application of noise invalidationdenoising in mri

Application of Noise InvalidationDenoising in MRI

Pegah Elahi

2012-09-18

LiTH-IMT/MASTER-EX�12/019�SE

Application of Noise InvalidationDenoising in MRI

Pegah Elahi

2012-09-18

LiTH-IMT/MASTER-EX�12/019�SE

Supervisor: Soosan Beheshti, Associate Professor,

Department of Electrical and Computer Engineering, Ryerson University

Examiner: Hans Knutsson, Professor,

Department of Biomedical Engineering, Linkoping University

i

Abstract

Magnetic Resonance Imaging (MRI) is a common medical imaging tool that have beenused in clinical industry for diagnostic and research purposes. These images are sub-ject to noises while capturing the data that can e�ect the image quality and diagnos-tics.Therefore, improving the quality of the generated images from both resolution andsignal to noise ratio (SNR) perspective is critical. Wavelet based denoising technique isone of the common tools to remove the noise in the MRI images. The noise is eliminatedfrom the detailed coe�cients of the signal in the wavelet domain. This can be done byapplying thresholding methods. The main task here is to �nd an optimal threshold andkeep all the coe�cients larger than this threshold as the noiseless ones. Noise InvalidationDenoising technique is a method in which the optimal threshold is found by comparingthe noisy signal to a noise signature (function of noise statistics). The original NIDeapproach is developed for one dimensional signals with additive Gaussian noise. In thiswork, the existing NIDe approach has been generalized for applications in MRI imageswith di�erent noise distribution. The developed algorithm was tested on simulated datafrom the Brainweb database and compared with the well-known Non Local Mean �l-tering method for MRI. The results indicated better detailed structural preserving forthe NIDe approach on the magnitude data while the signal to noise ratio is compatible.The algorithm shows an important advantageous which is less computational complexitythan the NLM method. On the other hand, the Unbiased NLM technique is combinedwith the proposed technique, it can yield the same structural similarity while the signalto noise ratio is improved.

KEYWORDs: Magnetic Resonance Imaging, Noise Invalidation Denoising, UnbiasedNon Local Mean �ltering, Wavelet Transform Function

ii

Acknowledgements

I would like to take an oppurtunity to express my appriciation for all the people whohelped me through this work. At �srt, I am grateful to my supervisor Prof. SoosanBeheshti at Ryerson University not only for giving me the opportunity to work underher supervison but also for her support and guidance throughout this thesis work.I would like to acknowlege Prof. Hans Knutsson for accepting being my examiner and

evaluating my work which means a lot based on his knowldge and experience in medicalimaging. Also, many thanks to Prof. Göran Salerud director of studies in Biomicalengineering department for his patience and support during my thesis work.Last but not least, I would like to express my deepest gratitude to my parents for their

unconditional love and support not only during this project but in every steps of my life.I could not achieve my goals without them standing by my side and their encouragement.

iii

Contents

1 Introduction 1

2 Background and literature review 32.1 MRI imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 creating a MRI image . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 MRI signal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Signal distribution of the raw data . . . . . . . . . . . . . . . . . . 62.2.2 Signal distribution of the magnitude data . . . . . . . . . . . . . . 62.2.3 Signal distribution of the squared magnitude data . . . . . . . . . 7

2.3 Noise estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Estimating the noise variance from the background . . . . . . . . . 72.3.2 Estimating the noise variance based on local statistics . . . . . . . 8

2.4 Denoising methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 Gaussian �ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Bilateral �ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3 Anisotropic Di�usion Filtering . . . . . . . . . . . . . . . . . . . . 102.4.4 Wavelet denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.5 Non Local Mean (NLM) �lters . . . . . . . . . . . . . . . . . . . . 15

3 Method and Simulation Results 173.1 Method and material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Noise Invalidation Denoising for Rician distribution . . . . . . . . . 173.1.2 Noise Invalidation Denoising for chi-square distribution . . . . . . . 183.1.3 Noise variance estimation . . . . . . . . . . . . . . . . . . . . . . . 193.1.4 Evaluation criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.5 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 Results for NIDe for Rician distributed data . . . . . . . . . . . . . 213.2.2 Results for NIDe for chi-squared distributed data . . . . . . . . . . 313.2.3 Results for combining NIDe with Richian distribution with UNLM 39

4 Discussion and future works 414.1 Comparison based on evaluation criteria . . . . . . . . . . . . . . . . . . . 414.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Conclusion and future trends . . . . . . . . . . . . . . . . . . . . . . . . . 53

Bibliography 54

iv

List of Figures

2.1 Spin and its related angular momentum [15] . . . . . . . . . . . . . . . . . 42.2 Generated current signal because of applying RF wave [2] . . . . . . . . . 52.3 Local second order distribution for a: normal image and b: MRI image.

The dashed line shows the distribution related to the noisy images[20] . . 82.4 E�ect of sorting the absolute values: samples of unsorted noise, sorted

noise values and changed axes for top to bottom respectively[21] . . . . . 142.5 Finding threshold with NIDe. Solid line is the noisy data and dashed lines

represent the noise only boundaries[21] . . . . . . . . . . . . . . . . . . . . 15

3.1 Probability density function plots for noisy image, data in the backgroundand signal area top to bottom respectively . . . . . . . . . . . . . . . . . . 22

3.2 ANS function on noise only data with Rayleigh distribution . . . . . . . . 233.3 Noise con�dence area for the Rician distributed Noise signature. The blue

line shows the mean while the two red lines shows the upper and lowerboundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Finding the optimal threshold for the NIDe with Rician distribution method.The blue line shows the sorted noisy data and how it exists the con�dencearea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Denoising result of NIDe for Rician distribution with 3% noise . . . . . . . 253.6 Denoising result of NIDe for Rician distribution with 5% noise . . . . . . . 263.7 Denoising result of NIDe for Rician distribution with 7% noise . . . . . . . 273.8 NIDe approach based on Rayleigh distribution results on T2 weighted

image with 3% noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.9 NIDe approach based on Rayleigh distribution results on T2 weighted

image with 5% noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.10 NIDe approach based on Rayleigh distribution results on T2 weighted

image with 7% noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.11 PDF plot of the squared data . . . . . . . . . . . . . . . . . . . . . . . . . 313.12 E�ect of sorting on the noisy data with chi-squared distribution . . . . . . 323.13 Noise con�dence area for the chi-squared distribution noise. The blue line

shows the mean while the red lines are the limits of this area. . . . . . . . 333.14 Finding the threshold in the NIDe approach based on chi-squared distri-

bution. The blue line shows the sorted noisy data. . . . . . . . . . . . . . 333.15 Final Result of the denoising with NIDe on the squared data with 3% noise 343.16 Final Result of the denoising with NIDe on the squared data with 5% noise 353.17 Final Result of the denoising with NIDe on the squared data with 7% noise 36

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3.18 Final result of NIDe algorithm based on chi-squared distribution on theT2 weighted MRI with 3% noise. . . . . . . . . . . . . . . . . . . . . . . . 37



