3.1. Experiment Setup
The experiment setup consists of a photoelectric target, a motorized linear stage, a projector (Model: BENQ CP270) and its posture adjustment mechanism (a ball tilting) as shown in
Figure 4a. The electronic control subsystem (a computer) and some extension cards were applied to drive the motorized stage, collect the voltage signals and control the projector (not shown in Figures). PDs (Model: Osram SFH2701) were mounted on the front of the photoelectric transformation units (PTUs), as shown in
Figure 4b. At the back of the PTU, a two-stage amplification circuit (transimpedance amplifier circuit) was applied to convert the photocurrent signal of the PD to voltage signal and then a multichannel data collection card (Model: Advantech PCI-1713U 32 channels) was used to store all the amplified signals from the 32 PDs. The number of the PTUs depends on the installing density and the height of the projected image. To capture the pixel lines in horizontal direction, the total height of the photoelectric target should be longer than the height of the projected image.
Figure 4.
Experiment setup of the calibration system. (a) Relative positions of the DLP projector, the photoelectric target and the motorized stage; (b) The photoelectric target with the projected line pattern; (c) Schematics of the entire projector calibrating system (Only two PTUs are shown) and (d) the enlarged view of the photoelectric module.
Figure 4.
Experiment setup of the calibration system. (a) Relative positions of the DLP projector, the photoelectric target and the motorized stage; (b) The photoelectric target with the projected line pattern; (c) Schematics of the entire projector calibrating system (Only two PTUs are shown) and (d) the enlarged view of the photoelectric module.
The installation reference line of the PTUs should be perpendicular to the movement direction of the stage in order to ensure the orthogonality of the coordinate system on the virtual image plane. The perpendicularity between the stage and the photoelectric target was guaranteed by precise machining and redundant locating. All the coordinates of PTU locating holes and fixing holes can be obtained by a measuring microscope (Model: TESA VISIO 300 DCC) and then the reference line for PTU installing was generated. An “L”-shaped mechanical component was designed to ensure the perpendicularity between the slide stage and the photoelectric target and then the installation perpendicularity of photoelectric target can be guaranteed within ±20 arc seconds.
The measuring microscope was also applied to make all the 32 PTUs collinear and equidistant through fine adjustment. The PD was installed on the surface of the PTU with 45° with the horizontal direction. The photosensitive area of the PD is a square and the side length is 0.6 mm so the length of PD’s receiving area is 0.84 mm (diagonal). With the help of the adjustment screws on the photoelectric target, the installation straightness of all the 32PTUs can be guaranteed within 0.008 mm.
Figure 5 shows the image of the photodiode under the measuring microscope.
Figure 5.
Photodiode assembly under image measurement tool. The green dotted line is the reference line for photodiode assembly. Numbers in the figure indicate the coordinate of the image corner point. (XYZ, Unit: mm).
Figure 5.
Photodiode assembly under image measurement tool. The green dotted line is the reference line for photodiode assembly. Numbers in the figure indicate the coordinate of the image corner point. (XYZ, Unit: mm).
The stage which holds ±10 μm positioning accuracy and ±3 μm repetitiveness was calibrated by a laser interferometer (Model: Renishaw XL-80) to ensure the accurate positioning along the travelling line. We only employ unidirectional motion during one calibration procedure to avoid the backlash of the linear stage, so the final positioning accuracy of the stage can be guaranteed within 0.006 mm in a travel distance of 300 mm.
The projector to be calibrated has a resolution of 1024 × 768. In order to test the proposed approach independently, all the projector’s pre-included geometric correction functions were turned off. The projector was fixed on a ball tilting and the projector’s control panel was facing upwards to make sure the origin point of the object plane coordinate was located at the bottom right. The optical axis does not need to be exactly perpendicular to the virtual image plane because the linear distortion parameters have already been considered in the final calibration results in the proposed method.
A motion controlling card (Model: Advantech PCI-1240) programmed by the computer was used to control the motorized stage. A data collection card (Model: Advantech PCI-1713U) programmed by computer recorded the voltage values representing the optical power. The line patterns for the projector were generated by the computer. The graphics card of the computer was set up to drive two display devices simultaneously, one for a LCD monitor and the other for the DLP projector.
A horizontal coordinate list for the stops is generated by the computer with fixed distance in between. The number of stops can be increased to achieve higher accuracy. In each stop, the computer records the current position of the linear motorized stages, the scanned sequence as mentioned in
Section 2 and output voltage values of the PD. The measured data were stored in the computer for post-processing by MATLAB.
Another model of a projector with different resolution (Model: CASIO XJ-M255 1280 × 800) is applied to test and verify the proposed calibration method, which is only shown in the supplementary attachment (
Figures S1–S5).
3.2. Measurement Results
The vertical distance of two adjacent PTUs is 10 mm while the horizontal distance between two stops of the motorized stage is 8 mm. The width of projecting image was adjusted to be 280 mm by changing the working distance. The projector’s width-height ratio is 4:3 so the height of the projected image is about 220 mm (a little longer than 210 mm) and only 21 PTUs are used in this calibration procedure. The image width of one pixel is approximately 0.27 mm to meet the accuracy requirement. The calibration procedure costs about 40 min when the speed of the procedure is optimized.
