714 , sensing pixels  1732 , IR pass filter  1717 , CDS/ADC  1718 , timing controller  1720 , memory  1724  and depth estimator  1726  form an effective system. Here, the depth estimator uses Equation E1, where t is the time difference between the emitted light (EL) and the received light (RL), d is the depth information corresponding to the distance between emitter to object(s) to sensor, and c is the speed of light. 
     The RL can get through the additional lens or lens module located in front of the IR pass filter  1717 . The IR Emitter  1712  may emit modulated IR towards the outside, and can be configured with one or more Light Emitting Diodes (LEDs), Organic Light Emitting Diodes (OLEDs), or Laser Diodes (LDs). 
     Thus, a depth sensing pixel  1732  having a 1-tap pixel structure can measure pixel signals (A 0 ′,A 1 ′,A 2 ′,A 3 ′) in accordance with gate signals (Ga,Gb,Gc,Gd) which have 0-degree, 90-degree, 180-degree, 270-degree phase offsets, respectively. Sensing pixels  1732  accumulate photo-generated electrons introduced by reflected IR or RL incident through the IR pass filter  1717  for a predetermined time period, such as during an integration time, and output pixel signals (A 0 ′,A 1 ′,A 2 ′,A 3 ′) generated by the accumulations in accordance with Equation E2. 
     Pixel signals (A 0 ′,A 1 ′,A 2 ′,A 3 ′) are generated by each pixel  1732 , where Ak′ is as follows, When the phase difference or offset of the gate signal is 0-degrees, k is 0. When the phase difference or offset of the gate signal is 90-degrees, k is 1. When the phase difference or offset of the gate signal is 180-degrees, k is 2. When the phase difference or offset of the gate signal is 270-degrees, k is 3. 
     Here, ak,n is the number of photo-generated electrons in the depth sensor  1732  while inputting the nth gate signal with phase difference according to the ‘k’. N=fm*Tint, where fm is the modulated IR or EL, and Tint is the integration time. 
     Turning now to  FIG. 18 , a 1-tap pixel unit is indicated generally by the reference numeral  1800 . The pixel unit includes a pixel  1832 , such as one of the pixels  1732  of  FIG. 17 . The pixel includes a first region  1821 , which includes a first tap  1822 . 
     Turning to  FIG. 19 , a 1-tap sampling point plot is indicated generally by the reference numeral  1900 . The plot  1900  shows four sampling points, t 0 , t 1 , t 2 , and t 3 . At the zeroth sampling point t 0 , the zeroth phase-offset quadrant photon count is sampled. At the first sampling point t 1 , the first phase-offset quadrant photon count is sampled. At the second sampling point t 2 , the second phase-offset quadrant photon count is sampled. At the third sampling point t 3 , the third phase-offset quadrant photon count is sampled. 
     In operation of the 1-tap structure of  FIGS. 17 ,  18  and  19 , gate signals Ga, Gb, Gc, Gd, which have phase offsets of 0-degree, 90-degree, 180-degree, and 270-degree, respectively, are sequentially applied to the photo-electric converting device or photogate  1822  of the depth sensing pixel  1832 , each of  FIG. 18 . Thus, the photogate  1822  transfers the photo-generated electrons generated by reflected light (RL) to the floating diffusion region (FD) through a transfer gate. 
     On the signal of the timing controller  1720 , the digital CDS/ADC circuit  1718  executes correlated double sampling and analog-to-digital conversion for each pixel signal, including A 0 ′ at time t 0 , A 1 ′ at time t 1 , A 2 ′ at time t 2 , and A 3 ′ at time t 3 , and outputs each digital pixel signal A 0 , A 1 , A 2 , and A 3 . The memory  1724 , which is configured as buffers, receives and stores each digital pixel signal A 0 , A 1 , A 2 , and A 3  outputted from the CDS/ADC circuit  1718 . 
     The depth estimator  1726 , in turn, calculates the phase difference based on each digital pixel signal A 0 , A 1 , A 2 , and A 3 . The phase difference estimated by the depth estimator  1726  is derived from Equation F4 of Table F. 
     Turning now to  FIG. 20 , a 1-tap timing diagram illustrating the estimation process for a digital pixel signal according to the operation of the depth sensor at estimation time is indicated generally by the reference numeral  2000 . In operation, the depth estimator  1726  of  FIG. 17  estimates digital signals from a different estimation time by using a plurality of digital pixel signals (A 1 ( k− 1), A 1 ( k )) already measured and stored at the time when A 0 ′(k) is measured. Thus, first signal estimates may be made using A 2 ( k− 1) and A 2 ( k ), while second signal estimates may be made using A 3 ( k− 1) and A 3 ( k ). If the depth estimator  1726  uses a linear interpolation, then it can estimate each digital pixel signal at the time when the pixel signal (A 0 ′(k)) corresponding to the digital pixel signal (A 0 ( k )) is estimated. 
