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Digital image processing as an integral component of optical design Article in Proceedings of SPIE - The International Society for Optical Engineering · August 2008 DOI: 10.1117/12.798581
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Invited Paper
Digital image processing as an integral component of optical design Andrew R Harvey*a, Tom Vettenburga, Mads Demenikova, Betrand Lucottea, Gonzalo Muyoa, Andrew Woodb, Nicholas Bustinb, Amritpal Singhc, Ewan Findlayd a School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK; b Qioptiq Ltd, Glascoed Road, St. Asaph, Denbighshire LL17 0LL; c Development and Technology Sensors,SAAB Bofors Dynamics AB, SE-402 51,Göteborg, Sweden; d ST HPC PMG Imaging Div., ST Microelectronics R&D Ltd, 33 Pinkhill,Edinburgh,EH12 7BF, UK ABSTRACT The design of modern imaging systems is intricately concerned with the control of optical aberrations in systems that can be manufactured at acceptable cost and with acceptable manufacturing tolerances. Traditionally this involves a multiparameter optimisation of the lens optics to achieve acceptable image quality at the detector. There is increasing interest in a more generalised approach whereby digital image processing is incorporated into the design process and the performance metric to be optimised is quality of the image at the output of the image processor. This introduces the possibility of manipulating the optical transfer function of the optics such that the overall sensitivity of the imaging system to optical aberrations is reduced. Although these hybrid optical/digital techniques, sometimes referred as wavefront coding, have on occasion been presented as a panacea, it is more realistic to consider them as an additional parameter in the optimisation process. We will discuss the trade-offs involved in the application of wavefront coding to low-cost imaging systems for use in the thermal infrared and visible imaging systems, showing how very useful performance enhancements can be achieved in practical systems. Keywords: Pupil plane encoding, aberration tolerance, extended depth of field, zoom lens, thermal imaging, noise amplification, wavefront coding.
1. I1TRODUCTIO1 The salient aim in traditional approaches to optical design normally involves producing a compact, perhaps diffractionlimited, point-spread-function so that the image formed at the image plane is, as far as is possible, a facsimile of the object. The mitigation of all types of aberrations is accomplished through careful optimisation of lens configurations and surface profiles. Practical factors including lens volume, cost, manufacturability and manufacturing tolerances often impact on the ultimate achievable performance. It is generally the case that if costs dictate that a lens should be relatively simple, it is reasonably straightforward to design a lens using modern optimisation processes that produces good performance for a restricted set of parameters; for example, for a narrow field of view or at a specific temperature. There is normally a trade off, however, such that high performance imaging for more demanding parameters normally requires increased complexity in the optical design; for example additional lens elements to increase the field-of-view or to athermalise the lens. We describe here a modern approach to lens design that seeks to achieve acceptable imaging performance in the presence of significant aberrations and variations in the magnitude of these aberrations. We start by considering an imaging system as a Linear-Time-Invariant system for transmitting information about the scene, in the form of an image, through the filtering function formed by the lens, electronic detector, image processing electronics to the output formed on the display. Whereas conventional lens design considers the filtering function of only the lens, that is, its Modulation Transfer Function MTF), we consider here the filtering function of the complete information channel. The figure of merit for the complete imaging system is then determined, not by the quality of the optical image formed at the detector, but by the quality of the electronic image formed on the output display. Clearly this enables the interaction between the filter functions of the lens, detector and crucially the digital-signal-processor to be included holistically within the optimisation process. More importantly it enables aberrations introduced by the lens to be corrected by digital signal processing electronics. Correction can be achieved post hoc to correct for unavoidable aberrations, as was demonstrated by the *
[email protected]; phone +44 (0)131 451 3356; fax +44 (0)131 451 4155; http://www.ece.eps.hw.ac.uk/~arharvey
Novel Optical Systems Design and Optimization XI, edited by R. John Koshel, G. Groot Gregory, James D. Moore Jr., David H. Krevor, Proc. of SPIE Vol. 7061, 706104 · © 2008 SPIE CCC code: 0277-786X/08/$18 · doi: 10.1117/12.798581 Proc. of SPIE Vol. 7061 706104-1 2008 SPIE Digital Library -- Subscriber Archive Copy
