Error analysis of CCD-based point source centroid computation under the background light

Xiaoyu Ma; Changhui Rao; Hanqing Zheng

doi:10.1364/OE.17.008525

1. Introduction

In the field of target tracking [1], laser triangulation system [2], star tracker [3],wavefront sensing [4,5], etc., a Charge-Coupled-Device (CCD) is generally placed in the focal plane of an imaging lens or lens array to detect the centroid of a point source. As the CCD-based point source centroid computation (PSCC) technology has been widely used, the PSCC precision directly affects system precision, and some error sources can be eliminated when an optimum threshold is set. The purposes of this paper are to establish the analytical function to describe the CCD-based PSCC detecting precision and to find the optimum threshold to increase the PSCC precision.

There are many articles concerning the PSCC accuracy, however, most of them are based on a particular condition. In Ref. 6, Tyler and Fried discussed the arriving angle error when using a quadrant detector as the position detector, but the arriving angle error function in their work is merely suitable for the quadrant detector which is used in the condition of only photon noise existing. In Ref. 7, Cao analyzed the influence of photon noise, readout noise and sampling error to the centroid-detecting precision without the factors of background light and background level. Jiang introduced the influence of background level but neglected that of background light in Ref. 8.

When the systems based on PSCC are used in the realistic environment, the background noise is always present due to the existence of the sky background light and the stray light of the system. Thus, the precision of PSCC is related directly to the error sources of photon noise (caused by the random eradiating of signal light source), readout noise (caused by the readout circuit noise of CCD), background noise (caused by the scattered light of sky or system light sources), background level (caused by the offset voltage of CCD), and sampling error (caused by the discrete sampling of spot). In Section 2, the distributions of each error source are listed, as well as the introduction of PSCC algorithm formula.

Considering the distributing is not the same for each error source, in Section 3, the error sources are classified into two sorts: the 1^st class error source is defined as the error sources affecting each pixel within the detection window equally (including the background photon noise, the readout noise and the background level), and the 2^nd class error source is defined as the error sources which only affect the area of light spot (including the signal photon noise and the sampling error). These two kinds of error sources could bring the displacement error and the wobbling error to PSCC. The displacement error results from the non-zero average value of error sources, and the wobbling error results from the fluctuation of error sources. The CCD-based PSCC precision formula conducted in Section 3 shows the relationship between the PSCC error and all the diversified error sources. This formula is more universal than the precision formulas in Ref. 6, 7 and 8.

Since some error sources can be eliminated when an optimum threshold is set, in Section 4, how the threshold works on each error source has been analyzed one by one. The precision formulas changing with threshold are established. Having analyzed the theoretical formulas and the numerical calculation results, the best threshold under some given condition is put forward. In Section 5 and Section 6, the simulation and experimental methods are used to validate the theoretical conclusion, the simulation and experimental results perfectly match the theoretical conclusion.

2. The CCD-based PSCC algorithm and the error sources List of symbols

The symbols used in this paper are based on those conventionally employed in the optics literatures. Symbols are generally defined in the sections in which they appear, and their meaning should always be clear from the context.

The schematic diagram of the CCD-based PSCC is shown in Fig. 1. Centroid computation is based on a certain system of coordinates. The centroid point coordinate is the weighted mean value when the weight coefficients are the coordinate values. The PSCC formula on the x-axis is expressed by

x_{c} = \frac{\sum_{ij}^{L, M} x_{ij} N_{ij}}{\sum_{ij}^{L, M} N_{ij}} = \frac{U}{V} .

x_c is the centroid coordinate, x_ij is the coordinate of a pixel on CCD, N_ij is the received photon numbers of a pixel with coordinate (i, j), L×M is the size of aperture, and $U = \sum_{ij}^{L, M} x_{ij} N_{ij}, V = \sum_{ij}^{L, M} N_{ij} .$

Table 1. The list of symbols and their explanations

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Fig. 1. Schematic diagram of CCD-based PSCC

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The error sources of CCD-based PSCC and their distributions are listed in Table 2. [9,10]:

Table 2. The error sources of CCD-based PSCC and their distributions

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Here $\overline{N_{sij}}$ is the average signal photon number of a pixel with coordinate (i, j), $\overline{N_{bij}}$ is the background photon number of a pixel with coordinate (i, j), and σ_r is the readout noise of CCD.

3. Error analysis of the centroid computation

For the x-axis is symmetrical with the y-axis, only the PSCC error on x-axis is analyzed, which is similar with that on the y-axis.

The error sources can be classified into two categories by the difference in effect areas.

The 1^st class error source equally affects each pixel within the detection window, including the background photon noise, the readout noise and the background level.

The background photon noise obeys Poisson distribution. However, when the average value of photon numbers is greater than ten on each pixel [11], it can be considered to obey Gauss distribution with the average value and the variation both equaling to the average photon numbers. The background level of the CCD can be considered as a constant and the readout noise of CCD obeys Gauss distribution with mean value zero. Therefore, the 1^st class error source on one pixel N_Bij is equal to N_bij + N_dij + N_rij , here N_bij is the background noise of a pixel with coordinate (i, j), N_dij is the background level of a pixel with coordinate (i, j), N_rij is the readout noise of a pixel with coordinate (i, j). N_Bij can be considered to obey Gauss distribution of which the average value $\overline{N_{Bij}}$ is $\overline{N_{bij}} + \overline{N_{dij}} + \overline{N_{rij}}$ and the variation σ_B ² is $σ_{r}^{2} + \overline{N_{bij}}$ , here $\overline{N_{bij}}$ is the average value of background photon number of a pixel with coordinate (i, j), $\overline{N_{dij}}$ is the average value of background level of a pixel with coordinate (i, j), $\overline{N_{rij}}$ is the average value of readout noise of a pixel with coordinate (i, j) The 2^nd class error source only affects the area of light spot, including the signal photon noise and the sampling error.

