1. Bivariate Splines for Image Denoising*° *Grant Fiddyment University of Georgia, 2008 °Ming-Jun Lai Dept. of Mathematics University of Georgia

2. Signals A signal is an observable quantity that varies over time or space. Thus it is parameterizable. The process of sampling involves transforming a continuous signal to a discrete one. There are two methods for this: analog and digital. We focus on digital data sampling.

3. Image processing Digital data sampling incorporates signal processing based on quantifiable observances ﾐ measurements. A constant signal is broken into a set of collected data. Images can be sampled this way by measuring color on a 256-color gray spectrum and approximating these values at pixel locations (coordinate pairs in R 2 ). Like all sampling, however, image sampling can be disturbed by noise.

4. Removing noise Currently, low levels of noise ( σ =10 or 15) can be removed well enough from pictures that the resulting image is acceptable to the eye. Higher levels ( σ = 25 or 30) cannot be so easily removed. Here we deal with denoising images with σ = 30).

5. Original imageσ = 10 σ = 20σ = 30 Noised Images (1)

6. Original imageσ = 10 σ = 20σ = 30 Noised Images (2)

7. Original imageσ = 10 σ = 20σ = 30 Noised Images (3)

8. PSNR The peak signal-to-noise ratio (PSNR) is the typical method of measuring image quality improvement. MAX I is the maximum possible pixel value of the image (here 255). MSE is mean squared error.

9. Method There are many methods available for image denoising. One that is popular is wavelet denoising; another is based on PDE methods. We test a new spline-based method to improve upon already wavelet denoised images. This approach has been shown to locally improve PSNR (peak signal to nosie ratio) levels, and visual examination shows noticeably reduced noise levels.

10. Edge finding The spline method samples a collection of data points (neighboring pixels) within the image. Of course, however, considering large regions involves sampling large amounts of data. As such, we must divide the domain. We do this locally by separating independent regions and then determining their approximate boundaries.

11. Edge finding (cont.) This involves selecting a starting location and searching radially away from it. Boundary points are collected and the data set is thinned for faster, smoother computation.

12. Boundary point refinement Our method of refinement relies on testing the angle formed by three successive points on the boundary. This angle’s minimum absolute distance from the horizontal is thresholded. If it is too small, the point is excluded.

13. Triangulation With these points, we can create a triangulation of the domain. Across the triangulation we fit penalized least squares splines to approximate color values.

14. Original image LenaTriangulations Wavelet denoised image (PSNR = 2223) Further spline denoised image (PSNR = 2224)

15. Original imageTriangulations Wavelet denoised image (PSNR = 2223) Further spline denoised image (PSNR = 2224)

16. Original imageTriangulations Wavelet denoised image (PSNR = 2223) Further spline denoised image (PSNR = 2224)

17. Results While visual evidence confirms our spline method is successful in locally eliminating noise, it does not yet exhibit significant global PSNR improvement. More intricate image details and small objects demonstrably lose sharpness and contrast shading as a result of spline denoising, which impairs global PSNR improvement. However, with reasonably good triangulations over larger areas, the eye verifies the spline method’s effectiveness in deleting noise.

18. Original image (Lena)Noised image (σ = 30) Wavelet denoised image (PSNR = 26.8042) Further spline denoised image (PSNR = 26.8522)

19. Original image (Peppers)Noised image (σ = 30) Wavelet denoised image (PSNR = 28.8263) Further spline denoised image (PSNR = 28.9133)

20. Original image (Saturn)Noised image (σ = 30) Wavelet denoised image (PSNR = 34.1798) Further spline denoised image (PSNR = 34.3253)

21. PSNR Readings - Peppers Tdomain456812 Regions28.8263 [1]33.048833.029832.968632.974332.9352 33.047233.059532.961333.047133.0054 31.28731.237331.221231.166831.159 33.566533.332133.28433.187933.0606 31.096131.25631.151531.176831.2014 28.8298 28.829228.82928.8287 45612 [2]32.764732.66632.508732.3133 32.64732.571132.581132.4546 33.717433.873933.832933.4683 28.857128.8628.856428.8417 456812 [3]30.671630.672530.597530.611730.5984 32.841832.801832.789532.692632.6862 32.184832.185332.360832.602532.7512 28.822928.822728.824828.826928.8286