3.21 T1-weighted MRI image with 5% noise level �ltered with NIDe-UNLM . 40

4.1 Comparing �gures of each methods for T1 weighted with 3% noise . . . . 444.2 Comparing �gures of each methods for T1 weighted with 5% noise . . . . 454.3 Comparing �gures of each methods for T1 weighted with 7% noise . . . . 464.4 Comparing �gures of each methods for T2 weighted with 3% noise . . . . 484.5 Comparing �gures of each methods for T2 weighted with 5% noise . . . . 494.6 Comparing �gures of each methods for T2 weighted with 7% noise . . . . 504.7 Final results of NIDe-Rician combined with UNLM . . . . . . . . . . . . . 52

vi

List of Tables

3.1 Threshold values T1/T2 weighted MRI images at di�erent noise level forRician distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Threshold values T1/T2 weighted MRI images at di�erent noise level forchi squared distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Comparison table for noise level of 3% on T1 weighted MRI image . . . . 414.2 Comparison table for noise level of 5% on T1 weighted MRI image . . . . 414.3 Comparison table for noise level of 7% on T1 weighted MRI image . . . . 424.4 Speed di�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Comparison table for noise level of 3% on T2 weighted MRI image . . . . 474.6 Comparison table for noise level of 5% on T2 weighted MRI image . . . . 474.7 Comparison table for noise level of 7% on T2 weighted MRI image . . . . 474.8 The improved statistics for NIDe-Rician combined with UNLM . . . . . . 51

vii

1 Introduction

Magnetic Resonance Imaging (MRI) is one of the e�ective medical equipments that hasbeen proven to be less harmful for patients compared to other medical modalities. Thismethod is more accurate for imaging soft tissues such as brain and muscles. It is knownthat better resolution in MRI images results in less SNR (signal to noise ratio) and viceversa [2]. Therefore, applying a proper denoising technique provides a better opportunityfor accessing to an acceptable image quality and higher SNR.MRI image is generated by using a reconstruction method based on the sampling

process (e.g. using the inverse Fourier transform of the raw data in k-space in case ofregular Cartesian grid sampling). However, the resulted data is still complex values dueto the magnetic �eld or hardware de�ciencies. Therefore, the �nal image is calculated asthe magnitude of the complex data[2] .The noise in MRI signal is a white Guassian noise added to each part of the com-

plex data. The magnitude image which is computed as the square root of two squaredGaussian variables added together has Rician distribution. One can demonstrate thatthe distribution becomes Rayleigh in high SNR areas and Gaussian in low SNR (back-ground) regions of the image [5]. So, the developed �ltering methods which are appliedon the magnitude image should be compatible with this type of signal. The signal depen-dent nature of the Rician distribution results in low contrast images, so a new method inwhich the �ltering method is applied on the squared image is introduced. The squaredsignal has a non-central chi-square distribution and a signal independent noise bias. Thenoise bias can be easily removed after applying the �lters [13].There are two general categories for the denoising methods: acquisition-based and post

acquisition �ltering methods. The former methods involve increasing the scan times, per-forming several similar measurements and averaging them, increasing the voxel dimensionor using more advanced hardware. However, using this type of noise reduction increasesboth the acquisition time and the expenses. Therefore, the second group can be con-sidered as an e�cient and a�ordable technique for MRI image denoising [10]. Several�ltering methods have been developed in past decades. The most classical �lters havebeen used are Gaussian �lters. However, these �lters result in blurring edges. Hence,more edge preserving �lters such as bilateral �lter which functions as a weighted aver-aging �lter have been developed. This �lter has better results compared to linear �ltersbut still removes some detailed information [23]The other type of �lter that is applied for MRI images is Anisotropic Di�usion �lter

which keeps edges by averaging pixels in the orthogonal dimension of the local gradient.However, it still eliminates small featuressmall features [23].Recently, a new method has been proposed by Buades called Non-Local Mean �ltering

(NLM) which uses a weighted averaging method based on the similarities between dif-

1

ferent neighborhoods in the image. This method takes advantage of the fact that thereare many repeated structures in most natural images and removes the noise by �ndingand averaging these structures [13].One of the most common methods that have widely been used for MRI imaging is

wavelet based �ltering method. In this procedure, approximation and detailed subbandsof the image are generated in the wavelet domain and then the detailed subbands whichcontain the high frequency data including small features and mostly noise are modi�edby thresholding method. Here, the most important factor in success of the denoisingmethod is to �nd the optimal threshold by which the noise can be removed while thedetailed structures and specially the edges are preserved. The most common methods for�nding the optimal threshold are VisuShrink, SureShrink, BaysShrink and NeighShrink.Although, they still remove some detailed information [5].The aim of this project is to develop new method of noise removal. The main concen-

tration is on wavelet optimal thresholding for denoising MRI images. For this purpose,a new method called Noise Invalidation Denoising (NIDE) is investigated. This methoduses a noise signature extracted from noise order statistics for denoising the image. Itdoes not require making any assumptions about the noise free signal and can be appliedon noisy data in any type of basis transformation (orthogonal or non-orthogonal) [21].This method is originally proposed for additive Gaussian noise.Therfore, in this thesis the NIDe approach is furthur developed to be compatible with

the MRI images. For this purpose, the original method is modi�ed to be applicableon the non-central chi squared and Rician distribution of the magnitude and squaredmagnitude data.

2

2 Background and literature review

This chapter introduces some basic information about MRI imaging and the availabledenoising methods for this type of the image. In order to describe the denoising methods,it is necessary to provide some theoretical information about the MRI signal distributionin presence of noise. All these are explained in more details as following.

2.1 MRI imaging

Mangentic Resonance imaging (MRI) is one of the common methods have been usedfrom 1946 in wide range of applications. Some of these areas are molecular investigations,chemical information and imaging of living tissues and organs. For the imaging purposes,MRI imaging technique provides images from internal parts of the body from receivedsignals. This application of MRI is so important for the pathological purposes [7].Although, there are other biomedical imaging techniques available but MRI imaging

has some properties that makes it more suitable method. By this method, multi dimen-sional images can be produced. The main di�erence in using MRI comparing to otheravailable techniques here is its capability to produce two dimensional, volumetric threeor four dimensional images in any direction. These images can also provide temporalinformation without any speci�c mechanical adjustments [7].The technique creates images from measured signals coming directly from the object

without any needs to inject contrast agents. It does not have the problems regardingthe radiation as it works in the radio frequency range. Finally, it can creates di�erentimages focusing on one speci�c property of a same object by just changing some intrinsicparameters of the system. The problem with this technique is related to application ofmagnetic �elds in the encoding process. Therefore, there would be some limitation forusing this technique for example in treating patients with metal implants in their bodies[7, 2].As the main focus of this work is on improving produced MRI images, presenting a

general information about the process of the MRI images can be useful. This is describedin 2.1.1 section.

2.1.1 creating a MRI image

The MRI imaging is based on the behavior of Nuclei consisting odd number of protonsor neutrons when an object placed in magnetic �eld. These particles have a spin angularmomentum. They are called spins and the basics of the nuclear magnetic resonance(NMR). A spin and it's angular momentum is illustrated in �gure 2.1 [15].

3

Figure 2.1: Spin and its related angular momentum [15]

The process of creating a MRI image based on spins can be categorized in four steps.Each of these steps are explained in more details as following.

Placing the object in a strong magnetic �eld

At �rst the object will be placed in a magnetic �eld B0. This causes the spins alignthemselves to the magnetic �eld by precessing around that direction. This alignmentcan be parallel or anti-parallel to the magnetic �eld based on the energy level of thespin which is low or high respectively. The di�erence between these two energy levels isrelated to the magnetic �eld and creates a bulk magnetization vector in the direction ofthe magnetic �eld B0 [2, 7].

