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OPTICS LETTERS / Vol. 39, No. 13 / July 1, 2014
Wavefront-coding technique for inexpensive and robust retinal imaging Justo Arines,1 Rene O. Hernandez,2 Stefan Sinzinger,3 A. Grewe,3 and Eva Acosta2,* 1
Department of Applied Physics (Area of Optometry), Faculty of Optics and Optometry, Campus Vida, Universidad de Santiago de Compostela, 15782, Spain 2
3
Department of Applied Physics (Area of Optics), Faculty of Physics, Campus Vida, Universidad de Santiago de Compostela, 15782, Spain
Technical University of Ilmenau, Department of Engineering Optics, P.O. Box 100565, 98684, Ilmenau, Germany *Corresponding author: [email protected] Received March 17, 2014; revised May 20, 2014; accepted May 20, 2014; posted May 22, 2014 (Doc. ID 208079); published June 27, 2014
We propose a hybrid optical-digital imaging system that can provide high-resolution retinal images without wavefront sensing or correction of the spatial and dynamic variations of eye aberrations. A methodology based on wavefront coding is implemented in a fundus camera in order to obtain a high-quality image of retinal detail. Wavefront-coded systems rely simply on the use of a cubic-phase plate in the pupil of the optical system. The phase element is intended to blur images in such a way that invariance to optical aberrations is achieved. The blur is then removed by image postprocessing. Thus, the system can provide high-resolution retinal images, avoiding all the optics needed to sense and correct ocular aberration, i.e., wavefront sensors and deformable mirrors. © 2014 Optical Society of America OCIS codes: (330.4300) Vision system - noninvasive assessment; (330.7327) Visual optics, ophthalmic instrumentation; (110.7348) Wavefront encoding. http://dx.doi.org/10.1364/OL.39.003986
High-resolution images of the retina are needed to diagnose and monitor the progress of certain diseases and pathologies as well as to research the photoreceptor structure. Acquiring high-resolution in vivo images requires correction for eye aberrations including tear film and saccadic eye movements. In order to overcome these problems, adaptive optics (AO) and deconvolution from wavefront-sensing (DWFS) or blind-deconvolution (BD) systems were implemented in retinal imaging instruments [1–3]. AO systems come with a wavefront sensor to measure ocular aberrations and a deformable mirror to compensate for these aberrations. This is expensive, thus limiting the system’s availability to only a few hospitals and research laboratories able to take full advantage of the potential benefits of high-resolution retinal images. DWFS is a more economical approach but a wavefront sensor is still needed and, more importantly, the presence of a significant degree of defocus and astigmatism reduces DWFS effectiveness due to the low signal-to-noise ratio of the recorded image. As an alternative, partially compensated DWFS is proposed. In this case, the static aberrations of the eye are corrected with a static optical element in such a way that the image degradation induced by the remaining dynamic-aberration component is corrected by the deconvolution process [4]. The technique proposed in 1995 by Dowski and Cathey called wavefront coding (WFC) enables the depth of field of incoherent optical systems to be extended [5–7]. This method often involves a cubic-phase plate in the shape of Ax3 y3 at the exit pupil plane of the optical system yielding a blurred image nearly invariant to defocus. In other words, this optical element modifies the imaging system in such a way that the resulting point-spread function (PSF) and optical transfer function (OTF) are insensitive to defocus, which results in images with a well-defined and predictable amount of blur. Therefore, images can be digitally restored in order to produce a 0146-9592/14/133986-03$15.00/0
sharp final image by means of an appropriate deconvolution filter, unique to the object distances included in the improved depth of field of the WFC system. Thus, depth of focus can be extended by a cubic-phase plate, and the extension ratio is dependent on the peak-to-valley value (or strength) of the phase plate, A. This idea was recently extended to also correct the other second order aberrations, i.e., astigmatism [8]. In this work, we will show that implementing a cubic mask in the shape of a trefoil [9]Ar3 cos3θ in a retinal imaging system can provide clear images of the retina, which also means that the technique can be applied to correct high-order aberrations. For the experimental setup, a phase plate was made using an ultraprecision micromilling technique in transparent polymethylmethacrylate (PMMA) [10]. The dimensions of the element are 10 mm × 10 mm with maximum sag of ∼1.8 mm. The optical quality of the plate was characterized with a point diffraction interferometer (PDI) [11]. Figure 1 shows (a) the fabricated phase plate, (b) the recorded interferogram within a region of 7.0 mm diameter, (c) the resulting fitted phase, and (d) the corresponding PSF. Interferometric analysis shows that the fabricated phase plate not only generates trefoil but also small amounts of other aberrations, mainly spherical aberration, coma, and astigmatism, as shown in Fig. 2. Nevertheless, as will be shown in this work, these aberrations will have no practical effect on the final results, thus providing a certain degree of tolerance in the fabrication process. The experimental setup for the retinal imaging system is shown in Fig. 3. The artificial eye (AE) consists of a 1951 USAF resolution target as an artificial retina (AR) and a converging lens of 25.4 mm focal length with a diaphragm of 5 mm diameter. The retina is back-illuminated with an LED (at 632 nm) and focused through an optical system into a © 2014 Optical Society of America
July 1, 2014 / Vol. 39, No. 13 / OPTICS LETTERS
Fig. 1. (a) Trefoil phase plate, (b) phase fringes provided by point diffraction interferometry, (c) retrieved phase, and (d) simulated PSF.
