# Image denoising via the redundant wavelet transform

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This report summarizes work done as part of the Wavelet Based Image Analysis PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module introduces the redundant discrete wavelet transform as well as two level dependent estimators that could potentially be used for image denoising, the Bishrink algorithm and the Bayesian Least Squares-Gaussian Scale Mixture algorithm. A simulation designed to evaluate the efficacies of each of these methods for the purpose of denoising astronomical image data is described, and its results are presented and discussed.This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant DMS?0739420.

## Introduction

The redundant wavelet transform (RWT) is widely used in order to denoise signals and images. Here, we consider two denoising methods used in the literature, to attempt to denoise astronomical images with the aim of obtaining images in which we can search for very faint objects that are not noise.

The paper is organized as follows. In "Redundant Wavelet Transform" , we introduce a few algorithms used to compute the RWT. In "Denoising Algorithms based on the RWT" , we discuss some denoising methods based on the RWT. In "Denoising Simulation" , a description of the simulation and the results from the implemented methods can be found, which are further discussed in "Conclusions" .

## Undecimated algorithm

The redundant discrete wavelet transform, similar in nature to the discrete wavelet transform, decomposes data into low-pass scaling (trend) and high-pass wavelet (detail) coefficients to obtain a projective decomposition of the data into different scales. More specifically, at each level the transform uses the scaling coefficients to compute the next level of scaling and wavelet coefficients. The difference lies in the fact that none of the latter are discarded through decimation as in the discrete wavelet transform but are instead retained, introducing a redundancy. This transform is good for denoising images, as the noise is usually spread over a small number of neighboring pixels. The Rice Wavelet Toolbox used to compute the transform in the simulation implements the redundant wavelet transform through the undecimated algorithm, which as its name suggests is similar to the discrete wavelet transform but omits downsampling, also known as decimation, in computation of the transform and upsampling in computation of the inverse transform .

## A trous algorithm

Another method of computing the redundant wavelet transform, the $\stackrel{´}{a}$ trous algorithm differs from the undecimated algorithm by modifying the low-pass and high-pass filters at each consecutive level. The algorithm up-samples the low-pass filter at each level by inserting zeros between each of the filter's coefficients. The high-pass coefficients are then computed as the difference between the low-pass images from the two consecutive levels. To compute the inverse transform, the detail coefficients from all levels are added to the final low-resolution image . While inefficient in implementation, the $\stackrel{´}{a}$ trous algorithm provides additional insight into the redundant discrete wavelet transform.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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