Applying RF waves to the object

In the next step an RF signal is applied to the object. This causes some of the spinsabsorb the energy and become exited. Therefore, they go from the low energy state to thehigh energy state. This transformation can be seen in bulk magnetization vector in thisway that it �ips than to the x-y plane while precessing. The precessing frequency equalsto Larmor frequency (w = γB0) in which γ is the gyromagnetic ratio. This phenomenacreates a current signal in the receiver [15, 2]. This process is illustrated in �gure 2.2.

Spatial encoding the signal

In order to determine the spatial location of the signal, several gradient �elds are applied.This changes the magnetic �eld locally. The local magnetization now can be presentedby the equation 2.1 [2].

m (x, t) = m0 (x) exp−i(w0 + γgxTn

)t (2.1)

where g (x)is the applied gradient �eld. The received signal is the sum over the objectwhich equals to sampling in the frequency domain or k-space.

4

Figure 2.2: Generated current signal because of applying RF wave [2]

The Received signal can be formulated as equation 2.2 to 2.3 [2].

F {m (x)} =

ˆ ∞−∞

m (x) exp−i2πxTu dx (2.2)

where u = (kx, ky, kz)and

kx (t) =γ

2π

ˆ 2π

0gx (τ) dτ ky (t) =

γ

2π

ˆ 2π

0gy (τ) dτ kz (t) =

γ

2π

ˆ 2π

0gz (τ) dτ

(2.3)

Image reconstruction

At the �nal step, the image is reconstructed based on the applied sampling method. Thesimplest reconstruction technique is an inverse FFT which corresponds to the sampling onCartesian grid. The obtained data from the reconstruction is still complex valued. Thiscan be explained due to de�ciencies of the magnetic �elds or used equipments. Therefore,the magnitude or phase of the complex data is used commonly for representing the �nalimage [2].

2.2 MRI signal distribution

As it was mentioned in last section, the recorded data before reconstruction is complexvalued. This data is also a�ected by an additive complex Gaussian noise with zero meanand variance of σ2. The inverse Fourier transform does not change the characteristicsof the noise due to the transform linearity and orthogonality. A MRI signal a�ected bynoise can be formulated as the equation 2.4 [5, 7] .

5

y = (p+ nre) + i (q + nim) (2.4)

where p ,q, nreand nimare the image and noise real and imaginary parts.As the denoising methods are performed mostly on the raw data, magnitude data or

squared magnitude data, the characteristics of the MRI signal distribution is exploredbased on this classi�cation.

2.2.1 Signal distribution of the raw data

As it was mentioned, the inverse Fourier transform does not change the properties of thecomplex data. Therefore, the data after reconstruction still has a format the same asthe equation 2.4. Therefore, the probability density function (PDF) of the signal can bepresented by the equation 2.5 which describes a Gaussian PDF[7].

p (wr, wi |A,ϕ, σ ) =1

2πσ2exp−(wr −A cosϕ)2

σ2exp−(wi −A sinϕ)2

σ2(2.5)

where wrand wiare the real and imaginary parts of the data with amplitude A andphase value of ϕ.

2.2.2 Signal distribution of the magnitude data

The MRI image is most commonly the magnitude of the raw data and can be formulatedwith equation 2.6 [5]. Using the magnitude data makes the data una�ected by somefactors such as phase variations because of RF inhomogeneity, delay and the samplingmethod [7].

|y| =√

(p+ nre)2 + (q + nim)2 (2.6)

The PDF of the magnitude data is not anymore Gaussian because of the non-linearityof the square root function. It can be derived by �nding the joint PDF of the realand imaginary data and using the polar coordinates[7]. Therefore, The PDF can berepresented as equation 2.7 [5].

p (y |A, σ ) =y

σ2exp−(y2 +A2

)2σ2

I0

(Ay

σ2

)(2.7)

where A is the noiseless signal magnitude and I0is the zeroth order modi�ed Besselfunction [5].One important attribute of the Rician distribution is its signal dependency. It means

that the distribution is changed to Rayleigh distribution when the SNR is low. This canbe related to the areas of the image like background. The PDF of a Rayleigh distributioncan be represented by equation 2.8 [5, 7].

p (y |σ ) =y

σ2exp−(y2)

2σ2(2.8)

6

On the other hand, when SNR is high i.e. in image areas the signal distribution changesto Gaussian [5, 7].

2.2.3 Signal distribution of the squared magnitude data

The magnitude data can be considered as the sum of two Gaussian distributed dataeach squared. Therefore, its distribution is non central chi square and can be shown asequation 2.9 [7].

p (y) =1

2σ2

(y

µ2

)N−12

exp−y + µ2

2σ2IN−1

(√yµ

σ2

)(2.9)

It should be noted that the estimation of noise in the magnitude image is signal depen-dent, so it is di�cult to remove the bias from the data. But for the squared magnitudedata this bias becomes additive. Therefore, it is easy to remove the bias from the squaredmagnitude data. The bias equals to 2σ2which is shown in equation 2.10 [5].

E[y2]

= E[p2]

+ E[q2]

+ 2σ2 (2.10)

2.3 Noise estimation

Di�erent methods has been proposed to estimate the noise variance from the MRI mag-nitude data. They either use the background part of the image or some image statisticsto �nd the noise variance. These methods are explained here brie�y.

2.3.1 Estimating the noise variance from the background

The second moment of the Rayleigh distribution (the signal distribution of the back-ground) equals to E

{y2}

= 2σ2. Therefore, the noise variance can be obtained from theequation 2.11 [20].

σ2 =1

2N

N∑i=1

M2i (2.11)

Also the noise can be estimated from the �rst moment of the Rayleigh distribution.This can be seen in equation 2.12 [20].

σ =

√2

π

1

N

N∑i=1

Mi (2.12)

However, estimating the noise variance based on background areas has some drawbacks.In these methods, the main assumption is that the signal in the background area is alwayszero and the background area is not selected automatically. These factors a�ects theresult and make it inaccurate [20].

7

2.3.2 Estimating the noise variance based on local statistics

The �rst approach for estimating the noise based on local statistics is using the secondorder moment of the whole image (not only the background). It is proved that for anormal image the shape of the second-order moment distribution is not changed when itgets noisy by Rician noise. In MRI images the second order moment distribution has anunique property which is a maximum in the origin. This maximum appears in the imagedue to the presence of background in the image. Again, when the image is noisy a shiftin the maximum can be seen. This principle is illustrated in �gure 2.3 [20].

Figure 2.3: Local second order distribution for a: normal image and b: MRI image. Thedashed line shows the distribution related to the noisy images[20]

Therefore, the position of the maximum can be used to estimate the noise variance.Equation 2.13 shows this relationship [20].

σ2 =1

2mode (µ2ij) (2.13)

where µ2ij = E{I2ij

}is a selected neighborhood.

The same reasoning can also be applied for the local mean distribution of the MRIimage. Again, due to the background area a maximum can be seen in origin. This max-imum will be shifted when the image is noisy. Therefore, the variance can be estimatedby �nding the maximum of the local mean. This is presented in equation 2.14 [20].

σ =

√2

πmode (µ1ij) (2.14)

where µ1ij is the local mean of the image in a selected neighborhood.

8

2.4 Denoising methods

Di�erent denoising methods have been used to improve the quality and increasing theSNR of MRI images. The SNR of an MR image can be represented by the equation 2.15[23].

