# Improvement and Assessment of a Blind Image Deblurring Algorithm Based on Independent Component Analysis

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Developments and Methods

#### 2.1. Bi-Dimensional Gabor Filters

#### 2.2. A Blurred Image Model Based on Taylor Series Expansion

#### 2.3. Blind Deblurring by Independent Component Analysis

- Column-centering consists of making the columns of the data matrix X zero-mean. Let us decompose the matrix X into its ${n}^{2}$ columns as follows:$$X=[{x}_{1}\phantom{\rule{4pt}{0ex}}{x}_{2}\phantom{\rule{4pt}{0ex}}{x}_{3}\phantom{\rule{4pt}{0ex}}{x}_{4}\phantom{\rule{4pt}{0ex}}\cdots \phantom{\rule{4pt}{0ex}}{x}_{{n}^{2}}],$$$$m:=\frac{1}{{n}^{2}}\sum _{k=1}^{{n}^{2}}{x}_{k},$$$${\stackrel{\u02c7}{x}}_{k}:={x}_{k}-m.$$
- Row-shrinking is based on empirical covariance estimation and on thresholding the eigenvalues of the estimated empirical covariance matrix [31]. The empirical covariance matrix associated to the columns of the centered data matrix $\stackrel{\u02c7}{X}$ is defined by:$${C}_{x}:=\frac{1}{{n}^{2}}\sum _{k=1}^{{n}^{2}}{\stackrel{\u02c7}{x}}_{k}{\stackrel{\u02c7}{x}}_{k}^{\top}=\frac{1}{{n}^{2}}\stackrel{\u02c7}{X}{\stackrel{\u02c7}{X}}^{\top}.$$
- Column-whitening is a linear transformation applied to each column of the data matrix $\stackrel{\u02c7}{X}$ to obtain a quasi-whitened data matrix $Z=[{z}_{1}\phantom{\rule{4pt}{0ex}}{z}_{2}\phantom{\rule{4pt}{0ex}}{z}_{3}\phantom{\rule{4pt}{0ex}}\cdots {z}_{{n}^{2}}]$ whose columns exhibit a unit covariance. Such linear transformation is described by$$Z={\tilde{D}}^{-1/2}{\tilde{E}}^{\top}\stackrel{\u02c7}{X}.$$

#### 2.4. An ICA Learning Algorithm Based on Exponentiated Gradient on the Unit Hypersphere

## 3. Experimental Results

#### 3.1. Experiments on Deblurring Artificially Blurred Images

- An isotropic point-spread function with variance 1, denoted as Gaussian-(1,1): The clean image, the blurred image and the point-spread function are shown in Figure 4. In this case, the PSF has size $M=3$.
- An isotropic point-spread function with variance 2, denoted as Gaussian-(2,2): The clean image, the blurred image and the point-spread function are shown in Figure 5. In this case, the PSF has size $M=6$.

#### 3.2. Limitations of the Restoration Method on Artificially Blurred Images

#### 3.3. Experiments on Deblurring Naturally Blurred Images

#### 3.4. First Comprehensive Set of Experiments

`(.png)`format with a resolution of $240\times 240$ pixels. In total, 33 images were used in the present comprehensive test. Each image differs from the others by type (file format, resolution and distance between two subjects on the same image) or subject (books, plate tags, cars and text). The first 32 test images are shown in Figure 12.

- In general, the discussed deblurring method performed poorly on human faces, unless the level of blur was moderate.
- When a picture originated from a phone camera, the distance between the subject and the camera should range between 10 and 30 cm to achieve a good result (over 40 cm of distance, deblurring was not achieved successfully).
- Distance and defocusing level should be inversely proportional to one another: the farther the subject, the lower the defocusing level should be.
- In general, the level of defocusing should range between 1% and 40% to achieve a B or A result; however, there are exceptions. In fact, an excellent result was obtained on a 100% defocused large-sized text.
- Although most images were of size 240 pixels × 240 pixels, comparable results were obtained on images whose size ranged between 200 × 200 and 300 × 300 pixels.
- The file format (image encoding algorithm) did not seem to influence the final result.
- In general, objects in the foreground resulted to be more focused than objects in the background; according to our estimations, good results were achieved up to 7 cm of staggering with a maximum initial defocusing of 30%.

