## Single channel blind image deconvolution from radially symmetric blur kernels

Optics Express, Vol. 15, Issue 7, pp. 3791-3803 (2007)

http://dx.doi.org/10.1364/OE.15.003791

Acrobat PDF (2296 KB)

### Abstract

The multichannel exact blind image deconvolution theory tells us that *exact* recovery of unknown blur kernels is possible from multiple measurements of an identical scene through distinct blur channels. However, in many biological applications, there often exist technical difficulties in obtaining multiple distinct blur measurements, since the image content may vary for various reasons, including specimen drift between snapshots, specimen damage due to prolonged exposure, or physiological changes in live cell imaging. The main contribution of this paper is a new non-iterative *single channel* blind deconvolution method that eliminates the need of multiple blur measurements, but still guarantees an accurate estimation of the blurring kernel. The basic idea behind this novel method is to exploit the *radial symmetry* of a certain class of PSFs. This approach simplifies the PSF estimation to a 1-D channel identification problem with multiple excitations, which can be solved using a standard subspace method. Since radially symmetric PSFs are quite often encountered in many practical applications, such as optical imaging systems and electron microscopy, our theory may have great influence on many practical imaging applications. Simulation results as well as real experimental results using optical and electron microscopy confirm our theory.

© 2007 Optical Society of America

## 1. Introduction

*a priori*, there exist several well-known methods for image restoration, such as inverse filtering, Wiener filtering, and iterative reconstruction (see the excellent review of [1

01. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Processing Mag. **23**,32–45 May (2006). [CrossRef]

01. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Processing Mag. **23**,32–45 May (2006). [CrossRef]

02. D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Processing Mag. **13**,43–64 May (1996). [CrossRef]

01. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Processing Mag. **23**,32–45 May (2006). [CrossRef]

02. D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Processing Mag. **13**,43–64 May (1996). [CrossRef]

04. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A **12**,58–65 (1995). [CrossRef]

05. J.A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A **15**,2609–2620 (1998). [CrossRef]

**23**,32–45 May (2006). [CrossRef]

02. D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Processing Mag. **13**,43–64 May (1996). [CrossRef]

06. G. Harikumar and Y. Bresler, “Exact image deconvolution from multiple FIR blurs,” IEEE Trans. on Image Processing **8**,846–862, (1999). [CrossRef]

07. G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms,” IEEE Trans. on Image Processing **8**,202–219 (1999). [CrossRef]

*exact*reconstruction of the unknown PSF and the true image from multiple blurred images through a number of distinct blur channels. Harikumar and Bresler [6

06. G. Harikumar and Y. Bresler, “Exact image deconvolution from multiple FIR blurs,” IEEE Trans. on Image Processing **8**,846–862, (1999). [CrossRef]

07. G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms,” IEEE Trans. on Image Processing **8**,202–219 (1999). [CrossRef]

*perfect*estimation of the unknown PSF is almost surely guaranteed if there are at least two or three distinct measurements of the identical image through distinct blur channels. Unlike other blind deconvolution methods, the multichannel blind deconvolution approach does not require the existence of frequency nulls in OTF or parametric decomposition of PSF. Furthermore, the algorithm is in principle

*non-iterative*and computationally and statistically more efficient than the conventional blind deconvolution approaches [8

08. E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. on Signal Processing **43**,516–525 (1995). [CrossRef]

*exact*estimation of PSF. The basic idea behind this novel method is to exploit the fact that PSFs in many imaging systems are

*radially symmetric*. This simplifies the PSF estimation problem to a 1-D channel identification problem with multiple excitations, which can be solved by a subspace method [8

08. E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. on Signal Processing **43**,516–525 (1995). [CrossRef]

08. E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. on Signal Processing **43**,516–525 (1995). [CrossRef]

## 2. Multichannel exact blind deconvolution - a review

*y*(

**r**) through a blur channel is described by a 2-D convolution:

**r**= (

*x*,

*y*) and

**r**́ = (

*x*́,

*y*́) denote the 2-D coordinates of the image and measurement domains, respectively; ∗∗ denotes the 2-D convolution; and

*y*(

**r**),

*h*

^{2D}(

**r**),

*x*(

**r**), and

*n*(

**r**) denote the measured image, the 2-D point-spread function (or blur kernel), the true image, and the additive noise, respectively.