As defined in Eq. (1), N_ij is the received photon numbers of a pixel with coordinate (i, j) , so:

N_{ij} = N_{sij} + N_{bij} + N_{rij} + N_{dij} = N_{sij} + N_{Bij} .

N_sij is the signal photon numbers.

Equation (2) proves that N_ij is the sum of the signal photon numbers and the 1^st class error source.

Suppose that $U_{s} = \sum_{ij}^{L, M} x_{i} N_{sij} {, V}_{s} = \sum_{ij}^{L, M} N_{sij}, U_{B} = \sum_{ij}^{L, M} x_{i} N_{Bij}, V_{B} = \sum_{ij}^{L, M} N_{Bij},$ then:

U = \sum_{ij}^{L, M} x_{i} N_{ij} = \sum_{ij}^{L, M} x_{i} (N_{sij} + N_{Bij}) = U_{s} + U_{B} .

V = \sum_{ij}^{L, M} N_{ij} = \sum_{ij}^{L, M} (N_{sij} + N_{Bij}) = V_{s} + V_{B} .

The position of the detected centroid is [8]:

x_{c} = \frac{U}{V} = \frac{U_{s} + U_{B}}{V_{s} + V_{B}} = \frac{V_{s}}{V} x_{s} + \frac{V_{B}}{V} x_{B} .

x_s is the sensed centroid without background light noise and readout noise, $x_{s} = \frac{U_{s}}{V_{s}}, x_{B}$ is the centroid position of the 1^st class error resource, $x_{B} = \frac{U_{B}}{V_{B}} .$

However, x_s is not the actual centroid position of the spot due to the sampling error. Suppose the actual centroid position is x_p and the sampling error is σ_S, then:

x_{s} = x_{p} + σ_{s} .

Equation (5) can be expressed as:

x_{c} = \frac{V_{s}}{V} (x_{p} + σ_{s}) + \frac{V_{B}}{V} x_{B} .

Equation (7) shows that centroid computation error can be divided into two parts: the displacement error and the wobbling error.

3.1 The displacement error

The displacement error is defined as the difference between the actual centroid of spot x_p and the detected centroid x_c:

σ_{p} = x_{p} - x_{c} .

From the simultaneous Eqs. of Eq. (7) and Eq. (8), the displacement error σ_p can be expressed as:

σ_{p} = \frac{V_{B}}{V} (x_{s} - x_{B}) - σ_{S} .

According to the conclusion in Ref. 11 by Zhang, the sampling error is:

σ_{S} = \frac{\sum_{i = 0}^{L - 1} {[\erf (\frac{i + 1 - x_{p}}{\sqrt{2} σ_{A}}) - \erf (\frac{i - x_{p}}{\sqrt{2} σ_{A}})] (i + 0.5)}}{\erf (\frac{L - x_{p}}{\sqrt{2} σ_{A}}) + \erf (\frac{x_{p}}{\sqrt{2} σ_{A}})} - x_{p} .

i is the position of pixel, σ_A is the Gauss width of spot, erf (…) is the error accumulation function which is defined as $\erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} dt .$

3.2 The wobbling error

The wobbling error is defined as the variance of the detected centroid position and can be derived from Eq. (1) as [7]:

σ_{x_{c}}^{2} = \frac{U^{2}}{V^{4}} σ_{V}^{2} + \frac{1}{V^{2}} σ_{U}^{2} - \frac{2 U}{V^{3}} σ_{UV} .

σ_{x_c} ² is the variance of the detected centroid position, σ_v ² is the variance of V which is defined in Eq. (1), σ_U ² is the variance of U which is defined in Eq. (1), and σ_UV is the covariance of V and U which are defined in Eq. (1)

Since the error sources between each pixel are not correlative, the wobbling error can be expressed as:

σ_{x_{c}}^{2} = \frac{U^{2} \sum_{i, j}^{L, M} S_{ij}^{2}}{V^{4}} + \frac{\sum_{i, j}^{L, M} x_{i}^{2} S_{ij}^{2}}{V^{2}} - \frac{2 U \sum_{i, j}^{L, M} x_{i} S_{ij}^{2}}{V^{3}} .

S_ij ² is the variance of error sources of the pixel with coordinate (i, j).

Because the 1^st class error source is not correlative with the 2^nd class error source, the S_ij ² can be calculated by Eq. (13).

S_{ij}^{2} = σ_{p}^{2} + σ_{B}^{2} = \overline{N_{sij}} + \overline{N_{bij}} + σ_{r}^{2} .

From the simultaneous Eqs. of Eq. (12) and Eq. (13) we get:

σ_{x_{c}}^{2} = \frac{U^{2} \sum_{i, j}^{L, M} (\overline{N_{sij}} + \overline{N_{bij}} + σ_{r}^{2})}{V^{4}} + \frac{\sum_{i, j}^{L, M} x_{i}^{2} (\overline{N_{sij}} + \overline{N_{bij}} + σ_{r}^{2})}{V^{2}} - \frac{2 U \sum_{i, j}^{L, M} x_{i} (\overline{N_{sij}} + \overline{N_{bij}} + σ_{r}^{2})}{V^{3}} .