digital correction of spherical aberration introduced by the defective primary mirror of the Hubble Space Telescope[1], this is nearly always sub-optimal due to loss of image information by the aberrated lens associated with nulls in the lens MTF. With mathematical or numerical optimisation it is possible to design lenses with special aberrations, which although involving some suppression of the MTF, exhibit two very important properties: (1) no nulls in the MTF and (2) an approximate invariance of the MTF with respect to other aberrations, such as defocus [2, 3]. The first feature enables image restoration without excessive amplification of noise introduced during image detection and the second provides insensitivity to variations in the magnitude of an aberration. Dowski and Cathey[4] showed, using the stationary phase approximation of the ambiguity function, that a phase mask with a cubic sag located in the exit pupil of a lens provides optimal insensitivity to defocus and exhibits no nulls in the MTF, although this was restricted to the case of a onedimensional pupil. For a rectangular two-dimensional pupil, the sag is of the form , where x and y are normalised pupil coordinates and α is a factor to control the peak-to-valley magnitude of the phase difference introduced. In more recent years several phase mask functions have been evaluated using criteria such as the on-axis intensity[5], the invariance of the PSF and restorability[6]. As for the cubic phase mask, these phase function produce a PSF that is less sensitive to focusing errors. Other phase mask functions have been shown to alleviate coma and astigmatism[7], or spherical aberration[8]. The merits of a wide variety of phase masks have been discussed in the literature, often comparing them to the well-studied cubic phase mask. The examples presented here will therefore concentrate on the cubic phase mask. In the next section we briefly outline the principles of this approach. In section 3 we describe some applications where the benefits of this approach have been exploited and in section 4 we outline the quid pro quo.
2. CO1CEPT OF PSF-E1GI1EERI1G The joint design of the optics and digital post-processing has been termed wavefront-coding or pupil-phase-engineering. Since there is general the decoding is executed by use of the encoding point-spread-function rather than directly on the pupil or the wavefront, we prefer the term point-spread-function-engineering as a more general and apposite. The traditional approach to optical design is summarised in Fig. 1 below. The general design aim is to focus the light emanating from points in the object plane to a compact PSF disk on the image plane for a range of defocus, field-angles, chromatic aberrations and system tolerances. By iteratively altering the lens variables and comparing the resulting PSFs, it is possible to enhance the optical design until a satisfactory image is obtained under all operating conditions.
Fig. 1: Schematic of the traditional optical design process. The system’s performance is measured directly on the optical image.
In the absence of tolerances the traditional approach to optical design can yield the system with the optimal performance, subject to the fundamental limitation that imaging performance varies with variations in aberrations. Often this can be addressed with increased lens complexity (normally involving increased cost, volume and weight), however some fundamental aberrations, such as defocus, cannot be mitigated without sacrificing some other aspect of imaging performance, such as optical throughput. The incorporation of PSF engineering into the design process combined with a more holistic approach in which the complete imaging system is incorporated into the optimisation process provides
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scope for designing relatively simple lenses that are insensitive to aberrations and hence to changes in the aberrations so that the system PSF can be of a high quality for a larger range of system parameters. When digital image restoration becomes an integral part of the imaging process, it is no longer necessary to have a compact, near-diffraction-limited PSF at the optical image plane. A phase mask can introduce deviations from the perfect spherical wavefront, with the only limitation that sufficient information is retained in the image so it can be restored correctly. With PSF-engineering it is therefore crucial to include the image restoration step in the iteration loop as depicted in Fig. 2. Now the image at the detector plane is not assessed directly, rather the optical configuration is evaluated based on the restored image.
Fig. 2: Schematic of the hybrid optical/digital design approach. The performance is measured on the processed image.
The introduction of a phase mask can improve the overall system characteristics such as tolerance to defocus and higher order aberrations such as coma. This effect can be enhanced by increasing the peak-to-valley phase difference introduced by the phase mask. At the same time the point-spread-function at the optical image plane can cover a larger area. Based on traditional optical design metrics, these systems would be considered inferior. Indeed, without image restoration the system would be unusable due to the strong blurring. Nevertheless, this is no indication of the final restored image quality as is shown in Fig. 3 below.
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Fig. 3: Effect of PSF-engineering in a hybrid imaging system. Left: the diffraction-limited image; centre: the image as recorded by the detector array; right: the detected image after the deconvolution.