     The digital pixel signals produced by the depth estimator  1726  are set forth in Equations F5, F6 and F7 of Table F. Thus, the depth estimator  1726  estimates the phase difference based on the measured digital pixel signal (A 0 ( k )) and the estimated digital pixel signals according to Equation F8. Here, the depth estimator  1726  of the depth sensor  1710  can output estimated depth information. 
     Turning now to  FIG. 21 , plots of 1-tap simulation results are indicated generally by the reference numeral  2100 . Here, a simulation graph that represents phase difference calculation error in a case where the object  1711  moves at the speed of 1 mm/s˜1 m/s is shown in plot  2110 , and a simulation graph that represents phase difference calculation error in a case where the object  1711  moves at the speed of 1 mm/s˜1 m/s, but corrected by linear interpolation according to a preferred embodiment of the present disclosure is shown in plot  2120 . 
     Before compensation, in the case of using a conventional algorithm, the phase difference error that is estimation error increases as the integration time increases and/or the speed of a moving object  1711  increases. After compensation, in the case of using an algorithm according to an exemplary embodiment method of the present disclosure, the phase difference calculation error is significantly decreased even if the integration time and/or speed of the object increase. 
     Turning now to  FIG. 22 , a 1-tap comparative simulation graph is indicated generally by the reference numeral  2200 . Here, the plot  2210  represents the phase difference calculation error of a conventional method, while the plot  2220  represents the phase difference calculation error of an exemplary embodiment of the present disclosure. 
     Turning to  FIG. 23 , 1-tap comparative simulation graphs are indicated generally by the reference numeral  2300 . Here, the plot  2310  shows the phase difference calculation error without compensation. The plot  2320  shows the phase difference calculation error with conventional compensation. The plot  2330  shows the phase difference calculation error with compensation in accordance with an exemplary embodiment of the present disclosure. The plot  2340  shows the mean squared error for the conventional method  2342  versus an exemplary embodiment of the present disclosure  2344 . 
     Turning now to  FIG. 24 , a method of depth estimation is indicated generally by the reference numeral  2400 . The method includes a step S 10  to estimate present values using measured previous values and measured next values, Here, the depth sensor including depth sensing pixels  716  of  FIG. 7  having the 2-tap pixel structure estimates values for an estimation point using values that were detected at two detection points adjacent to the estimation point. Next, at a step S 20 , the depth sensor estimates depth information using estimated values and two values detected by depth sensing pixels  716  at the estimation point. The depth sensor  710  determines the phase difference between two estimated values and the two detected values A 0 ( k ), A 2 ( k ). Next, at a step S 30 , the depth sensor estimates depth information based on the frequency of reflected light (RL) using fm, light speed c and phase difference. 
     As shown in  FIG. 25 , an exemplary three-transistor (3T) APS structure is indicated generally by the reference numeral  2500 . The 3T structure includes a photo-diode  2510 , an RX transistor  2520  connected to the photo-diode, a DX transistor  2530  connected to the photo-diode, and an SX transistor  2540  connected to the DX transistor. A preferred 3T structure of the present disclosure provides for reset and select transistor sharing, and has CDS operation enabled due to having floating diffusion (FD). An alternate embodiment shared structure is contemplated. 
     Turning to  FIG. 26 , an exemplary four-transistor (4T) APS structure is indicated generally by the reference numeral  2600 . The 4T structure includes a photo-diode  2610 , a TX transistor  2612  connected to the photo-diode, an RX transistor  2620  connected to the TX transistor, a DX transistor  2630  connected to the TX transistor, and an SX transistor  2640  connected to the DX transistor. An alternate embodiment shared structure is contemplated. 
