1 Motion Blur Identification in Noisy Images Using Fuzzy Sets IEEE 5th International Symposium on Signal Processing and Information Technology (ISSPIT 2005), Athens, Greece, Dec Mohsen Ebrahimi Moghadam, Mansour Jamzad Computer Engineering Dept, Sharif University of Technology, Tehran, Iran

2 Presentation Outline Introduction Motion Blur Attributes Parameter Estimation in Noise Free Images – Motion Direction – Motion Length Parameter Estimation in Noisy Images – Motion Direction – Motion Length Estimation Using Fuzzy Sets Experimental Results Conclusion

3 Introduction General Degradation Form: Blur Identification: finding parameters of degradation function h

4 Introduction (cont.) Goal : Uniform motion blur parameter estimation Related works: – Noise Free images: Solved! – Noisy Images: Solutions depends on noise variance: Noise Free Algorithms have been extended to noisy images –Canon [1976] –Chang [1992] –Qiang [1996] They used bispectrum or windowing technique

5 Introduction(cont.) Our Method is Simple and Robust. It supports Lower SNRs. We used Fuzzy concepts.

6 Linear Motion Blur degradation function The general form : This figure shows the frequency response of h(x,y) (that has a SINC form) Where L is the motion length is the motion direction

7 Linear Motion Blur Degradation function (cont.) The SINC form of frequency response of h causes parallel dark lines to appear in Image frequency response.

8 Noise Free Image: Motion Direction Detection

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10 Note at the inverse relation of motion length and width of central lobe of parallel lines

In Noise Free Image: We used Radon Transform to estimate Motion Direction 11

12 Noise Free Image: Using Radon Transform to estimate Motion Direction Using Radon Transform: – Finding Dark line direction in the frequency response of degraded image. – G(u,v) is the frequency response of degraded image. – Assume K(x,y) = log |G(u,v)| OR

13 Noise Free Image:Motion Direction(Cont.) We assume k(x,y) to be an image in spatial domain that is defined to be k(x,y)= log |G(u,v)| where G(u,v) is the frequency response of g(x,y): the degraded image Apply Radon Transform on K(x,y) Highest value of θ in Radon Transform output corresponds to dark lines direction Dark lines Direction corresponds to Motion direction ρ θ

How to estimate Motion Length 14

15 Noise Free Image: Motion Length estimation Our idea is based on using the SINC structure that is present in frequency response of degraded image, We supposed motion to be in horizontal direction – To achieve this assumption, we rotate the coordinate axis by (θ): motion direction

16 Noise Free Image: Motion Length estimation Assuming motion length to be L pixels, then, In horizontal direction, motion function is: Its Frequency response:

17 Noise Free Image: Motion Length estimation (cont.) To find motion length: L, we should solve the following equations: N: is the one dimensional vector size or the image size Finding 0 values for SINC function

18 Noise Free Image: Motion Length estimation (cont.) Where d is the distance between two successive zero lines in K(x,y) = log |G(u,v)| If we suppose u 0 and u 1 to be two successive 0 points in H(u) then:

19 Noise Free Image: Motion Length estimation (cont.) Those u for which G(u) = 0, are categorized into 2 groups: 1- Straight dark lines that are caused by motion blur ( H(u) = 0) 2- The dark lines created because of image pixel values ( G(u) = 0). To find distance d we should find the u from category 1.

20 Noisy Images: Motion Direction estimation

21 Noisy Images: Motion Direction estimation We used the same idea like Noise free images: – Applying Radon Transform on log of Fourier spectrum, on log |G(u,v)| - Highest value in Radon transform output corresponds to motion direction. In noisy images we usually have 2 strong peaks

22 Noisey Image: An example of degraded noisy image and its Fourier spectrum

23 Noisy Images: Motion Length estimation Dark lines disappear in frequency response of degraded noisy image Darker lines are better candidate! A Question: Which pixels belong to disappeared dark lines? Because of additive noise, there is an uncertainty to make such decision. This uncertainty guides us to use the fuzzy set theory.