22. Peppers (Cont.) 4561012 [4]33.22433.183533.174832.972932.9371 33.783533.838533.701333.547733.5037 31.774631.609531.59231.469831.495 30.921330.811131.588432.395432.1817 30.844231.399931.260931.435231.4184 28.815528.83228.832728.837528.834 456812 [5]30.183330.124530.07630.029630.0346 31.602831.561931.568331.585431.5743 33.036333.098232.959332.734432.6265 32.461632.444532.402632.261332.1804 31.630631.686231.482831.541231.5296 28.842728.842828.8428.836428.834 4441012 [6]32.1649 32.145932.14858 32.3787 32.358432.30406 28.8317 28.831228.83017 Global:28.867828.887428.883228.871228.8656

23. PSNR Readings - Lena Tdomain456789 Regions26.8042 28.520230.066130.934330.897231.635231.9146 28.98329.729230.497430.614630.860130.684 34.14234.084734.08233.914833.873433.7936 27.465929.651429.911931.352631.627431.8836 30.740230.89531.079930.97831.156831.152 32.38232.400232.289832.361232.338532.3939 29.039230.419330.64431.068231.149431.1255 30.136830.122830.122930.107730.08530.0631 32.362432.322332.319132.233232.274332.2636 29.562429.544229.512729.665429.523229.5513 23.142123.176423.153123.206823.210223.2083 31.813331.758331.730531.682731.659531.6445 30.206730.232430.172630.056830.077729.7697 30.19430.19130.104430.098530.107630.0956 29.307929.418329.532329.699629.702429.7697 Global26.2626.583426.686226.760626.819526.8409

24. PSNR Readings - Saturn Tdomain456789 Regions34.1798 35.64835.485435.423735.319435.30335.2643 33.409233.359833.254733.20233.175333.1426 28.194928.215528.186728.177728.126928.1327 (Rings)30.559730.394830.506130.401530.367230.3559 33.296233.44533.471533.395333.482133.5231 31.117931.523431.484631.31631.426631.3639 30.928231.024430.890130.903630.941330.9809 34.197934.19934.19834.195334.196134.1948 34.295234.404934.712734.356334.075434.034 32.299632.686332.341732.543832.648732.4957 (Planet)34.215333.804733.598634.746634.67934.4951 32.75332.751932.75733.190433.219133.0553 34.264834.251134.245334.309334.299734.281

25. Original imageNoised image Wavelet denoised imageFurther spline denoised image

26. Original imageNoised image Wavelet denoised imageFurther spline denoised image

27. Local spline denoising (T=4)Local spline denoising (T=6) Local spline denoising (T=5)Local spline denoising (T=7)

28. Local spline denoising (T=4)Triangulation (T=4) Local spline denoising (T=7)Triangulation (T=7)

29. Improvements The variety of results from spline denoising with fixed degree and smoothness indicate the signifiance of the triangulation of the splines’ domain. Thus it seems the most improvement could be exhibited by creating better triangulations. These give the splines better initial guesses at what they should be predicting. Also, we would like to automate this step as much as possible.

30. Improvements (cont.) Our next focus will be refining and trimming triangulations so that triangles maximize area – giving each one the appropriate number of sample points to analyze the distributions. We will write and employ MATLAB code to change the triangulations in this way, then denoise regions and images once again using splines.

31. Improvements (cont.) Once we’ve done this, we can focus on adjusting the degree and smoothness of the spline functions in order to observe their full power in analyzing complex data.

32. Bibliography Awanou, G. and M.J. Lai, The Multivariate spline method for scattered data fitting and numerical solutions of partial differential equations. Wavelets and Splines (2005), 24-74. Lai, M.J. and K. Nam, Image denoising using box spline tight-frames. Lian, Q. et al., Wavelets on invariant sets for image denoising. (2007)