SNR = C.f (Ob) .g (Im) (2.15)

Where C is a constant depending on the physical properties of the system like magnetic�ux, f (Ob) is related on the dimensions of the target, and g (Im) is based on selectedimaging factors. The last function can be measured knowing readout frequency ω0, voxeldimension Vh and the imaging time T (g (Im) = ω0Vh

√T ). It is obvious that having

better resolution leads to less SNR. Therefore, applying a proper denoising techniqueprovides the opportunity of accessing to an image with acceptable quality along withhigher SNR [23].There are two general categories for the denoising methods: acquisition-based and post

acquisition �ltering methods. The former methods involve increasing the scan times, per-forming several similar measurements and averaging them, increasing the voxel dimensionor using more advanced hardware. However, using this type of noise reduction methodsincreases both the acquisition time and the expenses. Therefore, the second group canbe considered as an e�cient and a�ordable technique for MRI image denoising [10].One important di�erence in post acquisition-based denoising methods is the data they

are applied to (raw, magnitude, squared magnitude). Some common methods of MRIdenoising are described here.

2.4.1 Gaussian �ltering

This is one of the most basic methods for denoising MRI images. However, it removessharp edges and �ne details as it is a smoothing �lter [23].

2.4.2 Bilateral �ltering

This method can be described as weighted averaging method. Two weight factors areused in this method. One of them is based on the spatial distance di�erence while theother is based on the intensity di�erence. An output of such a �lter can be described asequation 2.16 [23, 22].

h (x) =

∑ε f (ε) c (ε, x) .S (f (ε) , f (x))∑

ε c (ε, x) .S (f (ε) , f (x))(2.16)

where c (ε, x)is the spatial distance measuring factor, f (x)is the original image andS (f (ε) , f (x))is the intensity di�erence measuring factor. The similarity measuring fac-tors can be described by equations 2.17 and 2.18 in which both of them are Gaussian[23, 22].

9

c (ε, x) = exp 1/2

(‖ε− f (x)‖

σd

)2

(2.17)

S (ε, x) = exp 1/2

(‖fε − f (x)‖

σr

)2

(2.18)

where σd and σr are spatial and photometric spread respectively.The bilateral �ltering performs better than linear �ltering methods in case of smoothing

and keeping the edges. However, it still removes some details [23].

2.4.3 Anisotropic Di�usion Filtering

This method is based on Linear MinimumMean Square Error (LMMSE) estimator that iscombined with a Partial Di�erential Equation (PDE). The LMMSE is de�ned as equation2.19 [14, 16, 24].

ˆA2 (x) =

⟨M (x)2

⟩− 2σ2 +K (x)

(M2 (x)−

⟨M (x)2

⟩)(2.19)

where M (x)is the magnitude data, A (x)is the noiseless signal value and K (x)is afactor de�ned as equation 2.20 [14].

K (x) = 1−4σ2

(⟨M (x)2

⟩− 2σ

)⟨M (x)4

⟩−⟨M (x)2

⟩2 (2.20)

where 〈.〉describes the expected value.The PDE function is also can be described as equation 2.21 [14].{

u (x, 0) = g∂u(x,t)∂t = div (c∇u (x, t))

(2.21)

where c (x, t) = 1−K (x, t)is the di�usion function.Although this �lter preserve edges by averaging orthogonal to the local gradient, but

it still removes detailed information. It also modi�es the image statistics because of itsedge improving characteristic [13].

2.4.4 Wavelet denoising

The wavelet transform function is used to analyze a signal in its di�erent scales i.e.detailed and approximation structures. In contrast to the Fourier transform that is usedto present the spectral content of a signal, wavelet transform can be used to evaluate thelocal properties of a signal. Furthermore, the wavelet transform can be used to detectedges[18].To perform a wavelet transform, a speci�c wavelet is selected. The wavelet is brought

to a speci�c scale and also shifted. Then the correlation of the analyzed signal and the

10

wavelet is explored. The result determines the properties of the signal in that scale. Thewhole process can also be de�ned as a digital �lter bank consisting low and high pass�lters. These �lters are applied on the low pass results until the desired level. The outputof the �nal low pass �lter presents the approximation of the analyzed signal while theoutput of the high pass �lter can be considered as the detailed part (the high frequencypart) of the signal [18].The wavelets are generated by dilation and translation of a general wavelet function

called mother wavelet ψ (x). This is shown by equation 2.22 [18] .

ψa,b (x) =1√aψ

(x− ba

)(2.22)

The simplest form of the wavelet transform is the continuous wavelet transform. How-ever, this method is redundant and shift invariant. They are used often for signal char-acterization. Therefore, the other class of wavelet transforms i.e. Discrete wavelet trans-form (DWT) are introduced. This class of waveforms can be considered as simple sam-pling of the Continuous one. A common DWT wavelet coe�cient and approximationcoe�cient basis can be represented in equation 2.23 and 2.24 respectively [18].{

ψj,k (x) =1√2jψ

(x− 2jk

2j

)}(j,k)∈Z2

(2.23)

{ϕj,k (x) =

1√2jϕ

(x− 2jk

2j

)}kεZ

(2.24)

A signal can be decomposed with this basis according equation 2.25 [18].

f (x) =∞∑

j=−∞

∞∑k=−∞

ωj,kψj,k (x) +∞∑

k=−∞sJ,kϕJ,k (x) (2.25)

where the wavelet coe�cients ωj,k and approximation coe�cients sj,kcan be obtainedas equation 2.26 [18].

ωi,j =⟨f, ψi,j

⟩sj,k = 〈f, ϕi,j〉 (2.26)

the . function represent the dual basis function.The DWT method is not shift invariant. Therefore, it is not suitable for some appli-

cations such as pattern identi�cation and makes some problems for denoising due to notenough redundancy. Therefore, Non-decimated Discrete Wavelet Transform (nDWT)can be used to ful�ll the requirements for denoising and pattern recognition [18].In nDWT, the number of wavelet coe�cients is not decreased in each scale. They

are achieved by sampling the CWT at all integer locations at each translation. Thewavelet coe�cient basis for this function are de�ned as 2j/2ψ

(2−j (x− k)

)and the signal

is achieved from the equation 2.27 [18].

f (x) =

∞∑j=−∞

∞∑k=−∞

⟨f, ˜ψj,k

⟩ψj,k (x) (2.27)

11

The basis here are not anymore linearly independent and they form a frame. Thismethod is more redundant and also shift invariant. Therefore it could be a good optionfor image denoising [18].

Denoising in Wavelet domain

Di�erent method have been used to denoise an image in wavelet domain. They eitherbased on estimation techniques or thresholding [11, 19, 4, 1]. In the thresholding method,the coe�cients that are less than a given threshold are eliminated. There are two generaltypes of thresholding techniques: Soft and Hard Thresholding. The hard thresholdingthat is not optimal for discontinuities and remains unchanged for observations is de�nedas equation 2.28 [6].

ηH (ω, t) =

{ω |ω| ≥ t0 |ω| < t

(2.28)

On the other hand, the soft thresholding approach in which the data shrinks in eachtry and is a better option for discontinuities can be be de�ned as equation 2.29 [6].

ηS (ω, t) =

w − t ω ≥ t0 |ω| < t

ω + t ω ≤ −t(2.29)

The main task in this part is to �nd an optimal threshold. Di�erent methods have beenused. In the next section one of the methods that is used to �nd an optimal thresholdingand is a basis for the purpose of this project is introduced.