charge-coupled device (CCD) camera, a Hamamatsu ORCA R2 with 6.45 μm × 6.45 μm pixel size. In the light path, two sets of afocal 4f optical systems (each with lens focal lengths of 100 mm) are used to image the AE pupil plane in two different intermediate planes: PP1 and PP2. Eye aberration is added in PP1 by placing a phase plate fabricated in photoresist corresponding to the eye aberration of subject JA [12]. In Fig. 4 we show the interference fringes provided by the PDI. In order to better appreciate the astigmatism and high-order aberrations, 0.4
40
Aberrations (Waves)
Trefoil Aberration (Waves)
45 35 30 25 20 15 10 5 0 0.5
1
1.5
2 2.5 3 Radius (mm)
3.5
4
0.2 0 -0.2 Spherical Aberration
-0.4
Astigmatism Coma
-0.6 -0.8 0.5
1
1.5 2 2.5 3 Radius (mm)
3.5
4
(b)
(a)
Fig. 2. Aberrations of the fabricated phase plate for different pupil radii: (a) trefoil aberration and (b) other significant aberrations.
Fig. 3.
Experimental setup.
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Fig. 4. Interferogram of the phase plate used to generate JA eye aberrations within a pupil of 7 mm diameter. Spherical power was compensated with a spherical carrier wave in the PDI.
the interferogram in Fig. 4 is recorded with a spherical carrier wave to compensate for spherical power. This aberration consists of −1.5D spherical power, −1.5D astigmatism and high-order aberrations with a root-mean-square value of 0.686 μm. The trefoil aberration for the WFC is added in PP2 by adding the phase plate described above. The peak-to-valley value of the trefoil aberration corresponds to 30 waves for the 5 mm diameter of the artificial pupil used for the experiment. We used a 100-mm-focal-length lens placed at the exit pupil of the system to image the retina in the CCD camera. A collimated laser beam (CL) was used to obtain the experimental PSFs. The pixelated PSF used in the restoration was computed numerically using the magnitude of the WFC measured with the PDI and compared with the experimental PSF in order to adjust orientation and size. Next, we introduced the phase plate in PP1 to generate eye aberrations and recorded the retinal image to show the degradation. Then the WFC phase plate was placed in PP2 and the recorded image (which we will call the intermediate image) is restored by means of a regularized Wiener filter as explained below [13]. In both cases, we searched for the best focus providing the clearest image. This image contains the blurring effects of the eye aberration and the WFC plate which, as explained above, also generates a small degree of astigmatism, coma, and spherical aberration. All images are recorded with a low level of light within the limits of permissible radiation exposure [14]. Figure 5(a) shows the PSF corresponding to the AE. Figures 5(b) and 5(c) show the PSF for the WFC phase plate and for AE WFC, respectively. The similarity of the two images demonstrates the feasibility of using the PSF for the WFC phase plate to restore the image obtained in the presence of unknown ocular aberrations. Once the intermediate images have been taken, median filtering is performed to remove noise, and the image is then restored by deblurring with a Wiener filter. Figure 6(a) shows the image of the AR blurred by eye aberrations and Fig. 6(b) shows a high-contrast image recovered with WFC after deconvolution is applied, showing that WFC does provide sharp images (up to 10 μm resolution) with no need to measure or compensate for eye and setup aberrations. (We show in this work only
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OPTICS LETTERS / Vol. 39, No. 13 / July 1, 2014
Fig. 6. (a) Image blurred by AE aberrations and (b) wavefrontcoded image.
