## Abstract

A random target method for fast MTF inspection is proposed. The setup includes a random target, lens under test and a CCD camera with focus adjustment. The target consists of a random black and white pattern of a flat spectrum. The MTF of the lens is acquired by imaging the random target on the CCD using the lens under test, and then analyzing the spatial frequency content of the image. Frequency range up to about 50 cycles/mm is possible using commonly available CCD imagers. Measurement speed and precision depend on the sample matrix size used in calculation. A matrix of 128*128 samples per measured field point provides better than 2% precision and a few second’s total execution time (ordinary PC-computer) per lens including best focus evaluation and the measurement of tangential and sagittal MTF curves of 5 field points. Thus fast MTF inspection of low to medium quality lenses seems possible.

© 2004 Optical Society of America

## 1. Introduction

Increasing amounts of consumer products including optical lenses and camera modules are sold each year. These products include various kinds of digital cameras and camera phones, for example. Fast go/no-go MTF (modulation transfer function) tests are needed for efficient production of the lenses and modules. In this paper a method based on random targets is proposed for such task.

The method for MTF inspection includes a random target with random black and white pattern, lens under test and an imager such as CCD detector. The MTF of the lens is acquired by imaging
the random target on the CCD using the lens under test, and then analyzing the spatial frequency content of the captured image. The idea of using a random target for MTF measurement was
originally introduced in the late 1950s by Kubota and Ohzu, who used random targets created by a laser speckle [1]. Several works applying basically
the same method have been presented since. Daniels *et al*. [2] used random transparency targets generated by a computer for system
MTF measurements. Levy *et al*. [3] adapted the same principle for lens measurements. None of them, however, have considered speed
issues.

The random target method is estimated to be suitable for fast lens MTF inspection since both on and off-axis field points in each focus position can be measured without any mechanical adjustments or movements, which again is typical for other MTF measurement methods and instruments. It was also estimated that it would not be necessary to use magnifying optics, again typical for MTF instruments, since fairly dense and large imagers are available for image analysis. The lack of a magnifying lens results in a simpler implementation of the test instrument. The method was also estimated to be flexible so that changes in the inspection procedure could be made even online, if needed. Off-axis field point height and measurement precision could be altered by changing the center position and size of the sample matrix (number of pixels) used for MTF analysis. Targets could be application specific and they could be fast and easily changed using an image projector for target generation. In this paper the overall functionality, speed and precision of the random target method are studied.

## 2. Random target method

The modulation transfer function (MTF) is an universal measure describing the quality of image formation. It indicates to what extent information, described by its spatial frequency content, is transferred from the object to the image by the lens. When considering a sinusoidal input with unity contrast, the MTF is given by

where *A _{max}* and

*A*are the maximum and minimum irradiances.

_{min}Continuous random targets are wideband targets with a flat spatial power spectrum. These targets can be created with a computer random number generator and manufactured by direct printing,
using a projector or a display. Note however that the optical properties of the used material/projector may alter the power spectrum of the target. Daniels *et al*. [2], for example, suggests that a binary random target should be printed using a high-resolution printer and then photographically reproduced using a
high contrast film. The resulting continuous target contains black or white dots and has a uniform, band-limited noise spectrum whose upper limit is

where *l* is the size (spacing) of a random target dot in the image plane. The random target dot size in the image plane can be matched to measuring setup where the number
of target dots in a row is *N*, target size h and system magnification *M* through

Dividing the number of dots by size gives the “dot density”. To avoid aliasing the highest dot density in the image plane should be less than half of the detector pixel density and thus pixel density of the imager sets the limit for the usable measurement bandwidth. Filtering can obviously also be used to limit the bandwidth of the target and to reduce possible aliasing.

The MTF analysis is based on evaluating the frequency content of the image formed by the lens under test (LUT). The relation between the measured MTF and the target and image noise power spectral densities (PSDs) is

where *MTF _{total}* includes the

*MTF*

_{LUT}The MTF of the LUT can be calculated using

in which

In Eq. (7)
*PSD _{sys}* is the noise PSD of the imaging system including temporal and spatial noise of the imager and illumination.

