A Fixed-Pattern Noise Correction Method Based on Gray Value Compensation for TDI CMOS Image Sensor

In order to eliminate the fixed-pattern noise (FPN) in the output image of time-delay-integration CMOS image sensor (TDI-CIS), a FPN correction method based on gray value compensation is proposed. One hundred images are first captured under uniform illumination. Then, row FPN (RFPN) and column FPN (CFPN) are estimated based on the row-mean vector and column-mean vector of all collected images, respectively. Finally, RFPN are corrected by adding the estimated RFPN gray value to the original gray values of pixels in the corresponding row, and CFPN are corrected by subtracting the estimated CFPN gray value from the original gray values of pixels in the corresponding column. Experimental results based on a 128-stage TDI-CIS show that, after correcting the FPN in the image captured under uniform illumination with the proposed method, the standard-deviation of row-mean vector decreases from 5.6798 to 0.4214 LSB, and the standard-deviation of column-mean vector decreases from 15.2080 to 13.4623 LSB. Both kinds of FPN in the real images captured by TDI-CIS are eliminated effectively with the proposed method.


Introduction
Time-delay-integration (TDI) camera is a special kind of line-scan camera, which captures images through pixels arranged in an area array and works in line-scan mode. It is widely used for high-quality and low-noise imaging even under low illumination and at high scanning speed. TDI cameras play a key role in the remote push-broom sensing system to improve its low light level capability [1]. The TDI technique can be easily applied on charge-coupled device (CCD) which allows noiseless accumulation of signals in charge domain [2]. However, CCD requires high operating voltage and it is difficult to integrate signal processing circuits in it. CMOS technology is gaining attention in this application because of its low-power, low-cost and high integrated density [3]. However, it is more difficult to implement on-chip low-noise accumulation in analog domain and synchronous signal capturing of the image for all pixels in the same column with CMOS than with CCD. Therefore, the key technique in TDI-CMOS image sensor (TDI-CIS) is the signal accumulator.
Several reports on TDI-CIS have been published recently [4][5][6][7][8]. In our previous work, a 128-stage TDI-CIS with on-chip analog accumulator was proposed [9]. However, the parasitic resistors and capacitors of the accumulator can lead to row fixed-pattern noise (RFPN) when TDI-CIS works at high line rate. RFPN appears as obvious and periodic horizontal stripes in the output image. In addition, the structure of on-chip column-parallel readout circuit can lead to column fixed-pattern noise (CFPN) because of the technology fabrication deviation of TDI-CIS. CFPN appears as vertical stripes of different brightness in the output image. The stripes caused by RFPN and CFPN can lead to failure in data classification and incorrect retrievals of useful information. Therefore, both kinds of FPN degrade the output image quality seriously, thereby limiting the application of TDI-CIS.
The RFPN caused by the specific circuit structure of the accumulator is a special kind of FPN and has not been reported in the literature. The CFPN of CIS is usually suppressed by on-chip circuit design [10]. However, the FPN of manufactured image sensors can only be eliminated by image postprocessing. Most existing destriping algorithms are designed for specific image sensors or data types-infrared imagery [11], hyperspectral/multispectral imagery [12,13], and moderate resolution imaging spectroradiometer (MODIS) [14,15]. The stripes to be removed in the literature are either horizontal or vertical, but horizontal stripes and vertical stripes interweave with each other in the output image of TDI-CIS. Therefore, the existing destriping algorithms are not suitable for TDI-CIS.
In order to remove both kinds of stripes in the output image of TDI-CIS, a FPN correction method based on gray value compensation is proposed. In the proposed method, RFPN and CFPN are estimated and corrected based on the row-mean vector and column-mean vector of a large number of images captured under uniform illumination respectively. The remainder of this paper is organized as follows. Section 2 analyzes the sources and characteristics of FPN and establishes a noise model of TDI-CIS. Section 3 describes the proposed method. Section 4 presents the experimental results, and Section 5 provides the conclusions.

Analysis and Modeling of Sensor Noise
TDI-CIS is mainly comprised of a pixel array, column-parallel accumulators and column-parallel analog-to-digital converters (ADCs). In Figure 1, a block diagram of TDI-CIS is depicted [16]. The scanning direction of TDI is called "along-track direction", and the direction which is orthogonal to the along-track direction is called "across-track direction". The noise of TDI-CIS can be simply divided into two types, one is FPN, and the other is random noise. In this paper, we only consider the correction of RFPN and CFPN.