Fig. 5. Experimental PSFs: (a) AE aberrations, (b) WFC phase plate, and (c) AE WFC aberrations.
one result for JA’s eye aberrations and for a fixed orientation of the phase plate that generates these aberrations. The same results have been obtained not only for different orientations and centering of the plate, but also for other phase plates with different eye aberrations.) Here we need to stress the following facts: (1) In practice, pupil size may vary, therefore, the pupil size must be controlled in order to recalculate the restoring trefoil PSF for the corresponding pupil diameter to be used in the deconvolution process. (2) No further filtering [15] has been used to compensate for noise or eliminate restoration artifacts in order to improve the restored images. Moreover, USAF targets have high contrast. Real retinas will have low contrast. Therefore, algorithms must be adapted not only for lower contrast images, but also for different signal-to-noise ratios. (3) The peak-to-valley value of the trefoil mask is not fully optimized. Moreover, other cubic designs could also improve the performance. This implies that while the proposed method performs reasonably well, it is still far from optimal and could be further improved. In this connection, future work could include the joint process of optimizing the phase-mask design and digital processing for maximum image enhancement in real situations. This work proves that implementing wavefront-coding techniques in a fundus camera can provide highresolution images of the retina. The experimental setup does not contain either wavefront sensors or deformable mirrors to detect and compensate for high-order aberrations, which significantly reduces the complexity, size, weight, and cost of the device. Moreover, the device does not need full correction for defocus, astigmatism, and extra aberrations caused by fabrication flaws in the WFC mask. All of these are dealt with in the deconvolution process.
We present a different approach to achieving quality retinal images that appears to be an advantageous alternative to adaptive optics in terms of robustness and cost, leading to the early detection of retinal diseases and a greater volume of research in this field. These are just some of the most significant potential benefits of an affordable fundus camera like the one presented here. This work was supported by the Spanish Ministerio de Economia y Competitividad grant FIS2012-38244-C02-01 and Xunta de Galicia grant CN 2012/156. Dr. Justo Arines thanks the Xunta de Galicia (Spain) Isidro Parga Pondal 2009 Programme for financial support. References 1. E. J. Fernandez, I. Iglesias, and P. Artal, Opt. Lett. 26, 746 (2001). 2. D. Catlin and C. Dainty, J. Opt. Soc. Am. A 19, 1515 (2002). 3. V. Nourrit, B. Vohnsen, and P. Artal, J. Opt. A Pure Appl. Opt. 7, 585 (2005). 4. J. Arines, Opt. Commun. 284, 1548 (2011). 5. E. R. Dowski and W. T. Cathey, Appl. Opt. 34, 1859 (1995). 6. S. Bradburn, W. T. Cathey, and E. R. Dowski, Appl. Opt. 36, 9157 (1997). 7. H. B. Wach, W. T. Cathey, and E. R. Dowski, Appl. Opt. 37, 5359 (1998). 8. A. B. Samokhin, A. N. Simonov, and M. C. Rombach, J. Opt. Soc. Am. A 26, 977 (2009). 9. J. R. Dowski, “Mechanically-adjustable optical phase filters for modifying depth of field, aberration-tolerance, antialiasing in optical systems,” U.S. patent 7,180,673 B2 (February 20, 2007). 10. S. Stoebenau, R. Kleindienst, M. Hofmann, and S. Sinzinger, Proc. SPIE 8126, 812614 (2011). 11. E. Acosta, R. Blendowske, and S. Chamadoira, J. Opt. Soc. Am. A 23, 632 (2006). 12. R. Navarro, E. Moreno-Barriuso, and S. Bara, Opt. Lett. 25, 236 (2000). 13. R. C. Gonzalez and R. E. Wood, Digital Image Processing (Pearson Education, 2008). 14. http://www.lia.org/publications/ansi/Z136‑3. 15. J. A. Guerrero-Colon and J. Portilla, “Two-level adaptive denoising using Gaussian scale mixtures in overcomplete oriented pyramids,” in 12th IEEE International Conference on Image Processing, 2005.