*y*

_{(i)}through distinct blur kernels

*h*

_{(i)}, 1 ≤

*i*≤

*p,p*≥ 2 [6

06. G. Harikumar and Y. Bresler, “Exact image deconvolution from multiple FIR blurs,” IEEE Trans. on Image Processing **8**,846–862, (1999). [CrossRef]

07. G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms,” IEEE Trans. on Image Processing **8**,202–219 (1999). [CrossRef]

*n*

_{(i)}denotes the noise from

*i*-th blur channel and ∗∗ denotes the 2-D convolution. Harikumar and Bresler [6

**8**,846–862, (1999). [CrossRef]

**8**,202–219 (1999). [CrossRef]

*h*

_{(i)}

^{2D}}

^{p}

_{i=1}can be estimated perfectly using subspace methods, after which the unknown image

*x*can be easily reconstructed using the conventional deconvolution methods.

## 3. A novel single channel blind deconvolution theory

### 3.1. Radially symmetric PSF

*h*

^{2D}∗∗

*x*)(

**r**) is usually larger than that of

*x*(

**r**). A full data scenario denotes those cases where our field of view (FOV) includes all the measurements from the extended supports, whereas in a partial data scenario we only measure a part of them. Our theory starts from the full data scenario, and then can be extended to partial data cases.

*Y*(

**k**),

*H*

^{2D}(

**k**),

*X*(

**k**) and

*N*(

**k**) are the 2-D Fourier transforms of

*y*(

**r**),

*h*

^{2D}(

**r**),

*x*(

**r**) and

*n*(

**r**) at the 2-D spatial frequency

**k**= (

*k*,

_{x}*k*), respectively. On the polar coordinate, Eq. (3) can be converted into

_{y}*H*

^{2D}(

**k**) is also radially symmetric. More specifically, since

*H*

^{2D}(

*k*,Θ) =

*H*(

*k*) for all Θ, Eq. (4) becomes

*y*

^{Θ}(

*r*),

*x*

^{Θ}(

*r*) and

*n*

^{Θ}(

*r*) correspond to inverse Fourier transforms of

*Y*(

*k*, Θ) ,

*X*(

*k*, Θ) and

*N*(

*k*, Θ) along the radial direction, respectively.

*h*

^{2D}

_{(i)}}

^{p}

_{i=1}, Eq. (6) only needs identification of a 1-D filter,

*h*, thanks to the power of radial symmetry. This implies that our problem becomes a 1-D blind deconvolution problem, which is much simpler than its 2-D counterpart is. Furthermore, as will be explained, only a

*single*noiseless blur measurement is required to obtain the

*exact*blur kernel.

### 3.2. Subspace based blind deconvolution

**43**,516–525 (1995). [CrossRef]

**43**,516–525 (1995). [CrossRef]

**43**,516–525 (1995). [CrossRef]

**43**,516–525 (1995). [CrossRef]

*i*ΔΘ represents the

*i*-th discretized angle, where ΔΘ is the angular step size and

*i*= 1, ⋯,

*P*. Accordingly, Eq. (6) can be rewritten in a matrix form:

**y**

^{(i)}and

**x**

^{(i)}are (

*N*+ 2

*M*)×1 measurement vector and

*N*× 1 the true signal vector at the

*i*-th angle, respectively:

**H**∈ ℂ

^{(N+2M)×N}of the filter kernel

**h**is given by

*σ*

^{2}

_{n}, then the covariance matrix of the observed signal

**R**

_{y}is given by

^{H}denotes the Hermitian transpose. Let λ

_{0}≥ λ

_{1}⋯ ≥ λ

_{N+2M-1}be eigenvalues of

**R**

_{y}. The signal subspace

**S**is then spanned by the eigenvectors corresponding to eigenvalues larger than σ

^{2}

_{n}, and the noise subspace

**G**is spanned by the remaining eigenvectors. Under a noiseless scenario, if

**R**

_{x}is full ranked, then the eigenvectors corresponding to λ

_{0}≥ λ

_{1}⋯ ≥ λ

_{N-1}span the signal subspace. More specifically, we have

*s*and

_{i}**g**

*denote the eigenvectors of the signal and the noise subspace, respectively. Since the noise and the signal subspace are orthogonal to each other, for a full-ranked*

_{i}**R**

_{x}, any vectors in the noise space are orthogonal to the columns of the convolution matrix:

**43**,516–525 (1995). [CrossRef]

**h**denotes the FIR filter given by Eq. (7) and

**g**

*= [*

_{i}*g*

^{i}_{0},

*g*

^{i}_{1}, ⋯ ,

*g*

^{i}_{N + 2M-1}]

*. The matrix ℊ*

^{T}*can be calculated as follows:*

_{i}**h**has even symmetry due to the radial symmetry of PSF, we can further impose the symmetry condition:

^{e}*= (ℊ*

_{i}*+*

_{i}*)/2 and*

_{i}*are constructed by reordering ℊ*

_{i}*from the last to the first rows. Therefore, Eq. (15) can be reformulated as follows:*