Noise Invalidation denoising Noise invalidation Denoising can be considered as a methodwhich works in the wavelet domain. As it was mentioned before, several methods areavailable for noise �ltering in wavelet domain. They are designed to estimate a thresholdvalue and then use a thersholding method to remove the noise coe�cients from an image.However, the main logic of these methods are based on some characteristics of the noise-free signal and they are often application based. On the contrary, NIDe approach fordenoising relies on the noise properties. It de�nes a con�dence noise area and considerthe data outside of this area as a noise free signal. It should be noted that the originalNIDe method performs on signals contaminated with additive Gaussian noise[21]. Thebasic principles of this method can be described as following.A signature function can be de�ned for any variables v and z as g (z, v)with �nite mean

GE (z) and variance Gvar (z) over the random noise vector V . Therefore, the signaturefor N samples of the random noise and its mean and variance can be de�ned as equations2.30 to 2.32 respectively[21].

g(z, vN

)=

1

N

N∑i=1

g (z, vi) (2.30)

12

E(g(z, vN

))= GE (z) (2.31)

var(g(z, vN

))=

1

NGvar (z) (2.32)

The noise signature function that is used for this method is based on Absolute NoiseSorting (ANS) de�ned as equation 2.33 [21].

g (z, vi) =

{1 |vi| ≤ z0 |vi| > z

(2.33)

The mean and variance of this signature can be formulated as equations 2.34and 2.35respectively [21].

E(g(z, V N

))= F (z) (2.34)

var(g(z, V N

))=

1

NF (z) (1− F (z)) (2.35)

where F (z) = 2φ(zσ

)− 1 is the cumulative distribution function of absolute value of

the additive noise.It can be realized from the proposed signature function, that for each z , g

(z, vN

)= m

Nin which m is the number of samples with absolute values less than z. On the other hand,the mthvalue in a sorted array is the largest vithat is less than z can be formulated asm = Ng

(z, vN

)[21]. The e�ect of ANS function can be illustrated in �gure 2.4.

As it is obvious in the �gure 2.4, the sorted data place in such a denser area (middleimage) than the original data (top image). Also, changing the axes shows that values arearound mean NF (z)with variance F (z) (1− F (z)). Therefore it is possible to de�nean area in which the data can be considered as noise. In other words, the probabilityof data being placed in a limited area around the mean value should be high (equation2.36) [21].

Pr {LN (z) ≤ g (z, vN ) ≤ UN (z)} (2.36)

in which the up and down boundaries can be calculated based on the equations 2.38and2.37respectively[21].

LN (z) = F (z)− λ√

1

NF (z) (1− F (z)) (2.37)

UN (z) = F (z)− λ√

1

NF (z) (1− F (z)) (2.38)

According to the equations 2.37 and 2.38, the boundaries are functions of mean andstandard deviation of the sorted noise only data. λis the controlling factor to increasethe probability and at the same time to prevent having very loose boundaries [21].

13

Figure 2.4: E�ect of sorting the absolute values: samples of unsorted noise, sorted noisevalues and changed axes for top to bottom respectively[21]

Considering a noisy signal with coe�cients θi = vi + θi with θias the noiseless data,the mean and variance of samples of a random process noisy data can be represented bythe equations 2.39 and 2.40 [21].

E(g(z,ΘN

))=

1

N

N∑i=1

H(z, θi

)(2.39)

var(g(z,ΘN

))=

1

N2

N∑i=1

H(z, θi

) (1−H

(z, θi

))(2.40)

where H(z, θi

)is the expected value of the signature function of noisy data.

As the noise only data that was described before, sorting the noisy data leads to a densearea of data. Comparing this area with the noise only con�dence area will determine apoint at which the the noisy data leave the noise con�dence area. This concept can beformulated as equation 2.41 [21].

14

T = maxz∀z ≤ x : LN (x) � g

(x, θN

)� UN (x) (2.41)

The explained behavior is also illustrated in �gure 3.2.

Figure 2.5: Finding threshold with NIDe. Solid line is the noisy data and dashed linesrepresent the noise only boundaries[21]

2.4.5 Non Local Mean (NLM) �lters

The NLM �lters is a weighted averaging �lter based on the intensity similarity of thepixels. As it was mentioned in section 2.4.2, Bilateral �ltering behaves also like this. Thefactor that can discriminate between these two methods or other similar methods is thecomparing region. In contrast to bilateral �ltering that �nd the similarity of a pixel withits neighborhood, the NLM �lter is based on region comparison. Therefore, the patternredundancy is not local anymore and the far pixels are not ignored [13, 3].For a NLM �lter on a noisy image Y , the output can be described as equation 2.42.

NLM (Y (p)) =∑∀q∈Np

w (p, q)Y (q) (2.42)

Here, the weights are calculated according to the similarity of the squared windowneighborhoods of two pixels p and q with Radius of Rsim. The weights w (p, q)can bedriven based on the equations 2.43 and 2.44 [13].

w (p, q) =1

Z (p)exp−d (p, q)

h2(2.43)

15

Z (p) =∑∀q

exp−d (p, q)

h2(2.44)

where h is a controlling factor and d which is the Euclidean distance of pixels in twoneighborhoods can be described as equation 2.45 [13].

d(p, q) = Gσ ‖Y (Np)− Y (Nq)‖2Rsim (2.45)

Here, Gσis a Gaussian function with zero mean and variance of σ2.The other important parameter in this method is Rsearchwhich is de�ned as the radius

of a window to compare the pixels on that neighborhood (Np) instead of the whole imagefor a speci�c pixel. This will decrease the complexity of the method [13].As it was mentioned before, the noise bias in the magnitude image is hard to remove.

But it can be removed easily from the squared magnitude image. Based on this assump-tion, a version of NLM �ltering called Unbiased Non Local Mean �ltering (UNLM) isdeveloped. It is described based on the equation 2.46 [13].

UNLM (Y ) =

√NLM (Y )2 − 2σ2 (2.46)

16

3 Method and Simulation Results

Based on the knowledge about MRI images, their signal distribution and also denoisingmethods that were explained in the previous chapter, It is possible to introduce thegeneralized NIDe method for MRI denoising. The new method improves the signalto noise ratio while keeping the image quality as good as possible. Therefore, in thischapter at �rst the proposed method for denoising MRI images is introduced. Thenafter introducing the criteria for evaluating the new method, used material are described.Finally the results of running implemented algorithm on two di�erent type of MRI images(T1/T2 weighted) are investigated.

3.1 Method and material

The NIDe approach that is described in chapter two is originally implemented for onedimensional signals a�ected with additive Gaussian noise. Although, some of its interest-ing characteristics such as using noise statistics instead of the noise free signal propertiesimplies that this method can also be used on two dimensional MRI signals with di�erentnoise distribution after applying some modi�cations. These modi�cations are applied toadjust the original method with MRI di�erent signal and noise distribution. The exam-ined MRI signal can either be an image signal (amplitude signal) or squared image withRician or non-central chi-squared distribution respectively. Therefore, two approachesbased on NIDe method are developed for MRI signals which are described in the follow-ing.It should be noted that the NIDe approach for MRI images works in a wavelet domain.

Therefore, it can be considered as one of the wavelet domain methods used for MRIdenoising. In this method, a non-decimated wavelet transform is used. The waveletfamily used here is Haar and the image signal is decomposed to three level. Then theNIDe approach will be applied on the detailed coe�cients to �nd the optimal thresholdand soft thresholding is the method used here to eliminate the unwanted coe�cients.