The wobbling error caused by the signal photon noise is

σ_{x_{s}}^{2} = \frac{\sum_{i, j}^{L, M} x_{i}^{2} \overline{N_{sij}}}{V^{2}} - \frac{U_{s}^{2}}{V^{3}} = \frac{V_{s} σ_{A}^{2}}{V^{2}} + \frac{1}{V_{s}} x_{s}^{2} - \frac{1}{V} x_{c}^{2} .

σ_A is the Gauss width of spot.

The wobbling error caused by the readout noise is [7]

σ_{x_{r}}^{2} = \frac{U^{2} \sum_{i, j}^{L, M} σ_{r}^{2}}{V^{4}} + \frac{\sum_{i, j}^{L, M} x_{i}^{2} σ_{r}^{2}}{V^{2}} - \frac{2 U \sum_{i, j}^{L, M} x_{i} σ_{r}^{2}}{V^{3}} = \frac{σ_{r}^{2} LM}{V^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) .

The wobbling error caused by the background photon noise is

σ_{x_{b}}^{2} = \frac{U^{2} \sum_{i, j}^{L, M} \overline{N_{bij}}}{V^{4}} + \frac{\sum_{i, j}^{L, M} x_{i}^{2} \overline{N_{bij}}}{V^{2}} - \frac{2 U \sum_{i, j}^{L, M} x_{i} \overline{N_{bij}}}{V^{3}} = \frac{V_{b}}{V^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) .

To sum up, the wobbling error is

σ_{x_{c}}^{2} = σ_{x_{r}}^{2} + σ_{x_{s}}^{2} + σ_{x_{b}}^{2}

3.3 The variance of the PSCC error

The variance of the PSCC error is the summation of the variance of the displacement error and the wobbling error. That is:

σ_{x}^{2} = σ_{p}^{2} + σ_{x_{c}}^{2}

The V, V_s, V_b and σ_r in Eq. (19) are expressed by photon counts, but usually, the output of CCD is expressed in ADU. While one photon count can produce κ ADUs in the output, Eq. (19) can be modified as:

σ_{x}^{2} = \frac{σ_{B}^{2} LM}{V^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) + \frac{κ V_{s} σ_{A}^{2}}{V^{2}} + \frac{κ}{V_{s}} x_{s}^{2} - \frac{κ}{V} x_{c}^{2} + {[\frac{V_{B}}{V} (x_{s} - x_{B}) - σ_{S}]}^{2} .

In Tyler and Fried’s Literature [6], they considered the average value of noise is zero, and the detector only has 2×2 pixels. Furthermore, the sampling error is shielded too. The angular measurement error is given by the expression:

σ_{θ} = \frac{π {[{(\frac{3}{16})}^{2} + {(\frac{n}{8})}^{2}]}^{\frac{1}{2}}}{SN R_{v}} \frac{λ}{D} .

${SNR}_{v} = \frac{V_{s}}{σ_{N}},$ σ_N is the mean-square error of two quadrant, D is the diameter of aperture, and λ is the wavelength of the light used.

When the object is very small (point source), n is very small in equation (21), so:

σ_{θ} = \frac{3 π}{16} \frac{1}{SNRv} \frac{λ}{D} .

We must have the cognizance of that, in Ref. 6, Tyler just wanted to determine the theoretical limits of a sensor design. To achieve the theoretical limits, only the signal photon noise in Eq. (19) should be considered, that is:

σ_{x_{1}} = \frac{σ_{A}}{\sqrt{V_{s}}} .

To compare Eq. (22) and Eq. (23), σ_θ should be transformed to σ_x. σ_x is equal to σ_θ multiplied by f and σ_A equaled to 0.431 $\frac{λ}{D} f$ for a circle aperture diffraction. As mentioned in Eq. (21), SNR_v equals to $\frac{V_{s}}{σ_{N}},$ , σ_N is the mean-square error of two quadrant, so σ_N equals to $\sqrt{\frac{V_{s}}{2}}$ . When only the signal photon noise exists and x_s is zero, Eq. (19) can be modified as:

σ_{x} = \frac{1.37}{{SNR}_{v}} σ_{A} = 0.97 \frac{σ_{A}}{\sqrt{V_{s}}} .

Compare Eq. (23) with Eq. (24), σ_x approximates to σ _x₁, the 0.03 difference comes from the difference between the Airy spot and the Gauss spot we used in Eq. (22) and Eq. (23).

If no background light and background level are present, and the sampling error is not considered either, which means N_bij and N_dij are both zero. Then, Eq. (20) can be modified as:

σ_{x_{2}}^{2} = \frac{σ_{r}^{2} LM}{V_{s}^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) + \frac{κ σ_{A}^{2}}{V_{s}} .

Equation (25) accords with Eq. (6) plus Eq. (15) in Ref. 7 which is analyzed by Cao.

When the background light noise, the sampling error and $\frac{κ}{V_{s}} x_{s}^{2} - \frac{κ}{V} x_{c}^{2}$ are ignored, Eq. (20) can be modified as:

σ_{x_{3}}^{2} = \frac{V_{b}^{2}}{V^{2}} [\frac{σ_{r}^{2} LM}{V_{b}^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) + \frac{κ V_{s}}{V_{b}^{2}} σ_{A}^{2} + {(x_{d} - x_{s})}^{2}]

Suppose that $sbr = \frac{V_{b}}{V_{s}},$ , Eq. (20) can be expressed as:

σ_{x_{4}}^{2} = \frac{1}{{(1 + sbr)}^{2}} [\frac{σ_{r}^{2} LM}{V_{b}^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) + {sbr}^{2} \frac{κ σ_{A}^{2}}{V_{s}} + {(x_{d} - x_{s})}^{2}] .