24 Noisy Images: Motion Length estimation (cont.) We define a fuzzy set A i for each line i, in log |G(u,v)| : Where i is the row number, x is the column number and N is the number of rows and columns in log |G(u,v)| Membership function is defined as follows that is a z-function. μ shows the possibility that a pixel lies on a disappeared dark line a and c are two constants determined heuristically, a =20, c = 230

25 Noisy Images: Motion Length estimation (cont.) μ u is the mean of log (|G(u,v)|) at row i, in another words u stands for n(x) in following equation.

26 Noisy Images: Motion Length estimation (cont.) Those columns x, that their membership values (μ u ) in all of these sets (there is one fuzzy set for each row) are higher, are the best candidate of dark lines. For simplicity we rotate the spectrum by blur angle to force the dark lines parallel to Y axis.

27 Noisy Images: Motion Length estimation (cont.) In another words, Those members (column number) of the defined fuzzy set, that have higher membership values in all sets (all rows), are better candidates to lie on a dark line! In each row we are determining the possibility that a column x, belongs to a dark line.

28 Noisy Images: Motion Length estimation (cont.) We used Zadeh-tnorm (i.e. min) to find the intersection of these sets as follows: M is number of rows in image, μ ix shows membership value of x in fuzzy set Ai and t is Zadeh t-norm.

29 Noisy Images: Motion Length estimation (cont.) All valleys (low values) of are candidates of dark lines The best valleys are the ones that correspond to the valleys of SINC structure of degradation function the possibility that column x does not belong to dark line We define

30 Noisy Images: Motion Length estimation (cont.) of a noise free image degraded with L= 3 pixels. Note that f(x) = 0 shows the column number where the dark line passes through.

31 Noisy Images: Motion Length estimation (cont.) of the same image with Gaussian additive noise (SNR = 25 db)

32 Noisy Images: Motion Length estimation (cont.) To find motion length, we found the valleys around central peak of f(x) To find valleys, we used cepstrum analysis. A cepstrum (pronounced /k ɛ pstrəm/) is the result of taking the Fourier Transform (FT) of the decibel spectrum (or log spectrum) as if it was a signal./k ɛ pstrəm/ Motion length is calculated by using the following equation, : distance between valleys

33 Experimental Results Motion Angle and Length estimated for Noise free images

34 Experimental Results(Cont.) Motion Angle and Length estimation for noisy images

35 Examples of images restored in motion blur Left. A camera man motion blurred (L = 20 pixel, θ= 60◦) image with no additive noise, the estimated parameters using our method was (L = 20.4 pixel, θ= 60.2◦). Right: The restored image

36 Examples of images restored in motion blur Left: A camera man motion blurred (L = 20 pixel, = 60◦) image with additive noise (SNR = 35 dB), the estimated parameters using our method was (L = 20.8 pixel, = 58.6◦). Right: The restored image

37 Examples of images restored in motion blur Left: The Crowed image degraded by linear motion blur with parameters (L = 15 pixel, = 20◦) with additive noise (SNR = 30 dB), the estimated parameters with our algorithm was (L = 13.8 pixel, = 21.1◦). Right: The restored image

38 Conclusion We proposed a precise and robust algorithm. It is based on Radon transform and Fuzzy sets. Radon transform overcomes the problems in Hough transform and Robust regression for gray scale images. Our algorithm works well for noisy images. The low values of errors in estimating motion parameters showed the high performance of our method.

39 Thank you very much for your attention.

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41 Comparison with related works Canon[1976] – Paper did not present exhaustive experimental results to compare. Change[1991] – It has presented a method that its lowest SNR support is 40 dB Qiang[1996] – The best precision of their method when the SNR was about 30 dB – In this paper, an image with SNR 23.3 dB were restored, but estimated parameters precision didn’t present Others: – Some other methods are presented with lower precision and Higher SNR support

42 Noise Free Image: Motion Length estimation In horizontal direction