Evolving Systems

, Volume 10, Issue 4, pp 707–723 | Cite as

Reconstruction of catadioptric omnidirectional images using dual alternating total variation minimization

  • Soraya Zenati
  • Abdelhani BoukroucheEmail author
  • Larbi Boubchir
Original Paper


This paper discusses the possibility to extend and apply the conventional two dimensional recovering images from blurry and noisy observation with the total variation regularization method to the catadioptric images. The principal in this special method is the stabilization of dual alternating minimization. The latter introduces two auxiliary half quadratic variables to transfer the system out of the ill-posed term. The main contribution of this paper is the use of the inverse stereographic projection and the spherical harmonics in order to adapt this proposed deconvolution with catadioptric omnidirectional images. The projection on the unit sphere of the omnidirectional image, is one way to alleviate the problem of the heterogeneous resolution and the negative effects of anamorphosis. In both anisotropic and isotropic deconvolutions, the experimental results conducted on synthetic as well as captured catadioptric omnidirectional images which are subject to various effects, confirm the performance of the proposed method to restore such images impaired by the blur and noise. Compared with several state-of-the-art approaches, the images resulted can achieve up to an acceptable and higher level of deconvolution quality.


Restoration of omnidirectional images–Spherical deconvolution Spherical total variation Isotropic and anisotropic spherical deconvolution Omnidirectional image Spherical image 



This research was supported by the Laboratory of Inverse Problems, Modeling, Information and Systems, University of Guelma. The author is indebted to his coauthors: A. Boukrouche and L. Boubchir for their encouragements and suggestions. Special acknowledgement is due to the former CREA (Caen) research labs for making available a set of images (Fig. 3b, c). Big thanks and a warm dedication to O. Elkadmiri for providing us with the synthetic image (Fig. 3a) and counselling us on restoration.