Equation (27) accords with Eq. (13) in Ref. 8 which is analyzed by Jiang.

Compared with the classical Eq. (22), Eq. (25) and Eq. (27), Eq. (20) is more universal and more accurately describes the PSCC error.

With the development of optoelectronic technology, such as the Electron-Multiplying CCD (EMCCD) [9] or the APD-based array [12], the readout noise and the background level can be reduced to zero, and the sampling error can be ignored when there are enough pixels in one aperture [7]. However, the photon noise and the background noise would exist stably if the systems based on PSCC are used in a realistic environment. Then, the PSCC error formula is:

σ_{x_{4}}^{2} = \frac{σ_{b}^{2} LM}{{(V_{s} + V_{b})}^{2}} (\frac{L^{2} - 1}{12} + x_{c}^{2}) + \frac{κ V_{s} σ_{A}^{2}}{{(V_{s} + V_{b})}^{2}} + \frac{κ}{V_{s}} x_{s}^{2} - \frac{κ}{{(V_{s} + V_{b})}^{2}} x_{c}^{2} + {[\frac{V_{b}}{(V_{s} + V_{b})} (x_{s} + x_{b})]}^{2} .

4. The optimum threshold chosen

From the analysis above, the 1^st class error source sharply deduces the detecting accuracy. As it affects each pixel in aperture equally, its influence can be eliminated by setting a threshold. Before calculating the centroid, the output values of each pixel are subtracted by the threshold and those originally negative ones are set to zero. Obviously, if the threshold is set too low, the influence of the 1^st class error source would not be degraded effectively. On the contrary, if the threshold is set too high, many available signal photons would be cut off. Therefore, there exists an optimum threshold where the lowest PSCC error could be achieved. In order to obtain this value, it is necessary to respectively discuss the relationships between V_s, σ ² _A, V_B, σ_B ², x_c, x_s, x_B, σ_S and the threshold.

4.1 The relationship between the threshold and the 1^st class error source

4.1.1 The relationship between the threshold and V_B

As mentioned above, the 1^st class error source of one pixel N_Bij can be considered to obey Gauss distribution with the average value $\overline{N_{Bij}}$ which is the summation of $\overline{N_{bij}}, \overline{N_{dij}}$ and $\overline{N_{rij}}$ .

The variation σ_B ² is σ_r ² plus $\overline{N_{bij}}$ . The probability density function of N_Bij is:

P (N_{Bij}) = \frac{1}{\sqrt{2 π} σ_{B}} \exp [- \frac{{(N_{Bij} - \overline{N_{B}})}^{2}}{2 σ_{B}^{2}}] .

When the threshold is T, the average value of the 1^st class error source in one pixel N_Bij is:

\overline{N_{Bij}} (T) = \int_{T}^{\infty} [(N_{Bij} - T) \cdot P (N_{Bij})] d N_{Bij} .

The sum of the 1^st class error source in an aperture is

V_{B} (T) = \overline{N_{Bij}} (T) \cdot LM .

4.1.2 The relationship between the threshold and σ_B²

When the threshold is T, the second-order moment of the 1^st class error source in one pixel is:

\overline{N_{Bij}^{2}} (T) = \int_{T}^{\infty} [{(N_{Bij} - T)}^{2} \cdot P (N_{Bij})] d N_{Bij} .

Since the first-order moment of the 1^st class error source in one pixel is shown as Eq. (30), the variance of the 1^st class error source in one pixel is

σ_{B}^{2} (T) = \int_{T}^{\infty} [{(N_{Bij} - T)}^{2} \cdot P (N_{Bij})] d N_{Bij} - {\int_{T}^{\infty} [(N_{Bij} - T) \cdot P (N_{Bij})] d N_{Bij}}^{2} .

4.1.3 The relationship between the threshold and x_B

Since the centroid of the 1^st class error resource is $x_{B} = \frac{U_{B}}{V_{B}}, x_{B}$ , x_B can be express as

x_{B} = \frac{U_{B}}{V_{B}} = \frac{V_{b}}{V_{B}} x_{b} + \frac{V_{b}}{V_{B}} x_{d} + \frac{V_{r}}{V_{B}} x_{r} .

x_b is the centroid of the background light, x_d is the centroid of the background level, x_r is the centroid of the readout noise.

Summarized in the discussion in Table 3, the centroid of the 1^st class noise source is [8]

x_{B} (T) = {\begin{array}{c} 0, T < \overline{N_{B}} - 3 σ_{B} \\ x_{B} = \frac{lm σ_{B}}{V_{B} (T) + lm σ_{B}} x_{s}, \overline{N_{B}} - 3 σ_{B} \leq T \leq \overline{N_{B}} + 3 σ_{B} \\ x_{s}, T > \overline{N_{B}} + 3 σ_{B} \end{array}

Table 3. The relationship between the threshold and x_B

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4.2 The relationship between the threshold and the 2^nd class error source

4.2.1 The relationship between the threshold and V_s

The signal photon counts N_sij, collected by a pixel of which the coordinate is (i, j), are expressed as

N_{sij} = \frac{V_{s}}{4} [\erf (\frac{i + 0.5 - x_{s}}{\sqrt{2} σ_{A}}) - \erf (\frac{i - 0.5 - x_{s}}{\sqrt{2} σ_{A}})] \cdot [\erf (\frac{j + 0.5 - x_{s}}{\sqrt{2} σ_{A}}) - \erf (\frac{j - 0.5 - x_{s}}{\sqrt{2} σ_{A}})] .