OPTICS LETTERS / Vol. 39, No. 13 / July 1, 2014
Wavefront-coding technique for inexpensive and robust retinal imaging Justo Arines,1 Rene O. Hernandez,2 Stefan Sinzinger,3 A. Grewe,3 and Eva Acosta2,* 1
Department of Applied Physics (Area of Optometry), Faculty of Optics and Optometry, Campus Vida, Universidad de Santiago de Compostela, 15782, Spain 2
3
Department of Applied Physics (Area of Optics), Faculty of Physics, Campus Vida, Universidad de Santiago de Compostela, 15782, Spain
Technical University of Ilmenau, Department of Engineering Optics, P.O. Box 100565, 98684, Ilmenau, Germany *Corresponding author: [email protected] Received March 17, 2014; revised May 20, 2014; accepted May 20, 2014; posted May 22, 2014 (Doc. ID 208079); published June 27, 2014
We propose a hybrid optical-digital imaging system that can provide high-resolution retinal images without wavefront sensing or correction of the spatial and dynamic variations of eye aberrations. A methodology based on wavefront coding is implemented in a fundus camera in order to obtain a high-quality image of retinal detail. Wavefront-coded systems rely simply on the use of a cubic-phase plate in the pupil of the optical system. The phase element is intended to blur images in such a way that invariance to optical aberrations is achieved. The blur is then removed by image postprocessing. Thus, the system can provide high-resolution retinal images, avoiding all the optics needed to sense and correct ocular aberration, i.e., wavefront sensors and deformable mirrors. © 2014 Optical Society of America OCIS codes: (330.4300) Vision system - noninvasive assessment; (330.7327) Visual optics, ophthalmic instrumentation; (110.7348) Wavefront encoding. http://dx.doi.org/10.1364/OL.39.003986
High-resolution images of the retina are needed to diagnose and monitor the progress of certain diseases and pathologies as well as to research the photoreceptor structure. Acquiring high-resolution in vivo images requires correction for eye aberrations including tear film and saccadic eye movements. In order to overcome these problems, adaptive optics (AO) and deconvolution from wavefront-sensing (DWFS) or blind-deconvolution (BD) systems were implemented in retinal imaging instruments [1–3]. AO systems come with a wavefront sensor to measure ocular aberrations and a deformable mirror to compensate for these aberrations. This is expensive, thus limiting the system’s availability to only a few hospitals and research laboratories able to take full advantage of the potential benefits of high-resolution retinal images. DWFS is a more economical approach but a wavefront sensor is still needed and, more importantly, the presence of a significant degree of defocus and astigmatism reduces DWFS effectiveness due to the low signal-to-noise ratio of the recorded image. As an alternative, partially compensated DWFS is proposed. In this case, the static aberrations of the eye are corrected with a static optical element in such a way that the image degradation induced by the remaining dynamic-aberration component is corrected by the deconvolution process [4]. The technique proposed in 1995 by Dowski and Cathey called wavefront coding (WFC) enables the depth of field of incoherent optical systems to be extended [5–7]. This method often involves a cubic-phase plate in the shape of Ax3 y3 at the exit pupil plane of the optical system yielding a blurred image nearly invariant to defocus. In other words, this optical element modifies the imaging system in such a way that the resulting point-spread function (PSF) and optical transfer function (OTF) are insensitive to defocus, which results in images with a well-defined and predictable amount of blur. Therefore, images can be digitally restored in order to produce a 0146-9592/14/133986-03$15.00/0
sharp final image by means of an appropriate deconvolution filter, unique to the object distances included in the improved depth of field of the WFC system. Thus, depth of focus can be extended by a cubic-phase plate, and the extension ratio is dependent on the peak-to-valley value (or strength) of the phase plate, A. This idea was recently extended to also correct the other second order aberrations, i.e., astigmatism [8]. In this work, we will show that implementing a cubic mask in the shape of a trefoil [9]Ar3 cos3θ in a retinal imaging system can provide clear images of the retina, which also means that the technique can be applied to correct high-order aberrations. For the experimental setup, a phase plate was made using an ultraprecision micromilling technique in transparent polymethylmethacrylate (PMMA) [10]. The dimensions of the element are 10 mm × 10 mm with maximum sag of ∼1.8 mm. The optical quality of the plate was characterized with a point diffraction interferometer (PDI) [11]. Figure 1 shows (a) the fabricated phase plate, (b) the recorded interferogram within a region of 7.0 mm diameter, (c) the resulting fitted phase, and (d) the corresponding PSF. Interferometric analysis shows that the fabricated phase plate not only generates trefoil but also small amounts of other aberrations, mainly spherical aberration, coma, and astigmatism, as shown in Fig. 2. Nevertheless, as will be shown in this work, these aberrations will have no practical effect on the final results, thus providing a certain degree of tolerance in the fabrication process. The experimental setup for the retinal imaging system is shown in Fig. 3. The artificial eye (AE) consists of a 1951 USAF resolution target as an artificial retina (AR) and a converging lens of 25.4 mm focal length with a diaphragm of 5 mm diameter. The retina is back-illuminated with an LED (at 632 nm) and focused through an optical system into a © 2014 Optical Society of America
July 1, 2014 / Vol. 39, No. 13 / OPTICS LETTERS
Fig. 1. (a) Trefoil phase plate, (b) phase fringes provided by point diffraction interferometry, (c) retrieved phase, and (d) simulated PSF.