The measuring process starts by acquiring two images, one without and one with a random target. These images are treated in an equivalent manner. From the image its mean value is
subtracted, thereafter each horizontal and vertical row/column is Fourier transformed separately and added to rows and columns to yield mean value estimates for vertical and horizontal
PSDs, respectively. *PSD _{imag}e* is acquired by subtracting

*PSD*(without the RT) from the

_{sys}*PSD*(with the RT). Removing half of the spectrum, taking the square root, normalizing against low frequency and finally fitting a 4

_{measured}^{th}order polynomial to datapoints gives

*MTF*.

_{total}*MTF*is then divided by

_{total}*MTF*to obtain

_{sys}*MTF*. The flowchart of the MTF measurement procedure is presented in Fig. 1.

_{LUT}The time needed for the MTF test depends on the speed of data analysis and especially the speed of Fourier transformations. The speed of transformation depends on the algorithm used and
the size of the sample matrix (number of pixels), which again affects the achievable measurement precision. Increasing number of rows in the sample matrix improves measurement precision
since more rows are participating on averaging. On the other hand, having more samples in each row increases the number of data points within the measured frequency band (which again is
determined by the density of samples or pixel spacing), making data fitting more accurate. Note that the size of the sample matrix also affects the spatial resolution of the MTF
measurement itself, since the acquired result of certain field point is, in fact, an average result of the “field area” (with a desired field point at the centre) covered by the sample
matrix in the image space. Thus, reducing the size of the sample matrix improves speed and allows larger number of field points (of different heights) to be measured, but on the other hand
decreases precision of the measured MTF. Algorithms like the Cooley-Turkey reduces calculations through combining and decomposing symmetry properties of ordinary Fourier transforms. This
means that optimal speed versus sample matrix size is achieved when N=2^{p}, (p an integer) thus limiting the optimal sizes of the sample matrices to …, 32*32, 64*64, 128*128,
etc.[4]

The functionality of the above algorithm was verified by simulations. The MTF of a lens was first analyzed using a built-in diffraction MTF analysis of the optical design software Zemax. The second MTF estimation was obtained using simulated image of a random target and the presented algorithm. The simulated image was formed by the optical design software using ray tracing. On-axis MTFs for horizontal and vertical directions using a 128*128 sample matrix are depicted in Fig. 2 together with the result of diffraction MTF analysis (dashed). Simulations show good agreement.

## 3. Experimental results

#### 3.1 Instrument setup

The instrument setup for MTF measurement of lenses and lens-camera combinations is presented in Fig. 3. The instrument includes a random target, lens under test (LUT), light source with wavelength selection, a CCD camera and a standard PC for MTF calculations. The CCD camera is attached on a translation stage, which is moved by accurate and fast motorized actuators to adjust focus. The lens under test and the random target are fixed at an appropriate distance from each other and the camera. Since the bandwidth of the measurement instrument is determined by the pixel spacing, a black and white CCD camera with the smallest possible pixel spacing should be used in order to maximize measurement bandwidth. The camera should also have adequate frame rate, high responsivity (100% fill factor), low noise and preferably the same MTF in horizontal and vertical directions. The latter is achieved by means of square pixels and signal processing (frame grabber hardware/software) that does not alter the inherent, “geometrical” MTF of the imager and that of the sampling process. Using a common available one megapixel CCD with 5 µm pixel spacing results in measurement bandwidth of 50 cycles/mm, which is adequate for go/no-go inspection of low to medium quality lenses.