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The image can be restored only when the optical transform from object space to image space retains all required information. For a diffraction-limited system all information up to the cut-off spatial frequency is passed to the image plane. In aberrated systems the magnitude of the modulation-transfer-function (MTF) is reduced and can contain zeros, even for moderate aberrations as is shown in Fig. 4 for defocus. Information in the region of the spatial frequencies is lost, preventing restoration of a high-quality image.
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Fig. 4: Left: the MTF of an optical system, shown for various degrees of defocus. Right: examples of the impulse response.
The general form of the MTF for a diffraction-limited system is due to the redundancy with which spatial frequencies are transmitted through the lens: the higher redundancy of low spatial frequencies which coherently add leads to higher MTF than for mid and high frequencies. Some redundancy is needed for a good signal-to-noise ratio, but in general it is possible to retrieve the required information when the redundancy and the associated MTF are reduced. In the presence of aberrations this redundancy is broken, such that spatial frequencies recorded from different parts of the pupil are not co-phased such that the vector sum is reduced and at the spatial frequencies for which nulls occur, sum to zero[9]. Image reconstruction is possible as long as the magnitude and phase can be retrieved accurately for all desired spatial frequencies. The fundamental principle of PSF engineering is that it introduces a phase modulation in the spatial frequencies transmitted by different parts of the pupil that both dominates that introduced by optical aberrations and prevents the vector sum of redundant spatial frequencies approaching zero. As can be seen in Fig. 5, it is possible to trade this redundancy for defocus tolerance with an appropriate choice of phase mask. —w20=O waves
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3. EXAMPLES In this section we discuss some applications that exploit the ability of PSF engineering to mitigate optical aberrations. 3.1 Athermalisation of a thermal imager One of the prime examples of PSF-engineering is aberration control in thermal imagers. Defocus caused by temperature changes is a significant problem for these types of systems[10]. Lenses for thermal imaging are predominantly made from semiconductors such as germanium, a material chosen for its high refractive index and low dispersion. Unfortunately this material is also very sensitive to temperature variations as can be seen from the MTFs of a thermal imager using two germanium elements, shown in Fig. 6. Temperature variations introduce defocus and cause significant reductions in the MTF and even nulls for small variations in temperature. Additional corrective lenses increase price and
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size of the imager. On the other hand, a phase mask can be incorporated in the front face of the lens to reduce the MTF variation with temperature[11, 12]. Post-detection digital image processing is applied to the detected image to remove the temperature-invariant blur and obtain near-diffraction-limited quality, Fig. 6 compares the MTFs of the system with and without PSF-engineering.
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Fig. 6: Top row: MTFs of a thermal imager with two germanium lenses operating at 0, 20 and 70°C; Bottom row: MTFs of a thermal imager with a cubic phase mask before (grey) and after restoration (black).
3.2 Simplification of a thermal imager: singlet The complexity and cost of imaging systems can be reduced by the combined application of digital image processing and optical manipulations. In this example, by using PSF-engineering it is possible to reduce the lens count of a traditional two-element germanium infrared imaging system to a singlet whilst maintaining optical performance. The germanium singlet unavoidably suffers from astigmatism and field curvature. The performance in terms of the modulation transfer function (MTF) is practically diffraction-limited on-axis but rapidly falls even for narrow fields-of-view, as shown in Fig. 7. Simulations suggest that PSF-engineering can be useful to correct for the aforementioned aberrations. As can be observed from Fig. 7 (right), the variation of the MTF is minimal after introduction of an optimised phase mask. The relative invariance of the modulation transfer function can be used for accurate and fast restoration with linear imagerestoration techniques.
Fig. 7: The MTF for various fields-of-view directions. Left: without phase mask at the entrance pupil; Right: with an assymetric phase mask at the entrance pupil.
These simulations have been experimentally verified by placing a germanium phase mask in front of a single-element thermal imager[13]. In Fig. 8 an example is shown of thermal images recorded with the original singlet and with the PSF-engineered singlet. The introduction of the phase mask and the digital post-processing ensures invariance against
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aberrations for a wider field-of-view and this translates directly in an improved image quality. No new lenses had to be introduced, thereby keeping the cost and the size of the imager to a minimum.
Fig. 8: Image taken with a singlet thermal imager. The image on the left is produced by the unaltered system, off-axis blurring is visible towards the sides of the image. On the right-hand side the same image is taken after a phase mask is introduced before the germanium singlet and the obtained image is digitally restored.
