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

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.

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.

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).

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

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

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

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.

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.

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.

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.

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.

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.

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

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

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

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.

Original image (Lena)Noised image (σ = 30) Wavelet denoised image (PSNR = ) Further spline denoised image (PSNR = )

Original image (Peppers)Noised image (σ = 30) Wavelet denoised image (PSNR = ) Further spline denoised image (PSNR = )

Original image (Saturn)Noised image (σ = 30) Wavelet denoised image (PSNR = ) Further spline denoised image (PSNR = )

PSNR Readings - Peppers Tdomain Regions [1] [2] [3]

Peppers (Cont.) [4] [5] [6] Global:

PSNR Readings - Lena Tdomain Regions Global

PSNR Readings - Saturn Tdomain Regions (Rings) (Planet)

Original imageNoised image Wavelet denoised imageFurther spline denoised image

Original imageNoised image Wavelet denoised imageFurther spline denoised image

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

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

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.

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.

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.

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), 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)