The signal photon numbers would not change along with the threshold T unless T is greater than $\overline{N_{B}}$ . And the relationship between the threshold T and V_s is

V_{s} (T) = {\begin{array}{c} V_{s}, T \leq \overline{N_{B}} \\ \sum_{i, j = 1}^{L, M} check [N_{sij} - T + \overline{N_{B}}], T > \overline{N_{B}} \end{array} .

y = check(f) is defined as: $y = {\begin{array}{c} 0, f < 0 \\ f, f \geq 0 \end{array} .$ .

4.2.2 The relationships between the threshold, x_c and σ_S

While the threshold T is greater than $\overline{N_{B}}$ , the distribution of the density of signal spot changes. The photon numbers in one pixel can be expressed as:

N_{sij} (T) = check (N_{sij} - T + \overline{N_{B}}) .

So, the centroid computation formula of the signal spot is:

x_{s} (T) = \frac{\sum_{ij}^{L, M} x_{ij} check (N_{sij} - T + \overline{N_{B}})}{\sum_{ij}^{L, M} check (N_{sij} - T + \overline{N_{B}})} .

If the threshold T is greater than $\overline{N_{B}}$ and less than $N_{Gauss} + \overline{N_{B}}$ , (N_Gauss represents the least collected photon numbers by one pixel of the signal spot within Gauss width) Eq. (35) can be expressed as:

x_{s} (T) = \frac{\sum_{ij}^{L, M} [N_{sij} - T + \overline{N_{B}}]}{\sum_{ij}^{L, M} [N_{sij} - T + \overline{N_{B}}]} = \frac{\sum_{ij}^{L, M} [x_{ij} \cdot N_{sij}] - \sum_{ij}^{L, M} [x_{ij} \cdot (T - \overline{N_{B}})]}{\sum_{ij}^{L, M} N_{sij} - \sum_{ij}^{L, M} (T - \overline{N_{B}})}

Let $STR = \frac{\sum_{ij}^{L, M} N_{sij}}{\sum_{ij}^{L, M} (T - \overline{N_{B}})}, x_{T} = \frac{\sum_{ij}^{L, M} [x_{i} \cdot (T - \overline{N_{B}})]}{\sum_{ij}^{L, M} (T - \overline{N_{B}})},$ then:

x_{s} (T) = \frac{STR}{STR - 1} x_{p} - \frac{1}{STR - 1} X_{T} .

Thus the relationship between the sampling error and the threshold could be expressed as:

σ_{S} (T) = x_{p} - x_{c} (T) = \frac{1}{STR - 1} (x_{T} - x_{p}) .

4.2.3 The relationships between the threshold and σ_{x_c}²

If the threshold T is greater than $\overline{N_{B}} + 3 σ_{B}$ , the detected centroid is defined by Eq. (41). As the threshold T is constant, the wobbling error of spot could be expressed as:

σ_{x_{c}}^{2} = {(\frac{STR}{STR - 1})}^{2} σ_{x_{p}}^{2} .

σ_{x_p} ² is the variance of the signal photon noise. If the zone of signal spot contains l × m pixels, σ_{x_p} ² can be expressed as:

σ_{x_{p}}^{2} = \frac{4 (σ_{r}^{2} + \overline{N_{B}})}{V_{s} {(T)}^{2}} [\frac{1}{4} - x_{C}^{2} (T)] + \frac{κ σ_{A}^{2}}{V_{s} (T)} .

4.2.4 The summary

To sum up the relationship between the threshold and the PSCC error, the threshold T is discussed according to three different conditions:

(1) T is less than $\overline{N_{B}}$

Only the 1^st class error source changes along with T, the PSCC error is

σ_{xa}^{2} = \frac{ML σ_{B}^{2} (T)}{{[V_{s} + V_{B} (T)]}^{2}} [\frac{L^{2} - 1}{12} + x_{c}^{2} (T)] + \frac{κ V_{s} σ_{A}^{2}}{{[V_{s} + V_{B} (T)]}^{2}} + \frac{κ x_{s}^{2}}{V_{s}} - \frac{κ x_{c}^{2} (T)}{V_{s} + V_{B} (T)} + {\frac{V_{B} (T)}{V_{s} + V_{B} (T)} [x_{s} - x_{B} (T)] - σ_{s}}^{2}

(2) T is greater or equal to $\overline{N_{B}}$ and less than $\overline{N_{B}} + 3 σ_{B}$

Both the 1^st class error source and the 2^nd class error source change along with T, the PSCC error is

σ_{xb}^{2} = \frac{ML σ_{B}^{2} (T)}{{[V_{s} (T) + V_{B} (T)]}^{2}} [\frac{L^{2} - 1}{12} + x_{c}^{2} (T)] + \frac{κ V_{s} {(T) σ}_{A}^{2}}{{[V_{s} (T) + V_{B} (T)]}^{2}} + \frac{κ x_{s}^{2} (T)}{V_{s}} - \frac{κ x_{c}^{2} (T)}{V_{s} (T) + V_{B} (T)} + {\frac{V_{B} (T)}{V_{s} (T) + V_{B} (T)} [x_{s} (T) - x_{B} (T)] - σ_{s} (T)}^{2}

(3) T is greater or equal to $\overline{N_{B}} + 3 σ_{B}$ and less than $N_{Gauss} + \overline{N_{B}}$

The average value and the variance of the 1^st class error source are zero. Only the sampling error and the wobbling error of spot are left. The PSCC error is

σ_{x_{c}}^{2} = \frac{1}{{(STR - 1)}^{2}} (x_{T} - x_{p}) + \frac{4 (σ_{r}^{2} + \overline{N_{B}})}{V_{s}^{2} (T)} [\frac{1}{4} - x_{C}^{2} (T)] + \frac{κ σ_{A}^{2}}{V_{s} (T)} .