The benefits of PSF-engineering are certainly not restricted to thermal imaging. Reduction of sensitivity to defocus and aberration is welcome for most low-cost imaging systems. Due to their particular design requirements, zoom-lenses are exceptionally good candidates for PSF-engineering. 3.3 Zoom lens An asymmetric two-lens system suffers from a rapid variation in optical aberrations during zooming. Two symmetric zoom-lens configurations with stationary lenses (A and C) and a moving lens B have been described in [14] and are shown in Fig. 9.
Fig. 9: Two zoom lens configurations with one moving element and a total of three elements showing the defocus related to the movement of lens element B. The defocus is exaggerated to show the principle and the direction. Typically, the defocus for the two zoom lens configurations is not identical in magnitude as indicated by the dashed curve.
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The above zoom-lenses need extreme lengths in order to have a defocus which is tolerable for imaging. Reducing the length of the zoom-lens would increase the image defocus further. The common solution to this problem is to compensate the defocus by moving lens C opposite to the defocus curve. By implementation of wavefront coding or other PSF-engineering techniques, the defocus can be tolerated such that lens C can be kept stationary. In this way, point-spread-function-engineering techniques can be used in a simplification and miniaturization process of zoom lenses. This allowed us to reduce the length of the zoom lens system from 62mm to 10mm using a cubic phase mask to correct for the 1.76λ defocus that arises in the reduced design [15]. 3.4 Microscopy Light from out-of-focus planes is often considered detrimental to microscopy images. Points in these planes are blurred and reduce the visibility of the in-focus objects. Confocal microscopy techniques aim to attenuate radiation from out-offocus planes. Still, the specimen of interest is often not perfectly confined to the focal plane. As an example, consider the two sections in Fig. 11 of a single high-resolution image of a lung with bronchus observed with an F/0.83 microscope at λ=640nm. While the image as a whole shown in Fig. 10 is well-focused, the enlarged image sections shown in the top row show artefacts and blurring. The depth-of-field extending effect of asymmetric phase masks can reduce this problem. The results presented in Fig. 11 compare the non-encoded images with the PSF-engineered images restored using a single kernel. The experiments are conducted using a generalised cubic phase mask with an optical path delay function with α =4.9λ, β=-3α=-14.7λ, the coordinates are normalised to the aperture stop size. The image sections on the second row of Fig. 11 on the next page show the images taken after the phase mask is introduced. The full-resolution image is restored with a Wiener filter and the respective restored details are shown in the bottom row. More detail is retained in these images and the artefacts of the image sections in the top row are no longer present in the PSF-engineered images in the bottom row.
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Fig. 11: Two sections of one high-resolution microscope image shown in Fig. 10. The sections of the image in the top row are taken without phase mask, the sections in the centre row are with phase mask, and the sections in the bottom row are taken from the Wiener filter restored image with a phase mask.
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4. CHALLE1GES As demonstrated with the previous examples, PSF-engineering has a wide range of applications. The gained design freedom also makes the design process a more complex task. This section discusses the new challenges that are faced. 4.1 Decrease in signal-to-noise ratio Without digital-image-restoration, the magnitude of the OTF of an optical system is reduced by the introduction of a phase mask(Fig. 5). If the measurement noise remains the same, the reduced redundancy in the optical image translates directly into a reduced signal-to-noise ratio. Although the image restoration process is able to restore the signal back to its normal level, it cannot improve the signal-to-noise ratio. The optimal phase mask will therefore often be a trade-off between noise-gain and improved aberration tolerance or any other desired characteristic. The additional design freedom gained by the introduction of phase masks is nevertheless significant, and further improvements in photo-electric sensor array sensitivity make this limitation less relevant. In order to reduce noise amplification in the restored image, it is common for digital deconvolution techniques to regulate the signal amplification as a function of the (expected) signal-to-noise ratio. The Wiener filter has a parameter explicitly for this purpose, and e.g. the Tikhonov filter can be regulated to favour smooth images by suppressing high frequencies using a Laplacian filter. A performance metric for a PSF-engineered system should also account for this regularization effect. 4.2 Calculation of the deconvolution kernel The task of the deconvolution process is to reverse any amplitude attenuations and phase changes in the optical transfer function. This can only be done if the optical transfer function is precisely known, not only in magnitude but also in phase. Given an amount of defocus or aberration, the optical transfer function can be calculated or measured precisely. Even though the introduction of a phase mask can give a significant improvement in the system’s tolerances, the impulse response will never be entirely invariant to changes. By consequence, images will show artefacts unless the correct deconvolution kernel is used.