3.1.1 Noise Invalidation Denoising for Rician distribution

As it was mentioned in chapter 2, MRI signals are most often displayed as magnitudesignal. This signal has a Rician distribution which transforms to a Rayleigh distributionin background areas and a Gaussian distribution in the image area with high SNR.Therefore, it is possible to �lter the image with the NIDe approach if the noise signaturehas a di�erent distribution. This distribution should be compatible with the signalproperties of the MRI image.

17

Accordingly, the ANS function in this case is de�ned for a random process VN withindependent identically distributed members. These members have a Rayleigh distribu-tion. Therefore the mean and variance of the ANS function g

(z, V N

)are functions of a

Rayleigh distribution CDF (instead of Gaussian distribution).In this case the CDF in equations 2.34 and 2.35 can be formulated as equation 3.1[7].

F (z) = 1− exp

(− z2

2σ2

)(3.1)

Like the original NIDe approach with additive Gaussian noise, the sorting of absolutevalues of the noise only data with Rayleigh distribution causes the data gather in acondense area around mean value. Therefore, it is possible to de�ne a noise con�dencearea around the mean.. This change of CDF also a�ects the up and down limits (equations2.37 and 2.38) of the noise con�dence region.The new threshold can be described as the point at which noisy signal (MRI signal

with noise) will have a di�erent distribution than Rayleigh. In the case of the MRI imagesignal, this new distribution is Gaussian. Therefore by eliminating the coe�cients underthe obtained threshold, the signal will be noise free.Using NIDe approach for Rician distribution, the noise bias in the image is still signal

dependent and can not easily be removed. However, the bias can be easily removed fromthe squared signal. Therefore, the �ltered image signal is transformed to the time domainand squared. Now the bias can be removed easily from the squared image. This methodis based on the method proposed in [13] to create an unbiased NLM �ltering method.After removing the noise bias, the square root of the signal is taken and the image canbe displayed. This process can be formulated based on the following formula.

I =

√NIDe (I)2 − 2σ2 (3.2)

3.1.2 Noise Invalidation Denoising for chi-square distribution

It is also possible to apply the the NIDe approach to the squared image. As it wasexplained in chapter 2, the signal distribution in a squared MRI image is not Riciananymore. It changes to the non-central chi-squared distribution as it can be consideredas sum of the two squared Gaussian distribution with a �nite mean and variance.To apply the NIDe method it is necessary to have a noise distribution with chi-square

distribution. Therefore, the comulitative distribution function can be de�ned as equation3.3[12].

F (z) = 1−Q k2

(√λ,√z)

(3.3)

where QM (a, b) is the Marcum Q-function, λis the non-centrality parameter and k isthe degree of freedom.Again this change of CDF will a�ect the mean and variance of the ANS function in

equations 2.34 and 2.35. Therefore, the sorted absolute value of noise coe�cients arepopulated around the new mean which is a function of the chi-squared CDF. The lower

18

and upper limits are also can be calculated as the equations 2.37 and 2.38 respectivelywith the di�erence that the new CDF of the sorted absolute values should substituted inthe equations.The algorithm �nd the optimal threshold for de�ning noiseless signal by comparing

the noisy squared data and the noise con�dence area. It is the point at which the sortedabsolute valued signal leave the noise con�dence area.The noise bias in this case is no longer signal dependent and as it was mentioned

before, it can be removed by easily subtracting it from the squared data coe�cientsdirectly. Finally, the root square of the data is the image can be displayed.

3.1.3 Noise variance estimation

The noise variance estimation method that is used in the implemented algorithm isbased on the Local second order moment approach. This approach is described in detailin chapter 2. However, the size of the window for the neighborhood in which mean iscalculated is seven.

3.1.4 Evaluation criteria

Di�erent metrics have been used for the purpose of evaluating. These criteria can becategorized in two main group of quantitative and qualitative metrics. Among the �rstgroup Root Mean Squared Error (RMSE), SSIM and PSNR and from the second categorycontrast value are common measures that have been used in many research[5]. Therefore,these metrics are also used in this research to compare and validate the proposed methodof denoising a MRI image.

Root Mean Squared Error

RMSE calculates the di�erence between original and estimated data to measure how theapplied method was useful. The less the result is, the method is more valid. RMSE canbe formulated as equation 3.4 [5].

RMSE =

√√√√∑i

∑j

[I (i, j)− I (i, j)

]2M ×N

(3.4)

where the I (i, j)is the noiseless data, ˆI (i, j) is the estimated data and M and Nrepresent the dimension of the image [5].

Contrast

It is a qualitative metric that indicates the di�erence in brightness between two regions.The more the contrast is, the method can keep more details. Therefore, it is important toshow that the denoising method keeps the details and also remove noise. To measure thecontrast, a region of interest (ROI) should be selected. The contrast can be formulatedas equation 3.5 [5].

19

C =Smax − SminSmax + Smin

(3.5)

Smaxand Sminrepresent the maximum and minimum intensity of the pixels in theselected area[5].

Structural similarity index (SSIM)

To be able to investigate the similarity in the visual sense, structural similarity index isa good criteria to be explored. As RMSE is not an optimal measure for measuring ifthe denoising method keeps the detailed structures, the SSIM index is measured in thiswork. It can be calculated to show the similarity between the denoised (Y) and noiseless(X) image in a neighborhood with radius 11 based on the equation 3.6 [5, 25].

SSIMN (x, y) =(2µxµy + c1) (2σxy + c2)(

µ2x + µ2y + c1) (σ2x + σ2y + c2

) (3.6)

where µis the mean, σis the standard deviation and c1 and c2 are some constantrelated to the maximum intensity in the image. The �nal SSIM is calculated by achievingthe mean of all the SSIMs for the N locals. The mean and standard deviation can becalculated based on the equations 3.7 to 3.9[5, 25].

µx =N∑i=1

ωixi (3.7)

σx =

(N∑i=1

ωi (xi − µx)2)1/2

(3.8)

σxy =

N∑i=1

ωi (xi − µx) (yi − µy) (3.9)

The value of the SSIM is between -1 to 1 and the more is this value the more thesimilarity is [25, 5].

3.1.5 Material

The proposed method is implemented in Matlab. The data for the simulation are takenfrom the Brainweb database [9]. The data consist of T1 weighted and T2 weighted MRIimage of normal brain with resolution 217 × 181. Moreover, data are contaminated byadding Rician noise to the raw data taken from the brainweb site. The noise is addedaccording to the equation 3.10 [5].

I =

√(I + n1)

2 + n22 (3.10)

where n1and n2are two random Gaussian distributed variables with N(0, σ2

)and I is

the original data [5].

20

The level of noise varies between 3%, 5% and 7% on each of the T1 and T2 weightedMRI images to show the e�ect of di�erent noise level on di�erent types of MRI images.

3.2 Simulation Results

Two NIDe methods i.e. NIDe with Rician and NIDe with chi-squared distribution areimplemented and runned on the existing MRI data. In the next step, The Nide-Ricianapproach is combined with UNLM method to investigate the e�ect of combing two meth-ods. The results are described separately for each method as following.

3.2.1 Results for NIDe for Rician distributed data

The NIDe algorithm at �rst executed on each MRI data with di�erent noise level. As itwas described before, the magnitude MRI data has a Rician distribution which is signaldependent i.e. it is Rayleigh in low SNR and Gaussian in high SNR areas respectively.This fact is illustrated in �gure 3.1.