Source and Analysis of RFPN
As shown in Figure 1, the image sensor works in TDI mode with signal accumulation in analog domain. The circuit structure of the column-parallel accumulator is shown in Figure 2 [9]. The accumulator is mainly comprised of a fully-differential operational amplifier (OPA) and (M + 1) integrators. When TDI stage is M, the stage of the accumulator is (M + 1). The photoinduced charges are firstly stored in the integrators where they can be accumulated, then read out into the corresponding column ADC under the control of the sequential circuit. TDI-CIS works in rolling-shutter readout mode in along-track direction with temporal oversampling rate of ( 1)/ M M + , implementing synchronous signal capturing of the image for all M pixels in the same column [9,17]. The correspondence between the integrator stage and the pixel row number in the pixel array is shown in Figure 3. The charges stored in the (M + 1)-th-stage integrator, the M-th-stage integrator … and the 1st-stage integrator are read out in sequence and periodically under the control of the sequential circuit. Therefore, the gray value of the P-th row in the output image corresponds to the voltage signal read out from the (M + 2 -P)-th-stage integrator. The periodicity of the readout mode of the accumulator leads to the periodicity of the correspondence between the integrator stage and the pixel row number in the output image, and the period T is equal to the stage of the accumulator. Therefore, T can be expressed as: As shown in Figure 2, the "input bus" and "output bus" of the accumulator are not ideal wires. There are parasitic resistors and capacitors in the layout of the buses, as shown in Figure 4. The parasitic parameters of an integrator vary with its corresponding length of buses. When TDI-CIS works at high line rate, the readout voltage values of different stages of integrator are different even under uniform illumination, and the readout voltage value of an integrator increases with increasing integrator stage. This is the reason why RFPN exists. In the output image, RFPN gray values of all pixels in the same row are equal, while RFPN gray value of a row changes periodically and increases with increasing row number in the along-track direction, and the period T satisfies Equation (1).

Source and Analysis of CFPN
In Figure 1, there is a mismatch in both accumulators and ADCs of different columns because of the technology fabrication deviation of TDI-CIS. The mismatch leads to CFPN, which appears as vertical stripes of different brightness in the output image, just as shown in Figure 5.
In the output image, CFPN gray values of all pixels in the same column are equal, while CFPN gray values of pixels in different columns obey a zero-mean Gaussian distribution under the condition that the image size in across-track direction-N-is very large.

Noise Model of TDI-CIS
According to the analyses above, both RFPN and CFPN are a kind of spatial nonuniformity, and do not vary with illumination intensity. Both FPN and random noise are assumed to be additive noise, and they are uncorrelated. The FPN of a given manufactured TDI-CIS is assumed to be constant under the same working environment. The random noise obeys a zero-mean Gaussian distribution.
As shown in Figure 1, the size of the pixel array is M N × -M rows and N columns. The row number is defined as i, and the column number is defined as j. Assuming that the photoresponses of different pixels are uncorrelated, the gray value ( , ) y i j of pixel ( , ) i j in the output image can be expressed as: where L is the image size in along-track direction, ( , ) x i j is the ideal gray value of pixel ( , ) i j , ( ) a i is the RFPN gray value of the i-th-row pixel, ( ) b j is the CFPN gray value of the j-th-column pixel, and ( , ) r i j is the sum of all kinds of noises except FPN. According to the analyses in Section 2.1, ( ) a i satisfies the following Equation:

Method Description
According to the analyses in Section 2, the FPN correction method based on gray value compensation is proposed. Firstly sample data is acquired, and then RFPN and CFPN are estimated and corrected successively.

Sample Data Acquisition
In order to ensure effectiveness and accuracy of the estimation results of FPN, a large number of images should be captured with an image acquisition system. The image resolution of TDI-CIS is assumed to be 8-bit. Firstly K uniform-light images whose average gray values are all about 127-half-saturation-are captured under the same test environment [18]. Then the collected K images defined as 1 (2) can be expressed as:

Estimation and Correction of RFPN
The row-mean vector of k Y is defined as k U , and the average gray value of the i-th-row pixels is defined as k ( ) i U . As shown in Figure 6, due to the existence of RFPN, the curve of k U fluctuates strongly with periodic jumps in the locations of horizontal stripes. However, the locations of horizontal stripes in different images are not all the same, just as shown in Figure 7. Therefore, when calculating the row-mean vector of all collected images, the result obtained by averaging directly will be incorrect. In order to calculate the effective row-mean vector used for estimating RFPN, the locations of the first horizontal stripe in along-track direction of all collected images should be detected firstly, and this kind of operation is called "first line detection".  The ratio of two adjacent elements in the row-mean vector of k Y is defined as k ( ) i D , which can be expressed as: As shown in Figure 6, the average gray value of a row in which the horizontal stripes do not locate changes gradually and decreases with increasing row number in each period. Therefore, the locations where the jumps in the curve of k U take place can be easily got by threshold detection of k ( ) i D (such as larger than 1.15), i.e., the last row of every period can be detected. The row number of the last row of the first period is defined as k S .
A new image C,k Y is constructed by cutting t periods from k Y starting from the ( k S + 1)-th row. The image size of C,k Y is defined as c L N × , then the following equation can be deduced from Equation (1): C U is calculated by averaging based on multi-sample, so that the impacts of CFPN and random noise are eliminated. Therefore, ( ) a m ( c 1 m L ≤ ≤ ) can be estimated based on the row-mean vector C U .
According to the analyses above, both ( ) a m and the elements in C U change periodically and therefore only one period of RFPN needs to be estimated. The estimated RFPN gray value is defined as ( ) a r (1 r T ≤ ≤ ), then ( ) a r can be estimated based on C ( ) r U ( ) a r increases with increasing row number, and it is the relative differences of ( ) a r between different rows that lead to RFPN, especially the horizontal stripes. Therefore, assuming that (1) 0 a = will not reduce the effectiveness of RFPN correction. C is exactly the estimated RFPN gray value of the r-th row. In consideration of the fact that the gray values are all integers, ( ) a r can be calculated according to the following Equation: where round is a MATLAB function which stands for rounding to the nearest integer. The estimated RFPN gray values of one period are shown in Figure 8. According to Equation (2), RFPN can be corrected by adding the estimated RFPN gray value to the original gray values of pixels in the corresponding row. The operation of "first line detection" is required before RFPN correction, because the correspondence between ( ) a i in Equation (2) where ( , ) z i j is the gray value of pixel ( , ) i j with RFPN corrected. When ( , ) 0 y i j = , the correction procedure is not carried out, because it cannot be judged whether the gray value ( , ) y i j satisfies Equation (2).

Estimation and Correction of CFPN
According to Equations (2) and (9), the gray value ( , ) z i j of pixel ( , ) i j in the image with RFPN corrected can be expressed as: Carried out RFPN correction procedure for all collected images 1 V is calculated by averaging based on multi-sample, so that the impacts of random noise are eliminated. Therefore, ( ) b j (1 j N ≤ ≤ ) can be estimated based on the column-mean vector V . Due to the existence of CFPN, the curve of V fluctuates strongly, as shown by the green line in Figure 9. In addition, the average gray values of central columns are larger than those of peripheral columns. A lens is used when capturing images. In order to clarify the reasons of this kind of shading, another group of sample data has been captured under dark condition. The column-mean vector of all collected images under dark condition is defined as dark V . The experimental results show that, the distribution of dark V is nearly a flat curve. Therefore, the use of the lens is the only reason of the shading [19].
In order to estimate ( ) b j based on V , an ideal column-mean vector C V whose distribution is a nearly smooth curve should be constructed firstly, as shown by the red line in Figure 9. Then V minus C V is exactly the estimated CFPN vector. averaging-filter (NNAF). The simplest NNAF is implemented by taking the average of two adjacent elements, but an ideal curve cannot be got by using the simplest NNAF. In order to construct C V , a parametric NNAF expressed as the following Equation is designed: According to the analyses in Section 2.2, the elements in CFPN vector obey a zero-mean Gaussian distribution when N is very large, and therefore the average gray value of all elements in estimated CFPN vector should be zero or nearly zero. Then the W B whose corresponding MA,W B is minimal is the best estimation of CFPN vector. The best W B is defined as S B , and the elements in S B are defined as ( ) b s (1 s N ≤ ≤ ), as shown in Figure 10. According to Equation (10), CFPN can be corrected by subtracting the estimated CFPN gray value from the original gray values of pixels in the corresponding column, as shown in the following Equation:

Experimental Results
In order to validate the effectiveness of the proposed method, a TDI CMOS imaging system based on a 128-stage TDI-CIS with on-chip analog accumulator is designed, and the following experiments are conducted. The TDI-CIS used in the experiments is the same device as that in our previous work [9]. Both RFPN and CFPN also exist in the previous work, but they had been corrected with the proposed method in this paper. This is the reason why FPN was not visible in [9]. The specifications of the TDI-CIS are listed in Table 1. The TDI CMOS imaging system is mainly comprised of seven parts-a ring light source used for providing uniform illumination, a lens, a conveyor belt on which original papery-photos are pasted, a servo-motor with controller used for scanning motion control, a printed-circuit-board (PCB) with the TDI-CIS, a field-programmable-gate-array (FPGA) development-board, and a monitor used for image display. The schematic diagram and photograph of the imaging system are shown in Figures 11  and 12 respectively.
In the experiments, the TDI-CIS works at the line rate of 3875 lines/s. During sample data acquisition, a piece of white paper is used as the scene to be photographed to ensure that TDI-CIS works under uniform illumination. When papery-photos are selected as the scene to be photographed, the experiment is called "real-test experiment", and the output image is called "real-test image". In total, one hundred uniform-light images are collected as the sample data used for estimating FPN.
The resolution of all output gray-level images is 8-bit. The experimental results are evaluated both subjectively and objectively. The subjective evaluation methods consist of "image visual effect" and "row-mean curve". The objective evaluation indexes consist of "standard-deviation of row-mean vector (SDRMV)" and "standard-deviation of column-mean vector (SDCMV)", both of which are given in the units of Least Significant Bit (LSB). The smaller the SDRMV is, the better the effectiveness of RFPN correction is. Similarly, the smaller the SDCMV is, the better the effectiveness of CFPN correction is.

FPN Correction for Uniform-Light Image
The comparison of visual effect of a uniform-light image is shown in Figure 13. The image size is 1024 768 × pixels. After correcting the FPN with the proposed method, the brightness of the corrected image is almost uniform in along-track direction, and the vertical stripes in across-track direction are significantly alleviated. The comparison of objective evaluation indexes of the uniform-light image in Figure 13 is listed in Table 2. According to the analyses in Section 3.3, the use of the lens leads to that the average gray values of central columns are larger than those of peripheral columns, which is known as lens shading, as shown by the green line in Figure 9. The lens shading correction is not included in our method. This is the reason why both the SDCMVs of the original image and the corrected image in Table 2

FPN Correction for Real-Test Image
The comparison of visual effect of a real-test image is shown in Figure 14. Due to the existence of FPN, there are obvious horizontal stripes in the conjunctive areas of two adjacent periods in the along-track direction, and vertical stripes of different brightness in across-track direction. After correcting the FPN with the proposed method, both kinds of stripes are removed.
The comparison of row-mean curve of the real-test image in Figure 14 is shown in Figure 15, where the curve of the original image is called "original curve", and the curve of the corrected image is called "corrected curve". The original curve fluctuates strongly with periodic jumps marked with ellipses in Figure 15a. According to Equation (1), the period of a 128-stage TDI-CIS is 129 rows. After correcting the FPN with the proposed method, the jumps disappear completely, and therefore the corrected curve is smoother than the original curve. In addition, both the form and change trend of the corrected curve are consistent with those of the original curve, demonstrating that the proposed method is able to correct FPN with useful information content of the original image well preserved.  In order to validate the robustness of the proposed method, the real-test experiment was conducted twenty times with the imaging system in Figure 12, and some of the experimental results are shown in Figures 16-19. The stripes in different kinds of papery-photos are removed after correcting the FPN with the proposed method.

Running Time
In the experiments, both FPN estimation and FPN correction are accomplished in MATLAB R2014a on a personal computer with an Intel Core 2 CPU at 2.93-GHz and 2-GB memory. Before FPN correction, the FPN gray values are estimated in advance. The running time of the FPN correction procedure in different figures is listed in Table 3. All the running times are given in the units of Seconds. It can be seen that the FPN correction procedure is very fast. Table 3. Running time of the FPN correction procedure in different figures.

Conclusions
In this paper, the sources of FPN and noise characteristics of TDI-CIS were analyzed, and then a FPN correction method based on gray value compensation was proposed. Experimental results based on a 128-stage TDI-CIS show that, both RFPN and CFPN in the output image are eliminated effectively and both image visual effect and objective evaluation indexes are improved; finally, the image quality becomes better. The proposed method is suitable for TDI-CIS, and is able to correct FPN with useful information content of the original image well preserved.