charge-coupled device (CCD) camera, a Hamamatsu ORCA R2 with 6.45 μm × 6.45 μm pixel size. In the light path, two sets of afocal 4f optical systems (each with lens focal lengths of 100 mm) are used to image the AE pupil plane in two different intermediate planes: PP1 and PP2. Eye aberration is added in PP1 by placing a phase plate fabricated in photoresist corresponding to the eye aberration of subject JA [12]. In Fig. 4 we show the interference fringes provided by the PDI. In order to better appreciate the astigmatism and high-order aberrations, 0.4
40
Aberrations (Waves)
Trefoil Aberration (Waves)
45 35 30 25 20 15 10 5 0 0.5
1
1.5
2 2.5 3 Radius (mm)
3.5
4
0.2 0 -0.2 Spherical Aberration
-0.4
Astigmatism Coma
-0.6 -0.8 0.5
1
1.5 2 2.5 3 Radius (mm)
3.5
4
(b)
(a)
Fig. 2. Aberrations of the fabricated phase plate for different pupil radii: (a) trefoil aberration and (b) other significant aberrations.
Fig. 3.
Experimental setup.
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Fig. 4. Interferogram of the phase plate used to generate JA eye aberrations within a pupil of 7 mm diameter. Spherical power was compensated with a spherical carrier wave in the PDI.
the interferogram in Fig. 4 is recorded with a spherical carrier wave to compensate for spherical power. This aberration consists of −1.5D spherical power, −1.5D astigmatism and high-order aberrations with a root-mean-square value of 0.686 μm. The trefoil aberration for the WFC is added in PP2 by adding the phase plate described above. The peak-to-valley value of the trefoil aberration corresponds to 30 waves for the 5 mm diameter of the artificial pupil used for the experiment. We used a 100-mm-focal-length lens placed at the exit pupil of the system to image the retina in the CCD camera. A collimated laser beam (CL) was used to obtain the experimental PSFs. The pixelated PSF used in the restoration was computed numerically using the magnitude of the WFC measured with the PDI and compared with the experimental PSF in order to adjust orientation and size. Next, we introduced the phase plate in PP1 to generate eye aberrations and recorded the retinal image to show the degradation. Then the WFC phase plate was placed in PP2 and the recorded image (which we will call the intermediate image) is restored by means of a regularized Wiener filter as explained below [13]. In both cases, we searched for the best focus providing the clearest image. This image contains the blurring effects of the eye aberration and the WFC plate which, as explained above, also generates a small degree of astigmatism, coma, and spherical aberration. All images are recorded with a low level of light within the limits of permissible radiation exposure [14]. Figure 5(a) shows the PSF corresponding to the AE. Figures 5(b) and 5(c) show the PSF for the WFC phase plate and for AE WFC, respectively. The similarity of the two images demonstrates the feasibility of using the PSF for the WFC phase plate to restore the image obtained in the presence of unknown ocular aberrations. Once the intermediate images have been taken, median filtering is performed to remove noise, and the image is then restored by deblurring with a Wiener filter. Figure 6(a) shows the image of the AR blurred by eye aberrations and Fig. 6(b) shows a high-contrast image recovered with WFC after deconvolution is applied, showing that WFC does provide sharp images (up to 10 μm resolution) with no need to measure or compensate for eye and setup aberrations. (We show in this work only
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OPTICS LETTERS / Vol. 39, No. 13 / July 1, 2014
Fig. 6. (a) Image blurred by AE aberrations and (b) wavefrontcoded image.