     Turning to  FIG. 27 , a first exemplary five-transistor (5T) APS structure is indicated generally by the reference numeral  2700 . The 5T structure includes a photo-diode  2710 , a TX transistor  2712  connected to the photo-diode, a GX transistor  2714  connected to the TX transistor, an RX transistor  2720  connected to the TX transistor, a DX transistor  2730  connected to the TX transistor, and an SX transistor  2740  connected to the DX transistor. An alternate embodiment shared structure is contemplated. 
     Turning to  FIG. 28 , a second exemplary 5T APS structure is indicated generally by the reference numeral  2800 . The second 5T structure includes a photo-diode  2810 , a PX transistor  2811  connected to the photo-diode, a TX transistor  2812  connected to the PX transistor, an RX transistor  2820  connected to the TX transistor, a DX transistor  2830  connected to the TX transistor, and an SX transistor  2840  connected to the DX transistor. An alternate embodiment shared structure is contemplated. 
     Turning to  FIG. 29 , a contactless 3D shape measurement system using a two-chip solution is indicated generally by the reference numeral  2900 . The contactless 3D shape measurement device  2900  uses depth compensation for modulated optical time-of-flight phase estimation, and includes an emitter for transmitting light, an object or reflector  2912  for reflecting light, a depth sensor  2914  for receiving reflected light from the object, a color sensor  2918  for receiving ambient light from the object, and a signal processor  2916  in signal communication with the depth sensor and the color sensor for subtracting the ambient light from the reflected light and providing 3D information. 
     Turning to  FIG. 30 , a contactless 3D shape measurement system using a single-chip solution is indicated generally by the reference numeral  3000 . The contactless 3D shape measurement device  3000  uses depth compensation for modulated optical time-of-flight phase estimation, and includes a light source for transmitting light, an object or reflector  3012  for reflecting light, a single-chip color and depth sensor  3014  for receiving reflected light from the object and for receiving ambient light from the object, and a signal processor  3016  in signal communication with the combined color and depth sensor for subtracting the ambient light from the reflected light and providing 3D information. 
     Turning now to  FIG. 31 , a contactless 3D shape measurement system is indicated generally by the reference numeral  3100 . The contactless 3D shape measurement system  3100  includes a light source  3110  for transmitting light, an object  3112  for reflecting light, a pixel array  3114  for receiving light, a control unit  3116  for controlling the light source, a row address decoder  3118 , and a column address decoder  3122 , a row driver  3120  connected between the row address decoder and the pixel array, a column driver  3124  connected between the column address decoder and the pixel array, a sample and hold (S/H) register connected to the column driver, and analog-to-digital converter (ADC) connected to the S/H register, and an ISP  3130  connected to the ADC. 
     As shown in  FIG. 32 , a contactless 3D shape measurement system is indicated generally by the reference numeral  3200 . The contactless 3D shape measurement system  3200  includes a central processing unit (CPU)  3210  connected to a system bus  3220 , a single or multi-chip sensor  3230  connected to the system bus, and a memory  3240  connected to the system bus. 
     Turning now to  FIG. 33 , a contactless 3D shape measurement partial circuit and schematic signal diagram are indicated generally by the reference numeral  3300 . The partial circuit  3310  includes a photo-diode  3312  connected to a reset transistor  3314  and a floating diffusion transistor  3316 , which, in turn, is connected to a selection transistor  3318 . A grounding transistor  3320  and an analog-to-digital converter  3322  are each connected to the selection transistor. The signal diagram  3330  includes a reset state  3332 , in which the PD, RG and RD signal levels are low; a right-after-reset state, in which the PD level is rising, the RG level is high, and the RD level is low; and an integration state, in which the PD and RG levels are high and the RD level is low. 