#### 3.5. Second Comprehensive Set of Experiments

#### 3.6. Experiments on Choosing a Suitable Learning Step Size

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the blurring process, the filtering process by 16 Gabor filters and of an ICA-based processing to recover a sharp image from a blurred image according to the method developed in [11].

**Figure 2.**Examples of 16 real-valued Gabor filters defined by Equation (1) corresponding to the parameters values $\nu \in \left\{0,1\right\}$ and $k\in \left\{0,1,2,3\right\}$. In this example, the size of the filters is $G=4$.

**Figure 4.**Original image (

**a1**); blurred image (

**a2**); Gaussian-(1,1) point-spread function (

**a3**) (colors denote different filter values); and deblurred image obtained by the exponentiated-gradient learning algorithm (

**a4**).

**Figure 5.**Original image (

**b1**); blurred image (

**b2**); Gaussian-(2,2) point-spread function (

**b3**); and (

**b4**) deblurred image obtained by the exponentiated-gradient learning algorithm.

**Figure 6.**(

**Left**) Time-evolution of the 10 components of the weight vector w during learning of the exponentiated-gradient ICA neural network. (

**Right**) Time evolution of the learning criterion $\phi $ during learning of the exponentiated-gradient ICA neural network.

**Figure 7.**Learning curves of the exponentiated-gradient method and of the adapt-and-project method, superimposed (horizontal axis in logarithmic scale).

**Figure 8.**Images containing fine details could not be deblurred by the first independent component analysis method.

**Figure 9.**Low-resolution natural images could not be deblurred by the first independent component analysis method.

**Figure 10.**Image naturally blurred taken frontally by a digital camera through an out-of-focus lens. From left to right: Recorded image, image deblurred by the adapt-and-project method and image deblurred by the exponentiated-gradient method.

**Figure 11.**Image naturally blurred taken non-frontally through an out-of-focus lens. From left to right: Recorded image, image deblurred by the adapt-and-project method and image deblurred by the exponentiated-gradient method.

**Figure 12.**Thirty-two (out of thirty-three) test images used in the first comprehensive set of experiments. The colored images were turned grey-level by keeping the first channel of their RGB representation while discarding the remaining two channels.

**Figure 13.**Result of deblurring on a comprehensive image set, in particular, on image 33_IM: (

**Left**) input image; and (

**Right**) output of the ICA neural system.

**Figure 14.**Result of deblurring on a comprehensive image set (Image 12_IM): (

**Left**) input image; and (

**Right**) output of the ICA neural system.

**Figure 15.**Result of deblurring on a comprehensive image set (Image 11_IM): (

**Left**) input image; and (

**Right**) output of the ICA neural system.

**Figure 16.**Result of deblurring on a comprehensive image set (Image 02_IM): (

**Left**) input image; and (

**Right**) output of the ICA neural system.

**Figure 17.**Twenty-eight test images (out of twenty-nine) used in the second comprehensive set of experiments. The colored images were turned grey-level by using the first channel of their RGB representation. (Two images marked by a red-color frame appear as outliers in the experiments described in the text.)

**Figure 18.**Learning curves resulting from a sequential presentation of 29 images belonging to a plate-tag dataset. In both panels, one may count exactly 29 plateaus, which correspond to 29 seemingly independent partial learning curves.

**Figure 21.**Further result on deblurring an image that does not belong to a training set obtained by reshuffling the dataset (namely, by modifying the order of presentation of the single images).

**Figure 22.**Further result on deblurring an image that does belong to a training set obtained by reshuffling the data set.