Fig. 5. Experimental PSFs: (a) AE aberrations, (b) WFC phase plate, and (c) AE WFC aberrations.
one result for JA’s eye aberrations and for a fixed orientation of the phase plate that generates these aberrations. The same results have been obtained not only for different orientations and centering of the plate, but also for other phase plates with different eye aberrations.) Here we need to stress the following facts: (1) In practice, pupil size may vary, therefore, the pupil size must be controlled in order to recalculate the restoring trefoil PSF for the corresponding pupil diameter to be used in the deconvolution process. (2) No further filtering [15] has been used to compensate for noise or eliminate restoration artifacts in order to improve the restored images. Moreover, USAF targets have high contrast. Real retinas will have low contrast. Therefore, algorithms must be adapted not only for lower contrast images, but also for different signal-to-noise ratios. (3) The peak-to-valley value of the trefoil mask is not fully optimized. Moreover, other cubic designs could also improve the performance. This implies that while the proposed method performs reasonably well, it is still far from optimal and could be further improved. In this connection, future work could include the joint process of optimizing the phase-mask design and digital processing for maximum image enhancement in real situations. This work proves that implementing wavefront-coding techniques in a fundus camera can provide highresolution images of the retina. The experimental setup does not contain either wavefront sensors or deformable mirrors to detect and compensate for high-order aberrations, which significantly reduces the complexity, size, weight, and cost of the device. Moreover, the device does not need full correction for defocus, astigmatism, and extra aberrations caused by fabrication flaws in the WFC mask. All of these are dealt with in the deconvolution process.
We present a different approach to achieving quality retinal images that appears to be an advantageous alternative to adaptive optics in terms of robustness and cost, leading to the early detection of retinal diseases and a greater volume of research in this field. These are just some of the most significant potential benefits of an affordable fundus camera like the one presented here. This work was supported by the Spanish Ministerio de Economia y Competitividad grant FIS2012-38244-C02-01 and Xunta de Galicia grant CN 2012/156. Dr. Justo Arines thanks the Xunta de Galicia (Spain) Isidro Parga Pondal 2009 Programme for financial support. References 1. E. J. Fernandez, I. Iglesias, and P. Artal, Opt. Lett. 26, 746 (2001). 2. D. Catlin and C. Dainty, J. Opt. Soc. Am. A 19, 1515 (2002). 3. V. Nourrit, B. Vohnsen, and P. Artal, J. Opt. A Pure Appl. Opt. 7, 585 (2005). 4. J. Arines, Opt. Commun. 284, 1548 (2011). 5. E. R. Dowski and W. T. Cathey, Appl. Opt. 34, 1859 (1995). 6. S. Bradburn, W. T. Cathey, and E. R. Dowski, Appl. Opt. 36, 9157 (1997). 7. H. B. Wach, W. T. Cathey, and E. R. Dowski, Appl. Opt. 37, 5359 (1998). 8. A. B. Samokhin, A. N. Simonov, and M. C. Rombach, J. Opt. Soc. Am. A 26, 977 (2009). 9. J. R. Dowski, “Mechanically-adjustable optical phase filters for modifying depth of field, aberration-tolerance, antialiasing in optical systems,” U.S. patent 7,180,673 B2 (February 20, 2007). 10. S. Stoebenau, R. Kleindienst, M. Hofmann, and S. Sinzinger, Proc. SPIE 8126, 812614 (2011). 11. E. Acosta, R. Blendowske, and S. Chamadoira, J. Opt. Soc. Am. A 23, 632 (2006). 12. R. Navarro, E. Moreno-Barriuso, and S. Bara, Opt. Lett. 25, 236 (2000). 13. R. C. Gonzalez and R. E. Wood, Digital Image Processing (Pearson Education, 2008). 14. http://www.lia.org/publications/ansi/Z136‑3. 15. J. A. Guerrero-Colon and J. Portilla, “Two-level adaptive denoising using Gaussian scale mixtures in overcomplete oriented pyramids,” in 12th IEEE International Conference on Image Processing, 2005.