Fig. 2. The curves of the PSCC error that change along with threshold

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The curves in Fig. 2 show that the PSCC error changes along with the threshold when the spot positions vary.

Table 4. The relationship between the threshold and the PSCC error

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From the discussion, Eqs., and Fig. 2 above, we can arrive at the conclusion that an optimum threshold could reduce the PSCC error to a minimum, the PSCC error reaches its minimum value only when threshold is $\overline{N_{B}} + 3 σ_{B} .$ Consequently, the optimum threshold is $\overline{N_{B}} + 3 σ_{B} .$

5. Simulation results

5.1 Spot image in simulation with noise input

The distribution of an ideal spot obeying Gauss distribution is shown in Fig. 3(a). With existence of the sampling error, the spot imaged by CCD is shown as Fig. 3(b).

We can use the function of random(‘poisson’,N,1) in the function lab of Matlab to simulate the signal photon noise or the background photon noise, the function of random (‘normal’, 0, ReadNoise, L, M) to simulate the readout noise of CCD, and a constant array to simulate the background level noise of CCD. One of the simulated spot images, which is the summation of the photon noise, the readout noise and the background level, is shown in Fig. 4.

Fig. 3. The images of ideal spot (a) and sampling spot (b)

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Fig. 4. One of the spot imaged by simulation

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5.2 The comparison between theory results and simulation results

Since the spot image can be simulated in different situations, the theory and simulation results can be compared in different situations. The contrast maps of the PSCC error curves in theory and by simulation in different situations are shown in Fig. 5-Fig. 9.

Fig. 5. The PSCC error changing with threshold when position of spot is different

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Fig. 6. The PSCC error changing with threshold when background noise is different

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Fig. 7. The PSCC error changing with threshold when the size of aperture is different

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Fig. 8. The PSCC error changing with threshold when readout noise is different

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Fig. 9. The PSCC error changing with threshold when background level is different

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As shown in Fig. 5-Fig. 9, we can see that the simulation points are in accordance with the curves in theory. And when the threshold equals $\overline{N_{B}} + 3 σ_{B}$ , the PSCC error is obviously reduced. This verifies the theoretical deduction in section 4.

6. Experimental results

Fig. 10. Schematic diagram of experimental system

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The schematic diagram of the system is shown in Fig. 10. The tilt mirror is used to change the positions of spots on the CCD. In this system, the parameters of the devices are as follows:

The Gauss width of spot	0.8pixel.
The number of sub-apertures	23×23.
The size of sub-aperture	9pixels×9pixels.
The size of CCD	512pixels×512pixels.
The readout noise of CCD	6 ADUs.
The photon-electron response coefficient of CCD	0.001 ADU/photon.

Since the size of CCD is 512×512pixels and the valid numbers are only 207×207pixels, the left pixels of CCD can be used to sense the 1^st class error source, as shown in Fig.11. The image of spot and noise in one sub-aperture is shown as Fig. 12.

The displacement position of spot can be calculated by Eq. (48)

x_{p} = \frac{2 θ_{p} \cdot f_{1}}{S_{CCD}} (pixel) .

Here θ_p (in radians) is the tilt angle of the title mirror, f_l is the effect focus length of the micro-lens, and s_CCD is the size of one pixel.

Fig. 11. The zones of signal and the 1^st class error source

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Fig. 12. Spot and noise in single aperture

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The curves of the PSCC error in theory and by experiment are shown in Fig. 13. The experimental points are in great accordance with the curves in theory. Additionally, the minimum PSCC error appears at the point where the threshold T is equal to $\overline{N_{B}} + 3 σ_{B} .$

Fig. 13. The curves of the PSCC error in theory and by experiment

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7. Conclusion

CCD-based PSCC algorithm is widely used in detecting point source position field, whose accuracy is affected by various kinds of error sources of the photon noise, the readout noise, the background noise, the background level and the sampling error. A comprehensive expression formula of the CCD-based PSCC error caused by the error sources is put forward in this paper. After analyzing the relationship between the error sources and the threshold, we found that the PSCC error caused by the 1^st class error source equally affects each pixel within the detection window, contains background photon noise, readout noise and background level, and can be reduced by setting a threshold. However, the detecting precision would not be improved entirely, or even be worse so long as the threshold is set too low or too high. Having analyzed how the threshold works on the error sources, the optimum threshold is located at $\overline{N_{B}} + 3 σ_{B}$ , where $\overline{N_{B}}$ is the average value of the error sources and σ_B is the mean-square value of the fluctuation of the error sources. Both the simulation and experiment results show that the PSCC error would be reduced to a minimum while the threshold is set at $\overline{N_{B}} + 3 σ_{B} .$ This conclusion is of great importance in many fields, which require the noise analysis and the threshold design of PSCC, such as an ATP system, laser triangulation system, satellite orientation systems, and the Shack-Hartmann wavefront sensor.

Acknowledgments

The authors would like to thank Prof. Wenhan Jiang from our laboratory for his helpful suggestions. This project was supported by the National High-Technology Research and Development program of China.