     As shown in  FIG. 34 , a contactless 3D shape measurement partial circuit and schematic signal diagram are indicated generally by the reference numeral  3400 . The partial circuit  3410  includes a photo-diode  3412  connected to a pass transistor  3413 , a reset transistor  3414  and a floating diffusion transistor  3416  connected to the pass transistor, and a selection transistor  3418  connected to the floating diffusion transistor. A grounding transistor  3420  and an analog-to-digital converter  3422  are each connected to the selection transistor. 
     The signal diagram  3430  includes an integration state in which the PD, TG, FD and RG levels are high while the RD level is low; an FD reset state  3432  in which the PD and TG levels are high while the FD, RG and RD levels are low; a right-after-reset state, in which the PD, TG and RG levels are high while the FD and RD levels are low; and a signal transfer state  3435  in which the FD and RG levels are high while the PD, TG and RD levels are low. The timing diagram  3440  includes a reset signal activated first, a TG signal activated second, and an output signal that is step-wise decreasing after the reset signal. 
     Turning to  FIG. 35 , APS structures are indicated generally by the reference numeral  3500 . An exemplary three-transistor (3T) APS structure  3530  includes a photo-diode  3531 , an RX transistor  3532  connected to the photo-diode, a DX transistor  3533  connected to the photo-diode, and an SX transistor  3534  connected to the DX transistor. The 3T structure features a simple process, a high fill factor, pixel reset noise, and a low signal-to-noise ratio. 
     An exemplary four-transistor (4T) APS structure  3540  includes a photo-diode  3541 , a TX transistor  3545  connected to the photo-diode, an RX transistor  3542  connected to the TX transistor, a DX transistor  3543  connected to the TX transistor, and an SX transistor  3544  connected to the DX transistor. The 4T structure features a process for a low shallow potential photodiode, a low fill factor, a low dark level, higher sensitivity, CDS operation, and impractical SFCM. 
     An exemplary five-transistor (5T) APS structure  3550  includes a photo-diode  3551 , a TX transistor  3555  connected to the photo-diode, a GX transistor  3552  connected to the TX transistor, an RX transistor  3553  connected to the TX transistor, a DX transistor  3556  connected to the TX transistor, and an SX transistor  3554  connected to the DX transistor. The 5T structure features an addressed readout, full random access, possible single CDS, and the lowest fill factor. 
     A photogate structure  3560  includes a photodiode  3561 , a PX transistor  3567  connected to the photodiode, a TX transistor  3565  connected to the PX transistor, an RX transistor  3562  connected to the TX transistor, a DX transistor  3563  connected to the TX transistor, and an SX transistor  3564  connected to the DX transistor. The photogate structure features a simple process, operation like the 4T structure, signal charge shifting by PG and TG pulses, an additional signal line, and low blue response. 
     In the claims, any means-plus-function clauses are intended to cover the structures described herein as performing the recited function, and not only structural equivalents but also equivalent structures. Therefore, it is to be understood that the foregoing is illustrative of the inventive concept and is not to be construed as limited to the specific embodiments disclosed, and that modifications to the disclosed exemplary embodiments, as well as other exemplary embodiments, are intended to be included within the scope of the appended claims. The inventive concept is defined by the following claims, with equivalents of the claims to be included therein. 
     These and other features of the present disclosure may be readily ascertained by one of ordinary skill in the pertinent art based on the teachings herein. Although illustrative embodiments have been described herein with reference to the accompanying drawings, it is to be understood that the present disclosure is not limited to those precise embodiments, and that various other changes and modifications may be effected therein by those of ordinary skill in the pertinent art without departing from the scope or spirit of the present disclosure. All such changes and modifications are intended to be included within the scope of the present disclosure as set forth in the appended claims. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE A 
               