21

Figure 3.1: Probability density function plots for noisy image, data in the backgroundand signal area top to bottom respectively

After estimating the noise variance and transforming to the wavelet domain, noisesignature is produced. Figure 3.2 shows how ANS function causes most of the noisecoe�cients are gathered around mean.

22

Figure 3.2: ANS function on noise only data with Rayleigh distribution

After calculating the up and down boundaries, a noise con�dence area is determined.The value of the landa which determine how wide is this con�dence area is 5 < λ < 9.However, the best estimate for landa value in the this experiment is set to eight. Thenoise con�dence area is illustrated in �gure 3.3.

23

Figure 3.3: Noise con�dence area for the Rician distributed Noise signature. The blue lineshows the mean while the two red lines shows the upper and lower boundaries

Finally the optimal threshold is obtained by comparing the sorted noisy image datawith the con�dence noise area. This process is illustrated in �gure 3.4.

Figure 3.4: Finding the optimal threshold for the NIDe with Rician distribution method.The blue line shows the sorted noisy data and how it exists the con�dencearea

The other important point here is the noise bias. As it was mentioned before, the noisebias is removed by squaring the denoised image and subtracting it from this data. Then

24

again the �nal image is obtained by taking the square root of the data.The �nal results of NIDe denoising on a T1 weighted MRI image with 3%, 5% and 7%

noise, is illustrated in �gures 3.5to 3.7.

Figure 3.5: Denoising result of NIDe for Rician distribution with 3% noise

25


26


To illustrate the e�ect of the proposed denoising method on di�erent types of MRIimage with di�erent noise contribution, The algorithm was performed on T2 weightedwith di�erent levels on noise too. Figures 3.8 to 3.10 show the �nal result of NIDe methodon T2 weighted image with 3%, 5% and 7% noise level.

27

Figure 3.8: NIDe approach based on Rayleigh distribution results on T2 weighted imagewith 3% noise level

28


29


The value of the threshold for T1/ T2weighted images with 3-7% noise level are listedin Table 3.1.

Noise level 3% 5% 7%

T1 weighted MRI image 39.9719 94.7414 147.9547

T2 weighted MRI image 223.0496 420.2845 434.4847

Table 3.1: Threshold values T1/T2 weighted MRI images at di�erent noise level for Ri-cian distribution

Other statistics regarding to other levels of noise are mentioned in next chapter forthe purpose of comparing the proposed method with other existing approaches. Thecomparison is made in terms of evaluation criteria explained in section 3.1.4.

30

3.2.2 Results for NIDe for chi-squared distributed data

In the second type of NIDe i.e. with the chi-squared distribution, on the squared dataof the image. As it was mentioned before, the data now have non-central chi squareddistribution. This is illustrated in �gure3.11.

Figure 3.11: PDF plot of the squared data

To apply the NIDe method, again the noise variance estimated and wavelet transformwith non decimated wavelet transform approach is applied on the data. Then the noisebias equal to 2σ2is subtracted from the approximation coe�cients. The bias removal canbe done in this step because in the case of squared data, the noise bias is not anymoresignal dependent. It is additive and can be easily removed by subtraction.Then, the noise signature for chi-squared distribution noise is generated based on the

ANS method. Figure 3.12shows how ANS function behave on the noise only data. Byapplying this function, the major part of the noise only data gather around the mean.

31

Figure 3.12: E�ect of sorting on the noisy data with chi-squared distribution

The noise con�dence area for noise with chi-squared distribution with regard to itsup and down boundaries is determined. The value of landa which de�nes the largenessof noise con�dence are in this approach is 3 < λ < 5. However, the best �t for theexperiment is 3.5. The result noise con�dence area based on the mentioned values isillustrated in �gure 3.13.

32

Figure 3.13: Noise con�dence area for the chi-squared distribution noise. The blue lineshows the mean while the red lines are the limits of this area.

Finally, the optimal threshold is found by comparing the sorted noisy squared datawith the noise con�dence area. When the sorted noisy data leaves the con�dence area,the amplitude of data on that point is considered as the threshold for �ltering the image.The process of �nding this threshold is illustrated in �gure 3.14.

Figure 3.14: Finding the threshold in the NIDe approach based on chi-squared distribu-tion. The blue line shows the sorted noisy data.

After soft thresholding the detailed coe�cients of the squared data with the optimalthreshold found in the previous step, data will be transformed to the spatial area. Thisis done by inverse non-decimated wavelet transform. Finally, squared root of the dataare used for the purpose of displaying the image.The �nal �ltered image of a T1-wieghted MRI image with 3%, 5% and 7% noise level

is illustrated in �gure 3.15 to 3.17.

33

Figure 3.15: Final Result of the denoising with NIDe on the squared data with 3% noise

34


35


The algorithm is also performed on T2-wieghted MRI image with di�erent noise levels.�gure 3.18 to 3.20 illustrate the results of �ltering on a T2 weighted MRI image with3%, 5% and 7% noise levels respectively.

36

Figure 3.18: Final result of NIDe algorithm based on chi-squared distribution on the T2weighted MRI with 3% noise.

37


38


The threshold values for T1/ T2weighted images with 3-7% noise level found by NIDeapproach based on chi-squared distribution are listed in Table 3.2.

Noise level 3% 5% 7%

T1-weighted MRI image 1.5436e+04 8.3615e+04 2.4690e+05

T2-weighted MRI image 0.9303e+05 1.6124e+06 1.2995e+06

Table 3.2: Threshold values T1/T2 weighted MRI images at di�erent noise level for chisquared distribution

Other statistics regarding the evaluation of this method is explained in the next chaptercomparing with other available methods of MRI denoising.

3.2.3 Results for combining NIDe with Richian distribution with UNLM

In this part, the Nide method for image data with rician distribution is combined withUNLM. The combination is in a way that �rst the NIDe approach applied and the detailed

39

coe�cients are soft thresholded with the achieved threshold. Then to take advantage ofthe weighted averaging e�ect of the UNLM, it is applied on the thresholded detailedcoe�cients.The result for T1 image 5% level is illustrated in �gure 3.21.

Figure 3.21: T1-weighted MRI image with 5% noise level �ltered with NIDe-UNLM

40

4 Discussion and future works

In this chapter, the proposed method is compared with a well-known method of NLMand UNLM for MRI denoising. Then the advantageous and limitations of the method isexplaind. Finally, some ideas that would be good to be considered for the future workson MRI image denoising are presented.

4.1 Comparison based on evaluation criteria

For the purpose of evaluating the proposed method, the NIDe approach (for both chi-squared and Rician distribution) is compared with Non Local Mean and Unbiased NonLocal Mean technique. The NLM tool box on Mathwork is used for the basic NLM [17].However, for the UNLM some modi�cations are applied.As it was mentioned in 3.1.4 segment, four evaluation criteria are used. Tables 4.1 to

4.3 show the result of each criteria for four di�erent denoising methods i.e. NIDe-Ricianand chi-squared, UNLM and NLM on T1 wieghted MRI image with 3%, 5% and 7%noise levels respectively.