Figure 6 shows the polynomial fitting evaluating result. Before fitting all the rows and columns of the VMPs, the degree of the polynomial should be determined. To simplify the data processing, 2–6 degree polynomials are tried to fit Row 10 of the VMPs in order to evaluate the accuracy of the fitting. Finally, the polynomial with degree 4 is chosen because it has the lowest RMSE while the coefficients are not less than 1 × 10
−14 (limit of the double precision float number).
Figure 7 shows the results after obtaining mark point coordinates and applying polynomial fitting on the object plane. The results in
Figure 7 show the distortion property of the projecting lens intuitively. Because the tested projection lens has an off-axis optical path, the distortion is up-down asymmetry. All the VMPs and polynomial curves in the figures show an obvious barrel distortion when the tested lens is used as the camera lens. In fact, a pillow distortion will appear when an ideal pattern is projected onto the image plane with this tested projection lens. The maximum non-linear distortion of the projector which appears at the top of the projection image is about three pixels.
Figure 6.
Final evaluation results of the polynomial fitting.
Figure 6.
Final evaluation results of the polynomial fitting.
Figure 7.
Results after obtaining VMPs’ coordinates and polynomial fitting (unit: pixel). (a) The points on the object plane corresponding to the VMPs on the image plane; (b) and (c) give the partial enlarged views of the Axis Y; (d,e) give the partial enlarged views of the Axis X. The curves indicate polynomial curve of these measuring points.
Figure 7.
Results after obtaining VMPs’ coordinates and polynomial fitting (unit: pixel). (a) The points on the object plane corresponding to the VMPs on the image plane; (b) and (c) give the partial enlarged views of the Axis Y; (d,e) give the partial enlarged views of the Axis X. The curves indicate polynomial curve of these measuring points.
3.3. Quantitative Evaluation
The calibration purpose of projection lens is to eliminate the non-linear distortion. If a corrected line pattern is projected onto a flat plate, the image of this line pattern should be straight. The performance of the calibrated system can be quantitatively evaluated by comparing with that of an uncorrected line pattern.
A low noise, precise imaging system is built up to evaluate the projector calibration system, which makes use of a high resolution CCD camera (Imperx B2520M, 2456 × 2058 pixels, BW) and a white plate with flatness error smaller than ±5 micrometers. The accuracy of this evaluation is affected by distortion of the camera lens. To minimize this effect of camera lens distortion, the CCD’s central area was used to record the image of the line pattern, as illustrated in
Figure 8b. The line pattern mapped from the ideal line on the image plane has a sequence of non-integral coordinates on the object plane. To maximize the evaluation accuracy, Gaussian function is used to generate the corrected line pattern to project with subpixel accuracy. Then, the subpixel coordinates of the captured line is also extracted by Gaussian function fitting method to determine the center position sequence of the projected line pattern.
Figure 8.
The line images captured by the CCD camera. (a) Image captured by the camera; and (b) Line pattern center extraction result of all the image rows (Unit: pixel); (c) Image of the checkerboard calibration plate.
Figure 8.
The line images captured by the CCD camera. (a) Image captured by the camera; and (b) Line pattern center extraction result of all the image rows (Unit: pixel); (c) Image of the checkerboard calibration plate.
The distortion is more severe at the boundary area of the projected image, so pixel columns near the edge of the projectored image were evaluated to validate the proposed method. Three kinds of line patterns were generated and projected onto the plate surface for comparison.
Uncorrected straight line: Pixel column 17 (Pixel columns are 1–768 from left to right).
Corrected line by model-based method as shown in Reference [
8]: Like the distortion correction of a photograph, the distortion correction of a projector can be achieved by rearranging the pixels on the projector’s object plane with the method based on conventional distortion representation. The ideal line (goal of the correction) on the image plane is the pixel column 17 in the linear (pinhole) model.
Corrected line by the proposed polynomial distortion representation: The ideal vertical line (goal of the correction) on the image plane is located at 17/768 of the image width.
All these three line pattern images were captured by the CCD camera. The center coordinate sequences of the line patterns were extracted by Gaussian fitting method from the captured images. Each sequence should be corrected by the reference line sequence that was generated previously. In order to show the correction performance on the non-linear deviation, the linear components of the three sequences should be removed. To calculate the deviations in metric, a ceramic checkerboard calibration plate is used to generate a metric reference. The distance between each corner point is 20 mm on the plate and 150 camera pixels in the photograph (
Figure 8c) so a camera pixel stands for 0.133 mm. The final results of projection distortion correction were illustrated in
Figure 9. In fact, the average non-linear deviations of the three images were 0.35, 0.24, and 0.05 pixels (0.047, 0.032, 0.007 mm) respectively, while the maximum non-linear deviation were 1.63, 1.36 and 0.38 pixels (0.217, 0.181, 0.051 mm) respectively. All the metric deviations are calculated with a projecting image dimension 280 × 210 mm. The experimental results clearly showed that the proposed method accurately calibrates the projector distortion characters and the corresponding corrected line effectively decreases the distortion.
Figure 9.
Evaluation results on non-linear residual.
Figure 9.
Evaluation results on non-linear residual.