               
                   
                   
               
             
             
               
                   
                 A 0  ≈ α + βcosθ 
                 (Eqn. A1) 
               
               
                   
                 A 2  ≈ α − βcosθ 
                 (Eqn. A2) 
               
               
                   
                 A 1  ≈ α + βsinθ 
                 (Eqn. A3) 
               
               
                   
                 A 3  ≈ α − βsinθ 
                 (Eqn. A4) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           t 
                           Δ 
                         
                         = 
                         
                           
                             2 
                             ⁢ 
                             d 
                           
                           c 
                         
                       
                     
                   
                 
                 (Eqn. A5) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           A 
                           k 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                               
                                 a 
                                 
                                   k 
                                   , 
                                   n 
                                 
                               
                               ⁢ 
                               N 
                             
                           
                           = 
                           
                             
                               f 
                               m 
                             
                             ⁢ 
                             
                               T 
                               int 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. A6) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           θ 
                           ^ 
                         
                         = 
                         
                           
                             2 
                             ⁢ 
                             
                               πf 
                               m 
                             
                             ⁢ 
                             
                               t 
                               Δ 
                             
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 - 
                                 
                                   A 
                                   3 
                                 
                               
                               
                                 
                                   A 
                                   0 
                                 
                                 - 
                                 
                                   A 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. A7) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           d 
                           ^ 
                         
                         = 
                         
                           
                             c 
                             
                               4 
                               ⁢ 
                               
                                 πf 
                                 m 
                               
                             
                           
                           ⁢ 
                           
                             θ 
                             ^ 
                           
                         
                       
                     
                   
                 
                 (Eqn. A8) 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
           
               
                   
                 TABLE B 
               
               
                   
                   
               
             
             
               
                   
                 
                   
                     
                       
                         
                           P 
                           
                             IR 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             source 
                           
                         
                         ∝ 
                         
                           
                             
                               N 
                               e 
                             
                             ⁢ 
                             
                               
                                 A 
                                 imagel 
                               
                               
                                 A 
                                 pixel 
                               
                             
                             ⁢ 
                             hc 
                           
                           
                             
                               p 
                               ⁡ 
                               
                                 [ 
                                 
                                   D 
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               k 
                               lens 
                             
                             ⁢ 
                             
                               QE 
                               ⁡ 
                               
                                 ( 
                                 λ 
                                 ) 
                               
                             
                             ⁢ 
                             
                               λT 
                               int 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. B1) 
               
               
                   
                   
               
               
                   
                 P opt (t) = P 0  + P 0  · sin(2πf mod t) 
                 (Eqn. B2) 
               
               
                   
                 N el (t) = B meas  + A meas  · sin(2πf mod t +           )  
                 (Eqn. B3) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         φ 
                         = 
                         
                           arctan 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 - 
                                 
                                   A 
                                   2 
                                 
                               
                               
                                 
                                   A 
                                   1 
                                 
                                 - 
                                 
                                   A 
                                   3 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
                 (Eqn. B4) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         L 
                         = 
                         
                           
                             c 
                             
                               4 
                               ⁢ 
                               
                                 π 
                                 · 
                                 
                                   f 
                                   mod 
                                 
                               
                             
                           
                           · 
                           φ 
                         
                       
                     
                   
                 
                 (Eqn. B5) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         ΔL 
                         ∝ 
                         
                           
                             B 
                           
                           
                             
                               f 
                               mod 
                             
                             ⁢ 
                             A 
                           
                         
                       
                     
                   
                 
                 (Eqn. B6) 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
           
               
                   
                 TABLE C 
               
               
                   
                   
               
             
             
               
                   
                 t 1  = t 0  + T int   
                 (Eqn. C1) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         θ 
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     0 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. C2) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         θ 
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     3 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     0 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. C3) 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
           
               
                   
                 TABLE D 
               
               
                   
                   
               
             
             
               
                   
                 Â 1 (k) = f(A 1 (k), A 1 (k−1)) 
                 (Eqn. D1) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     t 
                                     2 
                                   
                                   - 
                                   
                                     t 
                                     1 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     t 
                                     3 
                                   
                                   - 
                                   
                                     t 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               t 
                               3 
                             
                             - 
                             
                               t 
                               1 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. D2) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             θ 
                             ^ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. D3) 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
           
               
                   
                 TABLE E 
               
               
                   
                   
               
             
             
               
                   
                 
                   
                     
                       
                         
                           t 
                           Δ 
                         
                         = 
                         
                           
                             2 
                             ⁢ 
                             d 
                           
                           c 
                         
                       
                     
                   
                 