**Figure 23.**Image used for a comparison between AAP- and EG-based ICA learning systems. Such colored image was turned into a grey-level image by extracting the first channel from its RGB representation.

**Figure 24.**Comparison of learning curves of AAP and EG for a value of the learning step size $\mu =2\times {10}^{-4}$.

**Table 1.**Comparison of the original adapt-and-project (AAP) neural ICA method to the proposed exponentiated-gradient (EG) method: Coefficients of correlation between the original image and the blurred image and between the original image and the deblurred image.

PSF | Original/Blurred | Original/Deblurred (AAP) | Original/Deblurred (EG) |
---|---|---|---|

Gau-(1,1) | 0.9658 | 0.9682 | 0.9684 |

Gau-(2,2) | 0.8836 | 0.9484 | 0.9509 |

**Table 2.**Deblurring results on a set of 33 test images included in the first comprehensive set of experiments.

Image | Subject | Blur Type | Result |
---|---|---|---|

01_IM | Male face | 10% defocusing | D |

02_IM | Male face | 30% defocusing | D |

03_IM | Books on background | 10% defocusing, 40 cm away | D |

04_IM | Books on background | 20% defocusing, 20 cm away | A |

05_IM | Books on background | 30% defocusing, 20 cm away | C |

06_IM | Books on background | 37.5% defocusing | B |

07_IM | Books on background | 37.5% defocusing, 300 × 300 pixels | C |

08_IM | Books on background | 37.5% defocusing, 200 × 200 pixels | C |

09_IM | Books on background | 50% defocusing, 20 cm away | D |

10_IM | Books staggered of 10 cm | 10% defocusing, 40 cm away | D |

11_IM | Books staggered of 10 cm | 40% defocusing, 10 cm away | C |

12_IM | Books staggered of 10 cm | 10% defocusing, 10 cm away | B |

13_IM | Books staggered of 7 cm | 10% defocusing, 10 cm away | A |

14_IM | Books staggered of 7 cm | 25% defocusing, 10 cm away | A |

15_IM | Books staggered of 7 cm | 30% defocusing, 10 cm away | C |

16_IM | Books staggered of 5 cm | 30% defocusing, 10 cm away | C |

17_IM | Lined-up books | 30% defocusing, 20 cm away | C |

18_IM | Lined-up books | 30% defocusing, 10 cm away | D |

19_IM | Lined-up books | 20% defocusing, 10 cm away | D |

20_IM | Giant letter | 100% defocusing | A |

21_IM | White tag | 100% defocusing | D |

22_IM | White tag | 60% defocusing | A |

23_IM | White tag | 70% defocusing | B |

24_IM | White tag | 80% defocusing | D |

25_IM | Orange tag | 60% defocusing | B |

26_IM | Yellow tag | 60% defocusing | B |

27_IM | Black tag | 60% defocusing | D |

28_IM | Green-white tag | 60% defocusing | D |

29_IM | Blue tag | 60% defocusing | D |

30_IM | White-green tag | 60% defocusing | A |

31_IM | Canary yellow tag | 60% defocusing | A |

32_IM | Red-white tag | 60% defocusing | D |

33_IM | Car with passenger | 20% defocusing | A |

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Fiori, S.
Improvement and Assessment of a Blind Image Deblurring Algorithm Based on Independent Component Analysis. *Computation* **2021**, *9*, 76.
https://doi.org/10.3390/computation9070076

**AMA Style**

Fiori S.
Improvement and Assessment of a Blind Image Deblurring Algorithm Based on Independent Component Analysis. *Computation*. 2021; 9(7):76.
https://doi.org/10.3390/computation9070076

**Chicago/Turabian Style**

Fiori, Simone.
2021. "Improvement and Assessment of a Blind Image Deblurring Algorithm Based on Independent Component Analysis" *Computation* 9, no. 7: 76.
https://doi.org/10.3390/computation9070076