References and links

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4. J. Ares, “Position and displacement sensing with Shack-Hartmann wavefront sensors,” Appl. Opt. 39, 1511–1520 (2000). [CrossRef]

5. C. H. Rao, W. H. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002). [CrossRef]

6. G. A. Tyler and D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. 6, 804–808 (1982). [CrossRef]

7. C. Genrui and Y. Xin, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2321–2335 (1994).

8. W. Jiang, H. Xian, and S. Feng, “Detecting error of Shack-Hartmann wavefront sensor,” Proc. SPIE 3126, 534–544 (1997). [CrossRef]

9. M. Xiaoyu, R. Changhui, and Z. Xuejun, “Performance comparison of Photon-Multiplier-Tube-Based and CCD-Based photon-electronic devices,” Acta Opt. Sin. 5, 882–888 (2007).

10. Z. Ang, R. Changhui, Z. Yudong, and J. Wenhan, “Performance analysis of Shack-Hartmann wavefront sensor with variable sub-aperture pixels,” Porc. SPIE 5490, 1268–1277 (2004). [CrossRef]

11. Z. Zhicheng, “On the Subject of Gauss Approximation of Poisson Distribution,” Acta Scicentiarum Naturalum Universitis Pekinesis 5, 605–619 (1988).

12. B. F. Aull, M. J. Renzi, A. H. Loomis, D. J. Young, B. J. Felton, T. A. Lind, and D. M. Craig, “Geiger-mode Quad-cell array for adaptive optics,” CLEO/QELS, (2008).

Symbol	Explanation	Symbol	Explanation
x_c	The detected centroid position	σ_r ²	The variance of readout noise of CCD
x_ij	The coordinate of a pixel on CCD	σ_S	The sampling error
x_p	The actual centroid position	σ_p ²	The variance of displacement error
Symbol	Explanation	Symbol	Explanation
x_s	The centroid position with sampling error	κ	Photon-ADUs transform coefficient
x_B	The centroid position of the 1^st class error resource	T	The threshold
N_ij	The received photon numbers of a pixel with coordinate (i, j)	L×M	The total number of pixels within an aperture
N_bij	The background photon numbers of a pixel with coordinate (i, j),	l×m	The total number of pixels within the spot area
N_dij	The background level of a pixel with coordinate (i, j)	V	The number of photons collected by CCD, $V = \sum_{ij}^{L, M} N_{ij}$
N_sij	The signal photon numbers of a pixel with coordinate (i, j)	V_s	The number of signal photons collected by CCD $V_{s} = \sum_{ij}^{L, M} N_{ij}$
N_rij	The readout noise of a pixel with coordinate (i, j)	V_b	The number of background noise photons collected by CCD $V_{b} = \sum_{ij}^{L, M} N_{bij}$
N_Bij	The noise of a pixel with coordinate (i, j)	V_d	The number of background level collected by CCD $V_{d} = \sum_{ij}^{L, M} N_{dij}$
$\overline{N_{bij}}$	The background photon number of a pixel with coordinate (i, j)	V_r	The number of readout noise collected by CCD $V_{r} = \sum_{ij}^{L, M} N_{rij}$
$\overline{N_{dij}}$	The average value of background level of a pixel with coordinate (i, j)	V_B	The number of noise collected by CCD $V_{B} = \sum_{ij}^{L, M} N_{Bij}$
$\overline{N_{sij}}$	The average signal photon number of a pixel with coordinate (i, j)	U	$U = \sum_{ij}^{L, M} x_{i} N_{ij}$
$\overline{N_{rij}}$	the average value of readout noise of a pixel with coordinate (i, j)	U_B	$U_{B} = \sum_{ij}^{L, M} x_{i} N_{Bij}$
$\overline{N_{Bij}}$	The average value of noise of a pixel with coordinate (i, j)	U_s	$U_{s} = \sum_{ij}^{L, M} x_{i} N_{sij}$
σ_B ²	The variance of the 1^st class error resource	F(T)	The value of F when the threshold is T
σ_{x_c} ²	The variance of the detected centroid position

The error source	The distribution
The signal light photon noise	Poisson distribution, the variance σ_pij ² is $\overline{N_{sij}}$ .
The background light photon noise	Poisson distribution, the variance σ_bij ² is $\overline{N_{b}}$
The readout noise of CCD	Gauss distribution, the mean is 0 and the variance is σ_r ²
The background level of CCD	Without undulation, the variance σ_dij ² is 0
The sampling error of CCD	Directly correlated with the spot size in a sub-aperture

The threshold T	the centroid of the 1^st class error resource x_B
The threshold T is greater than $\overline{N_{B}} - 3 σ_{B}$ .	Since the sight field is small while detecting the centroid of a point source, the background light irradiates on CCD uniformly. If the coordinate origin is the center of aperture, x_b is zero. As the background level of each pixel is a constant, x_d is zero too. Also, the average value of the readout noise is zero and thus V_r is zero. Then, we can get that x_B is zero.
The threshold T is greater or equal to $\overline{N_{B}} - 3 σ_{B}$ and less than $\overline{N_{B}} + 3 σ_{B}$ .	The RMS value of the 1^st class noise in the area of light spot is σ_B , the summation of the 1^st class error source of the rest area is V_B (T) . The spot area is small, so the centroid of the 1^st class error source within the spot area is x_s , while its value in the rest place is zero. Thus, the average centroid of the 1^st class error source is $\frac{l·m· σ_{B}}{V_{B} (T) + l·m· σ_{B}} x_{s},$ here l × m is the total number of pixels within the spot area.
The threshold T is greater or equal to $\overline{N_{B}} + 3 σ_{B}$ :	N_Bij is not equal to zero only in the spot area. x_B is the center of spot area.