Comparison RMSE PSNR Contrast SSIM

NIDe-Rician 21.56 34.84 0.53 0.91

NIDe-chi square 25.12 33.51 0.53 0.88

UNLM 19.91 35.53 0.51 0.90

NLM 25.56 33.36 0.52 0.85

Table 4.1: Comparison table for noise level of 3% on T1 weighted MRI image


NIDe-Rician 29.72 32.06 0.55 0.87

NIDe-chi square 32.05 31.40 0.55 0.86

UNLM 28.45 32.41 0.53 0.86

NLM 40.03 29.47 0.52 0.79


41


NIDe-Rician 37.65 30.00 0.50 0.84

NIDe-chi square 39.29 29.63 0.50 0.82

UNLM 37.10 30.13 0.54 0.81

NLM 55.19 26.68 0.49 0.73


The statistics show that the UNLM method has the better results in terms of RMSEand PSNR for low levels of noise . However the structural similarity is always better forNIDe approach. This comparision proves that the visual perception in terms of detailedstructures is better in NIDe-Rician method. Although the images are more blurredwhich is not a case with UNLM method. The blurring e�ect could be due to the wavelettransformation. It was mentioned before that wavelet trnasformation acts as a �lter bankconsisting low and high pass �lters. The low pass �lter can results in blurring in sharptransitions such as edges. However, in UNLM the wieghts are functions of Guassiandistribution. Therefore, they prevent blurring e�ects. This e�ect of blurring can be seenin the di�ernece image in �gure 3.6 in the chapter 3. It is obvious that the NIDe approchcan not preserve edges as good as UNLM but it keeps the detailed information whereasthe UNLM can not preserve them.Furthuremore, the results from the NIDe method on chi-squared data is worse than

NIDe-Rician and UNLM method. The reseaon can be due to the high degree of similaritybetween chi-squared noise signature and non-central chi squared distribution of the image.This results in smaller optimal threshold. Therefore, the noise can not be removedproperly with this method. This problem is more obvious with the lower levels of noisethat the similarity between two distributions is more.The NIDe Rician has better results in all aspects comparing to the basic NLM method.

However, the NIDe chi gets better results than NLM by increasing the noise level.One obvious advantageous of running NIDe approach comparing to the NLM and

UNLM is the algorithm speed. Although the NLM method does not do the similaritycheck for each pixel on the whole image (just in a searching neighborhood), it is still slow.It has to investigate for all similar patterns for all the pixels in the image. Therefore, thespeed is decreasing noticably for higher resolution images. This di�erence in calculationspeed can be represented in table 4.4 for two methods on NIDe and UNLM in threedi�erent sizes.

Image size original Size half size image quarter sized image

UNLM 18.02s 4.37s 1.03s

NIDe approach 2.8s 0.83s 0.37s

Table 4.4: Speed di�erence

This advantageous can also be explained in a di�erent manner. The complex complex-ity can be represented by Time computational complexity (O(N)). It is proved that the

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time computational complexity of the UNLM method is O (N ×B × S) where N is thesize of the image, B is the size of each neigborhood and S is the size of the search neigh-borhood [8] while for the NIDe approach the computational complexity is only dependson the length of the data (size of the image) and can be represented by the O (N).The results for each level of noise in a T1-weighted MRI image is illustrated in �gures

4.1 to 4.3 .

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Figure 4.1: Comparing �gures of each methods for T1 weighted with 3% noise

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45


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The same conclusion can be obtained by invastigating the T2-wieghted MRI images.The statistics are presented in Tables 4.5 to 4.7while �gures 4.4 to 4.6 illustrated theresulted images for all four methods with 3%, 5% and 7% noise levels.


NIDe-Rician 110.57 32.44 0.71 0.93

NIDe-chi square 113.94 32.17 0.70 0.91

UNLM 108.72 32.58 0.72 0.92

NLM 116.4008 31.99 0.71 0.91



NIDe-Rician 167.64 28.82 0.72 0.86

NIDe-chi square 170.11 28.69 0.71 0.86

UNLM 143.96 30.14 0.67 0.85

NLM 189.93 27.74 0.69 0.83



NIDe-Rician 197.64 27.39 0.72 0.82

NIDe-chi square 201.38 27.23 0.70 0.81

UNLM 178.26 22.84 0.70 0.80

NLM 216.90 26.58 0.69 0.81


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48


49


50

As some of the noise is not removed by the NIDe approach, the UNLM method is ap-plied in wavelet domain after thresholding the detailed coe�cients with optimal thresh-old. The statistics resulted from this method for a T1 wieghted MRI image with 5%noise level is illustrated in table.It is obvious that the PSNR and RSME is improved while the criteria regarding the

detailed structures are still the same. These statistics are listed in table 4.8.


NIDe-Rician 29.72 32.06 0.55 0.87

UNLM 28.45 32.41 0.53 0.86

NIDe-Rician with UNLM 28.48 32.47 0.54 0.87

Table 4.8: The improved statistics for NIDe-Rician combined with UNLM

However the image is still blurred. This can be illustarted in �gure 4.7.

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Figure 4.7: Final results of NIDe-Rician combined with UNLM

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It should be noted that applying UNLM method on the NIDe approach decreases thespeed of the algorithm.

4.2 Limitations

As it was mentioned in the 4.1 section, the following list can be considered as somelimitations of proposed method.

• Blurring e�ect: This e�ect can be due to wavelet transform function.

• Lower PSNR: It is specially obvious for the NIDe with chi-squared distribution dueto the high level of similarity between noise signature and MRI signal distribution.

However, the latter drawback can be improved by using UNLM method or some similarmethods after the soft thresholding. To decrease the blurring e�ect, using some otherwavelet transform functions chould be useful.

4.3 Conclusion and future trends

MRI images are one of the useful and most common modalities have been used fordecades in the �eld of medicine. Its special characteristics such as its harmlessness,its capability to generate good visualization of some internal tissues and also functionalbehaviors emphasizes its critical role. However, the noise and artifacts in the �nal imagescan a�ect the diagnosis. Therefore, developing signal processing methods compared tohardware based techniques or methods such as averaging can be cost e�ective.NIDe method developed for MRI images tries to provide alternative solution for this

problem. There are two methods based on NIDe proposed. In the �rst method (NIDeRician) the NIDe approach which is originally based on additive Gaussian noise is furtherdeveloped to be consistent with the Rician noise. The second method (NIDe-chi squared)is compatible with the chi-squared distributed noise for the squared magnitude MRI data.The results of applying NIDe approach on simulated data shows that the NIDe-Rician

method gains better results than the other method (NIDe-chi squared). While the resultshave less PSNR and are more blurred comparing the well-known UNLM method, it keepsmore detailed structures. Combining the NIDe and UNLM improved the PSNR whilethe blurring e�ect still exists. The important advantageous method over exstings methodis its less computational complexity.Therefore, investigating some other wavelet transform functions can be useful to de-

crease the blurring e�ect. On the other hand, the proposed method is only performedon the simulated data. Hence, implementing the method on some real clinical data canbe a good idea to prove the e�ectiveness of NIDe approach.

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Nomenclature

ANS Absolute Noise Sorting

DWT Discrete wavelet transform

FFT Fast Fourier Transform

LMMSE Linear Minimum Square Error

MRI Magnetic Resonance Imaging

nDWT Non-decimated Discrete Wavelet Transform

NLM Non Local Mean

NLM Non Local Mean

NMR nuclear magnetic resonance

PDE Partial Di�erential Equation

PDF probability density function

RMSE Root Mean Squared Error

ROI region of interest

SNR Signal to Noise Ratio

SSIM Structural similarity index

UNLM Unbiased Non Local Mean �ltering

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application of noise invalidationdenoising in mri

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