In experiments a CCD with 658*497 pixels having a pitch of 7.4 µm was used. Targets were created using a random number generator and they were printed with a standard laser printer on ordinary white paper or overhead transparency. Experiments showed that there is no difference between the MTF results measured with the above “home made” targets and the commercially available random targets based on laser speckle [5]. It was found that target illumination should be reasonably uniform and stable in order to obtain accurate and repeatable results. This was accomplished using a diffuser or an integrating sphere in front of an incoherent light source (e.g. dc powered quartz halogen lamp). Wavelengths beyond the visible range (>700 nm) were filtered out to minimize the imager noise and its MTF deterioration due to diffusion of carriers born deep in the substrate.

In order to increase the measurement bandwidth beyond that determined by the pixel density of the imager a setup including an additional lens for magnifying the image formed by the LUT could be used. Note, however, that in order to operate properly, the magnifying lens must have a higher numerical aperture than the LUT, its field of view must be sufficiently large to cover the total image area (sample matrix) and its MTF should be as high as possible (diffraction-limited) so that it would not affect the MTF of the LUT. In case of wide FOV and high NA LUT these requirements are very hard to fulfill. Scanning could be used to avoid these difficulties but it would also slow down the measurement. Thus for fast measurements direct imaging without scanning is preferred.

#### 3.2 Measurement results

First the measured MTF results of a lens-camera combination were compared with the results obtained from single frequency methods utilizing sinusoidal- and USAF bar targets. The measurement setup presented in Fig. 3 and a sample matrix size of 128*128 was used. The results (Fig. 4) are in good agreement.

Experimental results of the achievable precision, i.e. standard deviation of ten measurements, as a function of sample matrix size at spatial frequencies of 10, 20 and 30 cycles/mm are presented in Fig. 5. The results show that 2 to 5% precision is achievable using the sample matrix size of 128*128 and 64*64, respectively. This means that approximately ten “independent” field point heights can be measured with sufficient precision using a one megapixel imager as the analyzer.

The speed of the algorithm was tested using ordinary 800 MHz PC-computer for calculation. Using 128*128 sample matrix the calculation of tangential and sagittal MTF for one field point took a 0.1 seconds. Thus full test including best focus evaluation by performing on-axis MTF at 5 different image focus positions, and the measurement of tangential and sagittal MTF curves of 5 field points at the best focus plane would take only few seconds at most and provide better than 2 % precision.

## 4. Conclusion

A method and instrument setup for fast MTF inspection is proposed. The setup includes a random target, lens under test and a CCD camera with focus adjustment. The target consists of a random black and white pattern of a flat spatial power spectrum. The MTF is acquired by imaging the random target on the CCD using the lens under test, and then analyzing the spatial frequency content of the image. Measurement range up to about 50 cycles/mm is possible using direct imaging setup and commonly available CCD imagers. Measurement speed and precision depend on the sample matrix size used in calculation. Precisions better than 2 and 5 % seem to be achievable with sample matrix sizes of 128*128 and 64*64, respectively. Using the 128*128 matrix per field point provided a few second’s total execution time (ordinary 800 MHz PC-computer) per lens including best focus evaluation, and the measurement of tangential and sagittal MTF curves of 5 field points at the best focus plane. The results suggest that the random target method provides a fast way of MTF inspection for low and medium quality lenses.

## References and links

**1. **H. Kubota and H. Ohzu, “Method of Response Function by Means of Random Chart,” J.
Opt. Soc. Am. **47**, 666–667 (1957). [CrossRef]

**2. **A. Daniels, G. Boreman, A. Ducharme, and E. Sair, “Random transparency targets for modulation transfer measurement in
the visible and infrared regions,” Opt. Eng. **34**, 860–868 (1995). [CrossRef]

**3. **E. Levy, D. Peles, M. Opher-Lipson, and S.G. Lipson, “Modulation transfer function of a lens with a random target
method,” Appl. Opt. **38**, 679–683 (1999). [CrossRef]

**4. **E.C. Ifeachor and B.W. Jervis, *Digital signal processing* (Prentice Hall,
2002), *Ch. 3*.

**5. **Sine Patterns LLC, 3800 Monroe Avenue, Pittsford, NY 14534, USA.