The threshold T	The effects of the threshold to the PSCC error
The threshold T is less than $\overline{N_{B}} .$	As the threshold T equally affects every pixel in the aperture, it can decrease the average value of the 1^st class error source and thus correct the displacement error.
The threshold T is greater or equal to $\overline{N_{B}}$ and less than $\overline{N_{B}} + 3 σ_{B} .$	The threshold T could cut off signal light, but, at the same time, the threshold T would reduce the displacement error and the wobbling error caused by the 1^st class error source. The detecting accuracy still can benefit from the threshold because the signal light is mainly distributed in the Gauss width and the 1^st class error source is distributed in the whole aperture.
The threshold T is greater or equal to $\overline{N_{B}} + 3 σ_{B},$	All the 1^st class error could be eliminated to zero. In this instance, the larger the threshold is, the bigger damage to the signal light, and the PSCC error.

Symbol	Explanation	Symbol	Explanation
x_c	The detected centroid position	σ_r ²	The variance of readout noise of CCD
x_ij	The coordinate of a pixel on CCD	σ_S	The sampling error
x_p	The actual centroid position	σ_p ²	The variance of displacement error
Symbol	Explanation	Symbol	Explanation
x_s	The centroid position with sampling error	κ	Photon-ADUs transform coefficient
x_B	The centroid position of the 1^st class error resource	T	The threshold
N_ij	The received photon numbers of a pixel with coordinate (i, j)	L×M	The total number of pixels within an aperture
N_bij	The background photon numbers of a pixel with coordinate (i, j),	l×m	The total number of pixels within the spot area
N_dij	The background level of a pixel with coordinate (i, j)	V	The number of photons collected by CCD, $V = \sum_{ij}^{L, M} N_{ij}$
N_sij	The signal photon numbers of a pixel with coordinate (i, j)	V_s	The number of signal photons collected by CCD $V_{s} = \sum_{ij}^{L, M} N_{ij}$
N_rij	The readout noise of a pixel with coordinate (i, j)	V_b	The number of background noise photons collected by CCD $V_{b} = \sum_{ij}^{L, M} N_{bij}$
N_Bij	The noise of a pixel with coordinate (i, j)	V_d	The number of background level collected by CCD $V_{d} = \sum_{ij}^{L, M} N_{dij}$
$\overline{N_{bij}}$	The background photon number of a pixel with coordinate (i, j)	V_r	The number of readout noise collected by CCD $V_{r} = \sum_{ij}^{L, M} N_{rij}$
$\overline{N_{dij}}$	The average value of background level of a pixel with coordinate (i, j)	V_B	The number of noise collected by CCD $V_{B} = \sum_{ij}^{L, M} N_{Bij}$
$\overline{N_{sij}}$	The average signal photon number of a pixel with coordinate (i, j)	U	$U = \sum_{ij}^{L, M} x_{i} N_{ij}$
$\overline{N_{rij}}$	the average value of readout noise of a pixel with coordinate (i, j)	U_B	$U_{B} = \sum_{ij}^{L, M} x_{i} N_{Bij}$
$\overline{N_{Bij}}$	The average value of noise of a pixel with coordinate (i, j)	U_s	$U_{s} = \sum_{ij}^{L, M} x_{i} N_{sij}$
σ_B ²	The variance of the 1^st class error resource	F(T)	The value of F when the threshold is T
σ_{x_c} ²	The variance of the detected centroid position

Error analysis of CCD-based point source centroid computation under the background light

Abstract

1. Introduction

2. The CCD-based PSCC algorithm and the error sources List of symbols

3. Error analysis of the centroid computation

3.1 The displacement error

3.2 The wobbling error

3.3 The variance of the PSCC error

4. The optimum threshold chosen

4.1 The relationship between the threshold and the 1^st class error source

4.1.1 The relationship between the threshold and V_B

4.1.2 The relationship between the threshold and σ_B²

4.1.3 The relationship between the threshold and x_B

4.2 The relationship between the threshold and the 2^nd class error source

4.2.1 The relationship between the threshold and V_s

4.2.2 The relationships between the threshold, x_c and σ_S

4.2.3 The relationships between the threshold and σ_{x_c}²

4.2.4 The summary

5. Simulation results

5.1 Spot image in simulation with noise input

5.2 The comparison between theory results and simulation results

6. Experimental results

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (13)

Tables (4)

Equations (50)

Optics Express

Abstract

1. Introduction

2. The CCD-based PSCC algorithm and the error sources List of symbols

3. Error analysis of the centroid computation

3.1 The displacement error

3.2 The wobbling error

3.3 The variance of the PSCC error

4. The optimum threshold chosen

4.1 The relationship between the threshold and the 1st class error source

4.1.1 The relationship between the threshold and VB

4.1.2 The relationship between the threshold and σB2

4.1.3 The relationship between the threshold and xB

4.2 The relationship between the threshold and the 2nd class error source

4.2.1 The relationship between the threshold and Vs

4.2.2 The relationships between the threshold, xc and σS

4.2.3 The relationships between the threshold and σxc2

4.2.4 The summary

5. Simulation results

5.1 Spot image in simulation with noise input

5.2 The comparison between theory results and simulation results

6. Experimental results

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (13)

Tables (4)

Equations (50)

Optics Express

4.1 The relationship between the threshold and the 1^st class error source

4.1.1 The relationship between the threshold and V_B

4.1.2 The relationship between the threshold and σ_B²

4.1.3 The relationship between the threshold and x_B

4.2 The relationship between the threshold and the 2^nd class error source

4.2.1 The relationship between the threshold and V_s

4.2.2 The relationships between the threshold, x_c and σ_S

4.2.3 The relationships between the threshold and σ_{x_c}²