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Fig. 12: The intensity of an edge imaged through a rectangular cubic phase mask and restored with an inverse filter. The in-focus deconvolution kernel is used for various degrees of defocus. The top left image shows the in-focus edge(w20=0λ), the top right image shows the edge out of focus with w20=1λ and restored with the same in-focus kernel. The bottom left and right plot show the edge out of focus with w20=2λ and w20=5λ respectively.
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In Fig. 12 the effects of deconvolution with the same in-focus kernel for various degrees of defocus is demonstrated with the image of an edge. Increased mismatch of the deconvolution kernel causes strong variations near the edge, in the image this will result in ringing artefacts. An example of this is shown in Fig. 13.
Fig. 13: Image of Lena simulated with w20=1(left) and 5 waves of defocus(right) and restored with the in-focus kernel.
A potential solution to the PSF variance is to choose the kernel that gives best contrast for the image at hand. Practically this can be achieved by iterative optimisation of the kernel against a cost function related to the artefacts introduced in an image. 4.3 Spatially variant PSF It is also possible that the impulse response is spatially-variant, if this is the case the image should be deconvolved with a spatially variant kernel for optimal results. The assumption that the optical impulse response is spatially invariant is often only partially true. As shown in Fig. 14, off-axis aberrations can become too large to ignore towards the sides of the image. Many image restoration algorithms rely on the spatially invariance for their efficiency, but deconvolution with a single kernel will only restore a part of the image optimally while the remainder shows artefacts. Still in most practical situations the variation is only gradual. In that case the deconvolution can be done by blocks where inside each block PSF-invariance is assumed.
Fig. 14: The spatial variation of the PSF is shown for a ~3λ phase mask. Each PSF box corresponds to a 1.95-by-1.95 mm square in the image plane. The distance between the PSF boxes is not to scale. The total field-of-view of the system is 9.2° by 7°.
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5. CO1CLUSIO1S Optical design has gained an additional degree of freedom by considering digital-image-processing as an integral part of the design. In this paper we have shown how this design freedom can be exploited and what trade-offs are involved. Its most appreciated application today is increased tolerance to aberrations, allowing larger operational ranges and/or lower production costs. This has been demonstrated with examples in thermal imaging, zoom-lens systems and microscopy. The challenges met in the design of hybrid optical/digital systems are discussed. An important factor is the reduction in the signal-to-noise ratio, it is therefore crucial to consider the noise gain as an extra parameter in the design process.
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R. L. White, "Restoration of Images and Spectra from the Hubble Space Telescope," ASP Conference Series. 25, 176–185. J. Ojeda-Castañeda, P. Andrés, and A. Díaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration.,” Optics Letters, 11, 487-489 (1986). Z. Zalevsky, A. Shemer, A. Zlotnik et al., “All-optical axial super resolving imaging using a low-frequency binary-phase mask,” Opt. Express, 14(7), 2631-2643 (2006). J. Edward R. Dowski, and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt., 34(11), 1859-1866 (1995). W. Chi, and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett., 26(12), 875-877 (2001). S. Prasad, V. P. Pauca, R. J. Plemmons et al., "Pupil-phase optimization for extended-focus, aberrationcorrected imaging systems," SPIE. 5559, 335-345. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A, 23(5), 1058-1062 (2006). S. Mezouari, and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett., 28(10), 771-773 (2003). G. Muyo, and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett., 30(20), 2715-2717 (2005). A. Mann, [Infrared Optics and Zoom Lenses] SPIE press, Bellingham, WA(2000). G. D. Muyo, and A. R. Harvey, "Wavefront coding for athermalization of infrared imaging systems," SPIE. 5612, 227-235. G. Muyo, A. Singh, and A. R. Harvey, "High performance thermal imaging with a singlet and pupil plane encoding," SPIE. 5987, 59870I.1-59870I.8. G. Muyo, A. Singh, M. Andersson et al., "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," SPIE. 6395, 63950M. K. Yamaji, [Progress in Optics] North-Holland, Amsterdam(1967). M. Demenikov, E. Findlay, and A. Harvey, "Miniaturization and simplification of zoom lenses using wavefront coding," Proceedings of the SPIE. 7061A. This conference.
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