Fast Algorithms for Restoration of Color Wireless Capsule Endoscopy Images

Transcription

1 Fast Algorithms for Restoration of Color Wireless Capsle Endoscopy Images Haiying Li, W.-S. L, and Max Q.-H. Meng School of Control Science and Engineering, Shangdon University, Jinan, China Dept. of Electrical & Compter Engineering, University of Victoria, Canada Agst 10, 011 1

2 Otline Introdction Backgrond and Related Work A Fast Algorithm for Denoising Color Images Performance Evalation Conclding Remarks

3 1. Introdction Wireless capsle endoscopy (WCE) is a state-of-the-art technology for diagnosing gastrointestinal tract diseases withot invasiveness. It acqires images dring a slow sqirm process and transmits them from inside of the body by a wireless transmitter. Raw WCE images are often blrred and noise corrpted de to less ideal environment and conditions for image transmission, imposing difficlties for accrate and effective diagnosis. This paper reports some new development for the denoising of color WCE images in a total variation (TV) minimization framework. 3

4 . Backgrond and Related Work Image Model 0 = A + w (1) with A a blrring operator and w Gassian white noise. The restoration problem is to recover image given the observation 0. The problem can be treated as an nconstrained convex problem minimize 1 μ + A () TV 0 F where the TV norm is defined by 4

5 TV n 1n 1 1 i= 1 j= 1 n 1 n 1 1 ( ) ( ) = + i, j i+ 1, j i, j i, j i, n i+ 1, n n1, j n1, j+ 1 i= 1 j= 1 (3) A Channel-by-Channel (CBC) TV Minimization Approach The approach leads to three independent EL eqations ( () i () i ) ( i ) ( i ) λ = (4) 0 0 For discrete images, the TV formlation related to these ELs is given by minimize () i 1 () i () i μ + TV 0 (5) F for i = 1,, 3, where μ is inversely related to the Lagrange mltiplier λ. 5

6 A TV Norm for Color Images Let { (1), (), (3) } be a color image with (1), (), (3) the red, green and ble components of. The color-tv (CTV) norm of is defined as 3 1/ () i = TV i= 1 (6) CTV The Eler-Lagrange eqations associated with CTV norm are ( i ) ( i ) ( ) TV () i () i ( i ) λ = CTV (7) 0 0 The EL eqations in (7) are copled via the ratio of the TV norms: r i ( ) = () i TV CTV 6 (8)

7 3.A Fast Algorithm for Denoising Color Images Analysis High correlation between channel components (i) exists, hence we are interested in a discrete formlation that is related to the CTV-based EL eqations (7), (8) which can be written as i.e. ( i ) λ ( () i () i ) ( i ) 0 = 0 r ( ) i λ ( ) i where λ ( ) = i r i λ ( ) λ i ( () i () i ) 0 ( i ) ( i ) = ( ) 0 (9) throgh which the EL eqs. are copled. 7

8 The EL in (9) leads to a TV minimization formlation as () i 1 () i () i minimize μ i( ) + TV 0 (10) F where μ ( ) 1 λ ( ) and assmes the form i i μ ( ) = μ r ( ) = μ i i () i TV CTV (11) Solving Minimization Problem (10) We take an iterative approach where in the kth iteration (10) is approximated by () i 1 () i () i minimize μi( k 1) + TV 0 (1) F for i = 1,, 3. Note that (1) can be solved efficiently sing a fast algorithm like MFISTA. 8

9 A Bisection Techniqe for Determining Optimal μ From image model 0 = + w, we have 0 nnσ F F 1 = w (13) Following (5), parameter μ controls the trade-off between the TV norm of the image and the closeness of to 0 in Frobenis norm: a μ too large pts a heavier weight on the TV norm leading to a too-large 1 0 F exceeding nnσ 1 ths violating (13); a μ too small pts a heavier weight on 1 0 F, leading to a smaller than 0 F nnσ 1. Conseqently as a fnction of μ is monotonic that 0 F increases with μ, and a near optimal vale of μ can be 9

10 identified as one that satisfies (13). Based on the analysis, a bisection techniqe is developed to identify sch a μ. Step 1: Set an initial iterate, say, to the noisy observation 0 and identify an interval [ μ, μ ] containing the optimal μ. Set a tolerance ε, and k = 1. L U Step : Set μ = ( μ + μ )/, solve (1) for i = 1,, 3, and form k = { (1) () (3),, }. k L U Step 3: If 0 1 k > nnσ, set μ F U = μ k ; otherwise set μl = μk. Step 4: If U <, otpt optimal soltion k and stop; μ μ ε L otherwise set k = k + 1 and repeat from Step. 10

11 4. Performance Evalation Data: A color WCE image of size 140 1, see Fig. 1a. The image was corrpted by additive Gassian white noise with σ = 0.05, see Fig. 1b. The initial interval [ μ, μ ] was identified as μ = 0 and μ = 0.. L U L U A modified MFISTA/FGP algorithm (A. Beck and M. Tebolle, 009) based on the proposed CTV formlation, eqipped with the proposed bisection techniqe was applied to the above noise-corrpted WCE image. 11

12 A Windows XP laptop PC with an Intel CPU GHz with GB of RAM, eqipped with MATLAB was sed to rn the code. It took the algorithm 10 iterations to converge with 1.4 seconds of elapsed time. The SNR was improved from db to db. The restored image is shown in Fig. 1c. 1

13 (a) (b) (c) (d) Fig. 1 13

14 Fig. shows profiles of { μ ( ) = μ r( ), for k = 1,...,10} three color channels. for the i k 1 k i k Fig. 14

15 To jstify the bisection techniqe, Fig. 3 depicts a profile of the vales of { μ, for k = 1,,...,10} generated in the iterations. We k observe that μ k converges to a vale of in 10 iterations Fig. 3 15

16 Fig. 4 shows the SNR of soltion with respect to parameter μ over an interval [ μ, μ ]. We see that the SNR of the soltion L U prodced by the proposed algorithm with μ = is very close to the maximm SNR achievable by a soltion of (10) which is db when μ is taken to be

17 Fig. 4 17

18 For comparison, a modified MFISTA/FGP algorithm based on CBC-TV formlation (5), eqipped with the bisection techniqe, was also applied to the above noise-corrpted image. It took the algorithm 10 iterations to converge to a soltion. The SNR achieved was fond to be , a gain that was db less than that obtained by the CTV-based algorithm. The restored image is shown in Fig. 1d. 18

19 5. Conclding Remarks An algorithm for removing within-channel white noise from mlti-channel images has been proposed. The algorithm is bilt on a concept of color total variation (CTV) in an MFISTA/FGP framework. The algorithm is enhanced by incorporating a bisection techniqe that helps identify a near optimal vale of the reglarization parameter μ. Simlation reslts have demonstrated the effectiveness of the proposed algorithm for restoring color WCE images. 19