## Translation- and Direction- Invariant Denoising of 2-D and 3-D Images: Experience and Algorithms

Thomas P.Y. Yu (Scientific Computing and Computational Math., Stanford University), Arne Stoschek (Max-Planck-Institute for Biochemistry, Germany) and David L. Donoho (Department of Statistics, Stanford University)

In Wavelet Applications in Signal and Image Processing IV, Proceedings SPIE 96

Surface Rendering before denoisingSurface Rendering after denoising

### Abstract

Removal of noise from 2-D and 3-D datasets is a task frequently needed in applications typical examples including in magnetic resonance imaging, in seismic exploration, and in video processing. We are currently interested in visualizing macromolecular structures of biological specimen , in which slices of very noisy (≈ 0 dB) electron microscopy (EM) images are volume rendered. Volume rendering of those datasets without any denoising normally gives very foggy'' results that are not very informative.

The wavelet and image processing communities have proposed in the past decade various multiscale image representations, many of which are of potential use for image de-noising. One of our goals here is to explore the importance of {\it translation and direction invariance} to the quality of reconstruction, which leads us to study the use of various tight frames for image reconstruction. We have developed 2-D translation invariant transforms for both the isotropic and anisotropic wavelet bases. These allow us to develop a 2-D analog of the 1-D translation invariant denoising algorithm proposed by Coifman and Donoho \cite{CD95}. We have also developed algorithms for implementing {\it directionally-invariant de-noising} for digital images. We have experiments to measure the relative importance of translation- and direction- invariance for both isotropic and anisotropic transforms. We also are exploring how to apply tight frames for linear inversion of noisy indirect data, which is what ultimately needed in EM tomography.

## Available Online

• Postscript version of the full paper: spie96.ps.gz
• Matlab code to reproduce all the figures (except Fig4.1) in the paper: ImageDenoise.tar.gz (NOTE: This code must be used together with WaveLab.)

• Excerpt

Which one do you think is the best? One of the following eight denoised images has the best MSE when compared to the original (clean) Barbara image, make a guess, and let me know by e-mail. How about the second best? Prize for making the right guess: Free Image Denoising Software. (Hint: In order see the difference, you have to use a monitor with a high enough resolution.)

Which one do you think is really the best? I understand that MSE doesn't mean too much for images, maybe you can tell me which image is the best according to your own criteria. Tell me your opinions.

Original Noisy
ISO,TI,DI ANISO,TI,DI ISO,ORTHO ANISO,ORTHO
ANISO,DI Steerable Pyramid ISO,TI ANISO,TI