_{i}**Q**= (∑

^{2M-1}

_{i=0}ℊ

^{e}*ℊ*

_{i}

_{eH}*). Under the constraint of ∥*

_{i}**h**∥

_{2}= 1, the minimization problem of Eq. (19) is to find the eigenvector

**Q**, which corresponds to the minimum eigenvalue. Additionally, we can define the cost function in terms of the signal subspace [8

**43**,516–525 (1995). [CrossRef]

**Q**̃ = (∑

^{N-1}

_{i=0}

*S*

^{e}

_{i}*S*

_{eH}*) and*

_{i}*S*

^{e}*= (*

_{i}*S*

^{e}*+*

_{i}*S*́

*)/2 are constructed using*

_{i}*S*

*, similar to Eq. (17). The optimal h of Eq. (20) can be easily obtained by calculating the eigenvector corresponding to the maximum eigenvalue of*

_{i}**Q**̃.

*exact*in the sense that the estimated filter

**h**̂ only differs by a constant scaling factor from the true filter [8

**43**,516–525 (1995). [CrossRef]

04. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A **12**,58–65 (1995). [CrossRef]

05. J.A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A **15**,2609–2620 (1998). [CrossRef]

## 4. Implementation issues

**y**

^{(i)}}

^{P}_{i=1}, could be obtained using the inverse Fourier transform of the radial lines in Fourier space, {

*Y*(

*k*,

*i*ΔΘ)}

^{P}_{i=1}, along the

*k*direction. However, the direct Fourier transform of an image to obtain the radial lines is prone to error, since the discrete Fourier transform (DFT) approximation of the continuous time Fourier transform (CTFT) assumes a periodic repetition of the images, which breaks the radial symmetry of the OTFs. In addition, there exists considerable errors originating from the interpolation between the Cartesian to polar coordinate transform.

^{(i)}corresponds to the projection data along the

*i*-th projection angle. Hence, we propose the following blind deconvolution algorithm:

- Apply a Radon transform to obtain the measured sinogram data {
**y**^{(i)}}^{P}_{i=1}. - Estimate the filter kernel
**h**using the sinogram by minimizing Eq. (20). - Obtain a restored sinogram by solving the inverse problem of Eq. (8) using a penalized least squares algorithm. In this paper, we employ the iterative coordinate descent (ICD) algorithm by Bouman and Sauer [9].
09. C. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. on Image Processing ,

**5**,480–492 (1996). [CrossRef] - Calculate a restored image using a filtered backprojection of the restored sinogram.

**8**,846–862, (1999). [CrossRef]

**8**,202–219 (1999). [CrossRef]

**8**,846–862, (1999). [CrossRef]

**8**,846–862, (1999). [CrossRef]

**8**,202–219 (1999). [CrossRef]

*a priori*. In practice, however, this should be estimated. There exist several methods for filter size estimation, for example, eigenvalue based techniques [10

10. M. Gurelli and C. Nikias, “EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals,” IEEE Trans. on Signal Processing ,**43**,134–149 (1995). [CrossRef]

**8**,202–219 (1999). [CrossRef]

**h**̂. In practice, measurements are noisy and the inverse filter becomes unstable. A penalized least square algorithm ameliorates the ill-posedness of the restoration process by imposing a penalty term. The resultant cost function can be minimized by any nonlinear optimization algorithm, and we have chosen the ICD algorithm - originally proposed for image reconstruction from noisy sinogram data [9

09. C. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. on Image Processing ,**5**,480–492 (1996). [CrossRef]

^{2}. In our implementation, the ICD algorithm is used for deblurring the sinogram in Radon space (unlike the original application of ICD in [9

**5**,480–492 (1996). [CrossRef]

## 5. Experimental results

### 5.1. Computer simulation

*H*(

*k*) is given by

*H*(

*k*) =

*CTF*(

*k*)

*E*(

*k*), where the envelop function is defined by

*E*(

*k*) =

*e*

*2*

^{-Bk}*B*, and the contrast transfer function

*CTF*(

*k*) is given by

*C*,

_{s}*λ*,Δ

*z*and

*Q*denote the spherical aberration constant, electron wavelength, defocus, and amplitude constant, respectively [3].

- Apply Radon transform of a true image at every ΔΦ angle to obtain the sinogram of a true image.
- Apply 1-D convolution along the radial direction of the sinogram to obtain a blurred sinogram.
- Apply an inverse Radon transform to obtain a blurred image.

- Zero-pad the true image.
- Multiply the frequency domain response of the zero padded image with 2-D OTF.
- Take the inverse Fourier transform to obtain the blurred image.

04. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A **12**,58–65 (1995). [CrossRef]

### 5.2. Optical microscopy experiments

*μm*is imaged using an Olympus BX51 optical microscope with a magnifying power of 200. During the experiment, we obtained focused and out-of-focus images, as shown in Fig. 6(a) and 6(b), respectively, by changing the focus of the microscope. The in-focus image was used as a ground-truth for evaluating the performance of our blind deconvolution algorithm. Figure 6(c) illustrates the restoration results using the proposed method from the blurred image in Fig. 6(b). Even though the reconstruction is not as perfect as the ground-truth, we can clearly see that the resolution is improved. Since the specimen is 60

*μm*thick, the true blurring process in Fig. 6(b) is three-dimensional, which makes our 2-D assumption invalid. However, the deconvolution result in Fig. 6(c) indicates that our algorithm is still useful in restoring transversal resolution. Another important observation from Fig. 6(c) is the artifact of partial data. As discussed before, artifacts due to the partial data appeared near the boundaries, while the central region of the image was not affected by the artifacts.

### 5.3. Cryo-electron microscopy experiments

*Å*diameter

*Paramecium bursaria chlorella*virus, type 1 (PBCV-1) [11

11. X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, “Structure and assembly of large lipid-containing dsDNA viruses,” Nat. Struct. Biol .,**7**,101–103 (2000). [CrossRef] [PubMed]

*Å*. Figures 7(a) and 7(b) show the defocused micrograph data and the restoration results obtained using the proposed blind deconvolution method, respectively. The image matrix size was 1073 × 1073.

## 6. Conclusion

**12**,58–65 (1995). [CrossRef]

*a priori*information about the PSF other than radially symmetry. Furthermore, the new method guarantees the

*exact*recovery of unknown blurring kernel in a noiseless measurement scenario. Additionally, compared to the existing multichannel exact blind deconvolution methods, our approach is simpler: since only a single blurred image is required and the estimation process is one-dimensional. Based on the Fourier slice theorem, the blur kernel estimation as well as the restoration procedure were implemented in Radon space, after which the final image was obtained using the filtered backprojection of the restored sinogram. In order to ameliorate the ill-posedness of the deconvolution, a regularized least squares method was employed. Simulation and real experimental results using an optical and an electron microscope demonstrated that reasonably good quality reconstructions were obtained even from noisy measurements.

## Acknowledgments

## Footnotes

2 | Note that the other computationally more efficient optimization methods rather than the ICD could be also used for deblurring the sinogram data. |

## References and links

01. | P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Processing Mag. |

02. | D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Processing Mag. |

03. | J. Frank, |

04. | D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A |

05. | J.A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A |

06. | G. Harikumar and Y. Bresler, “Exact image deconvolution from multiple FIR blurs,” IEEE Trans. on Image Processing |

07. | G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms,” IEEE Trans. on Image Processing |

08. | E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filters,” IEEE Trans. on Signal Processing |

09. | C. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. on Image Processing , |

10. | M. Gurelli and C. Nikias, “EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals,” IEEE Trans. on Signal Processing , |

11. | X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, “Structure and assembly of large lipid-containing dsDNA viruses,” Nat. Struct. Biol ., |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(100.2000) Image processing : Digital image processing

(100.3020) Image processing : Image reconstruction-restoration

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 16, 2007

Revised Manuscript: March 21, 2007

Manuscript Accepted: March 22, 2007

Published: April 2, 2007

**Virtual Issues**

Vol. 2, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Kwang Eun Jang and Jong Chul Ye, "Single channel blind image deconvolution from radially symmetric blur kernels," Opt. Express **15**, 3791-3803 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-7-3791

Sort: Year | Journal | Reset

### References

- P. Sarder and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Processing Mag.23, 32-45 May (2006). [CrossRef]
- D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Processing Mag.13, 43-64 May (1996). [CrossRef]
- J. Frank, Three Dimensional Electron Microscopy of Macromolecular Assemblies, (Academic Press, San Diego, CA, USA, 1996).
- D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, "Blind deconvolution by means of the Richardson- Lucy algorithm," J. Opt. Soc. Am. A 12, 58-65 (1995). [CrossRef]
- J. A. Conchello, "Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images," J. Opt. Soc. Am. A 15, 2609-2620 (1998). [CrossRef]
- G. Harikumar and Y. Bresler, "Exact image deconvolution from multiple FIR blurs," IEEE Trans. on Image Processing 8, 846-862 (1999). [CrossRef]
- G. Harikumar and Y. Bresler, "Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms," IEEE Trans. on Image Processing 8, 202-219 (1999). [CrossRef]
- E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995). [CrossRef]
- C. Bouman, and K. Sauer, "A unified approach to statistical tomography using coordinate descent optimization," IEEE Trans. on Image Processing 5, 480-492 (1996). [CrossRef]
- M. Gurelli and C. Nikias, "EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals," IEEE Trans. on Signal Processing 43, 134-149 (1995). [CrossRef]
- X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol. 7, 101-103 (2000). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.