  1. Abbas SA, Sun Q, Foroosh H (2017) An exact and fast computation of discrete Fourier transform for polar and spherical grid. IEEE Trans Signal Process 65(8):2033–2048MathSciNetCrossRefGoogle Scholar
  2. Azar AT, Ammar HH, Mliki H (2018) Fuzzy logic controller with color vision system tracking for mobile manipulator robot. In: Hassanien A, Tolba M, Elhoseny M, Mostafa M (eds) The International conference on advanced machine learning technologies and applications (AMLTA2018). AMLTA 2018. Advances in intelligent systems and computing, vol 723. Springer, ChamGoogle Scholar
  3. Barone S, Carulli M, Neri P, Paoli A, Razionale AV (2018) An omnidirectional vision sensor based on a spherical mirror catadioptric system. Sensors 18(2):408CrossRefGoogle Scholar
  4. Bovik ZAC, Sheikh EP (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612CrossRefGoogle Scholar
  5. Bradley D, Brunton A, Fiala M, Roth G (2005) Image-based navigation in real environments using panoramas. In: IEEE international workshop on haptic audio visual environments and their applications (HAVE), pp 57–59Google Scholar
  6. Chambolle A, Lions PL (1997) Image recovery via total variation minimization and related problems. Nmerische Math 76(2):167–188MathSciNetCrossRefGoogle Scholar
  7. Chambolle A, Tan P, Vaiter S (2017) Accelerated alternating descent methods for Dykstra-like problems. J Math Imaging Vis 59(3):481–497MathSciNetCrossRefGoogle Scholar
  8. Chan TF, Shen J (2007) Theory and computation of variational image deblurring. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Mathematics and Computation in Imaging Science and Information Processing, pp 93–130Google Scholar
  9. Chan TF, Golub GH, Mulet P (1999) A nonlinear primal-dual method for total variation based image restoration. SIAM J Sci Comput 20:1964–1977MathSciNetCrossRefGoogle Scholar
  10. Chikaraishi T, Minato T, Ishiguro H (2018) Development of an android system integrated with sensor networks. In: Ishiguro H, Dalla Libera F (eds) Geminoid studies. Springer, SingaporeGoogle Scholar
  11. Daniilidis K, Makadia A, Bulow T (2002) Image processing in catadioptric planes: spatiotemporal derivatives and optical flow computation. In: IEEE third workshop on omnidirectional vision (OMNIVIS), pp 3–10Google Scholar
  12. Driscoll JR, Healy DM (1994) Computing spherical transform and convolution on the 2-sphere. Adv Appl Math 15:202–250CrossRefGoogle Scholar
  13. El Jaouhari Z, Zaz Y, Moughyt S, El Kadmiri O, El Kadmiri Z (2018) Dual-axis solar tracker design based on a digital hemispherical imager. J Sol Energy Eng 141(1):011001. CrossRefGoogle Scholar
  14. El Kadmiri O, El Kadmiri Z, Masmoudi L (2013) A new corner detection method for omnidirectional images. J Theor Appl Inform Technol 58(2):282–290Google Scholar
  15. Faugeras O (2013) Panoramic vision: sensors, theory, and applications. Springer, BerlinGoogle Scholar
  16. Geyer C, Daniilidis K (2001) Catadioptric projective geometry. Int J Comput Vis 45(3):223–243CrossRefGoogle Scholar
  17. Gupta P, da Vitoria Lobo N, Laviola JJ (2013) Markerless tracking and gesture recognition using polar correlation of camera optical flow. Mach Vis Appl 24(3):651–666CrossRefGoogle Scholar
  18. Healy DM, Rockmore DN, Moore SB (1996) An FFT for the 2-sphere and applications. Proc ICASSP Atlanta USA 3:1323–1326Google Scholar
  19. Healy DM, Rockmore DN, Kostelec PJ et al (2003) FFTs for the 2-Sphere-Improvements and variations. J Fourier Anal Appl 9:340–385MathSciNetCrossRefGoogle Scholar
  20. Hrabar S, Sukhatme GS (2003) Omnidirectional vision for an autonomous helicopter. In: IEEE international conference on robotics and automation (ICRA), vol 1, pp 558–563Google Scholar
  21. Labutov I, Jaramillo C, Xiao J (2013) Generating near-spherical range panoramas by fusing optical flow and stereo from a single-camera folded catadioptric rig. Mach Vis Appl 24(1):133–144CrossRefGoogle Scholar
  22. Li Y, Liu L, Wang W et al (2013) Defocus deblurring for catadioptric omnidirectional imaging based on spatially invariant point spread function. J Mod Opt 60(6):458–466CrossRefGoogle Scholar
  23. Liu Y, Li Y, Lou J et al (2014) Omni-total variation algorithm for the restoration of all-focused catadioptric image. Opt Int J Light Electron Opt 125(14):3685CrossRefGoogle Scholar
  24. Liu Y, Li H, Li Y et al (2014) Coded aperture enhanced catadioptric optical system for omnidirectional image deblurring. Opt Int J Light Electron Opt 125(1):11CrossRefGoogle Scholar
  25. Lou J, Li Y, Liu Y et al (2014) Omni-gradient-based total variation minimisation for sparse reconstruction of omni-directional image. IET Image Process 8(7):397–405CrossRefGoogle Scholar
  26. Ly DS, Demonceaux C, Vasseur P et al (2014) Extrinsic calibration of heterogeneous cameras by line images. Mach Vis Appl 25(6):1601–1614CrossRefGoogle Scholar
  27. Phan TDK (2018) A triangle mesh-based corner detection algorithm for catadioptric images. Imaging Sci J 66(4):220–230CrossRefGoogle Scholar
  28. Puig L, Sturm P, Guerrero JJ (2013) Hybrid homographies and fundamental matrices mixing uncalibrated omnidirectional and conventional cameras. Mach Vis Appl 24(4):721–738CrossRefGoogle Scholar
  29. Rizkallah M, De Simone F, Maugey T, Guillemot C, Frossard P (2018) Rate distortion optimized graph partitioning for omnidirectional image coding. In: The 26th European signal processing conference (EUSIPCO), pp 1–5Google Scholar
  30. Rudin LI, Osher S (1994) Total variation based image restoration with free local constraints. Proc 1st IEEE ICIP a:31–35Google Scholar
  31. Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60:259–268MathSciNetCrossRefGoogle Scholar
  32. Sanders T, Gelb A, Platte RB, Arslan I, Landskron K (2017) Recovering fine details from under-resolved electron tomography data using higher order total variation 1 regularization. Ultramicroscopy 174:97–10CrossRefGoogle Scholar
  33. Sando Y, Barada D, Jackin BJ, Yatagai T (2018) Spherical-harmonic-transform-based fast calculation algorithm for spherical computer-generated hologram considering occlusion culling. Appl Opt 57(23):6781–6787CrossRefGoogle Scholar
  34. Shabayek A, Demonceaux C, Morel O et al (2012) Vision based UAV attitude estimation: progress and insights. J Intell Robot Syst 65(1–4):295–308CrossRefGoogle Scholar
  35. Singh J, Pliefke S, Hess H (2018) Vehicle vision system with calibration algorithm. U.S. Patent Application No 15/914,059, 12 juillGoogle Scholar
  36. Strauss O, Comby F (2005) Fuzzy morphology for omnidirectional images. In: International conference on image processing (ICIP), pp II-141Google Scholar
  37. Wandelt BD, Gorski KM (2000) Fast convolution on the sphere. Physical review D 63:123002 (6 pages) MathSciNetCrossRefGoogle Scholar
  38. Wang Y, Yang J, Yin W et al (2008) A new alternating minimization algorithm for total variation image reconstruction. SIAM J Imaging Sci 1(3):248–272MathSciNetCrossRefGoogle Scholar
  39. Wohlberg B, Rodríguez P (2007) An iteratively reweighted norm algorithm for minimization of total variation functionals. IEEE Signal Process Lett 14(12):948–951CrossRefGoogle Scholar
  40. Xiong Z, Cheng I, Basu A et al (2012) Efficient omni-image unwarping using geometric symmetry. Mach Vis Appl 23(4):725–737CrossRefGoogle Scholar
  41. Xue K, Liu Y, Ogunmakin G et al (2013) Panoramic Gaussian Mixture Model and large-scale range background substraction method for PTZ camera-based surveillance systems. Mach Vis Appl 24(3):477–492CrossRefGoogle Scholar
  42. Yakhdani MF, Azizi A (2010) Quality assessment of image fusion techniques for multisensory high resolution satellite images (case study: IRS-P5 and IRS-P6 satellite images). In: ISPRS TC VII symposium, vol 39, pp 204–209Google Scholar
  43. Zenati S, Boukrouche A (2010) Wiener filter improvement on the sphere. In: 7th international symposium on IEEE mechatronics and its applications (ISMA), pp 1–6Google Scholar
  44. Zenati S, Boukrouche A (2015) Classical processing for plane and omnidirectional images, In: Image Processing Theory, Tools and Applications (IPTA), 2015 International Conference on IEEE, 131-136Google Scholar
  45. Zenati S, Boukrouche A, Neveux Ph (2012) Deconvolution for slowly time-varying systems 3D cases. In: The 3rd international conference on IEEE image processing theory, tools and applications (IPTA), pp 121–126Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Inverse Problems, Modelling, Information and SystemsUniversity 8 Mai 1945 GuelmaGuelmaAlgeria
  2. 2.Department of Computer SciencesBadji Mokhtar UniversityAnnabaAlgeria
  3. 3.LIASD research Lab., University of Paris 8Saint-DenisFrance

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