                 (Eqn. E1) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           A 
                           k 
                           ′ 
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             a 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E2) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           θ 
                           ^ 
                         
                         = 
                         
                           
                             2 
                             ⁢ 
                             
                               πf 
                               m 
                             
                             ⁢ 
                             
                               t 
                               Δ 
                             
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 - 
                                 
                                   A 
                                   3 
                                 
                               
                               
                                 
                                   A 
                                   0 
                                 
                                 - 
                                 
                                   A 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E3) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           d 
                           ^ 
                         
                         = 
                         
                           
                             c 
                             
                               4 
                               ⁢ 
                               
                                 πf 
                                 m 
                               
                             
                           
                           ⁢ 
                           
                             θ 
                             ^ 
                           
                         
                       
                     
                   
                 
                 (Eqn. E4) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           θ 
                           ^ 
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     0 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     0 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E5) 
               
               
                   
                   
               
               
                   
                 Â 1 (k) = f(A 1 (k), A 1 (k−1)) 
                 (Eqn. E6) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     t 
                                     2 
                                   
                                   - 
                                   
                                     t 
                                     1 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     t 
                                     3 
                                   
                                   - 
                                   
                                     t 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               t 
                               3 
                             
                             - 
                             
                               t 
                               1 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E7) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             θ 
                             ^ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E8) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             θ 
                             ^ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E9) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           θ 
                           ^ 
                         
                         = 
                         
                           
                             2 
                             ⁢ 
                             
                               πf 
                               m 
                             
                             ⁢ 
                             
                               t 
                               Δ 
                             
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 - 
                                 
                                   A 
                                   3 
                                 
                               
                               
                                 
                                   A 
                                   0 
                                 
                                 - 
                                 
                                   A 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E10) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     int 
                                   
                                   + 
                                   
                                     T 
                                     read 
                                   
                                   + 
                                   
                                     T 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       al 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 T 
                                 int 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               2 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                             
                             + 
                             
                               T 
                               read 
                             
                             + 
                             
                               T 
                               cal 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E11) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             3 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     int 
                                   
                                   + 
                                   
                                     T 
                                     read 
                                   
                                   + 
                                   
                                     T 
                                     cal 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 T 
                                 int 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               2 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                             
                             + 
                             
                               T 
                               read 
                             
                             + 
                             
                               T 
                               cal 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. E12) 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
             
           
               
                   
                 TABLE F 
               
               
                   
                   
               
             
             
               
                   
                 
                   
                     
                       
                         
                           t 
                           Δ 
                         
                         = 
                         
                           
                             2 
                             ⁢ 
                             d 
                           
                           c 
                         
                       
                     
                   
                 
                 (Eqn. F1) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           A 
                           k 
                           ′ 
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             a 
                             
                               k 
                               , 
                               n 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. F2) 
               
               
                   
                   
               
               
                   
                 N = fm * Tint 
                 (Eqn. F3) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           θ 
                           ^ 
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     3 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     0 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. F4) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             1 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     3 
                                     ⁢ 
                                     
                                       T 
                                       int 
                                     
                                   
                                   + 
                                   
                                     T 
                                     read 
                                   
                                   + 
                                   
                                     T 
                                     cal 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 T 
                                 int 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               4 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                             
                             + 
                             
                               T 
                               read 
                             
                             + 
                             
                               T 
                               cal 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. F5) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             2 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       T 
                                       int 
                                     
                                   
                                   + 
                                   
                                     T 
                                     read 
                                   
                                   + 
                                   
                                     T 
                                     cal 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               4 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                             
                             + 
                             
                               T 
                               read 
                             
                             + 
                             
                               T 
                               cal 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. F6) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               A 
                               ^ 
                             
                             3 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     int 
                                   
                                   + 
                                   
                                     T 
                                     read 
                                   
                                   + 
                                   
                                     T 
                                     cal 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               3 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               4 
                               ⁢ 
                               
                                 T 
                                 int 
                               
                             
                             + 
                             
                               T 
                               read 
                             
                             + 
                             
                               T 
                               cal 
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. F7) 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             θ 
                             ^ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   3 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   
                                     A 
                                     ^ 
                                   
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 (Eqn. F8)