Title: Medical Image Processing: Image Slice Interpolation and Noise Removal in spatial and frequency domains. Shashank Singh Enrollment Number: 20000048. Semester 7 Indian Institute of Information Technology ALLAHABAD. Email: shashank@iiita.ac.inshashank_iiita@yahoo.com

Contents:

1. Certificate 2. Abstract. 3. Introduction. 4. Interpolation 5. Noise Removal 6. Results 7. Further Improvements 8. Acknowledgement 9. Further Improvements 10. References.

Certificate:

This is to certify that Mr. Shashank Singh has successfully developed the ‘Interpolation and Noise Removal’ modules for the Ministry of Human Resource Development funded Medical Image Processing Project. The work, carried out under my guidance, is his original and the results are satisfactory. Dr. Uma Shankar Tiwari. Date: December 8, 2003. Place: Indian Institute of Information Technology

Jhalwa - ALLAHABAD

Abstract:

This project is a part of the project titled ‘Fusion of Multimodal Medical Imagery Data’. The project is funded by MHRD. I have developed the modules for Noise Removal and Interpolation of slices from multiple input image slices. Introduction:

The medical imagery and the subsequent processing of the image data have come as boon to the medical diagnosis and treatment. Different modalities exist for imagery, ranging from MRI, CT, Spiral CT Scans and X-Ray to PET, SPECT, fMRI and Ultrasound Imagery.

The medical imagery data often consists of discrete slices representing the internals of the body. There are applications that demand in-between-slice views and that too in different plane orientations.

Next, the imaging techniques can introduces certain noises into the captured data. It needs to be removed without affecting the true data.

The two modules developed in this project are meant to address the above two issues to some extent.

Image Modalities:

1. Endoscope: visible light 2. Retina imaging: visible coherent light (laser) 3. Microscopes: laser (co focal imaging), in-coherent light

(standard) 4. X-ray: electro magnetic waves 5. NMR: magnetic field 6. PET: radio-nuclear waves 7. US: sound waves

Computed Tomography (CT)

1. 1917: Inverse Radon Transform 2. 1971: Godfrey Hounsfield and Allan Cormack built the

first computer tomography system 3. Dec. 10, 1979: Godfrey Hounsfield and Allan Cormack

received the Nobel Prize in Medicine. 4. 1990: Introduction of Spiral CT by W. A. Kalender, W.

Seissler, E. Klotz, and P.Vock 5. 2002: 16{slice CT and 4D reconstruction of the beating

heart. Magnetic Resonance Imaging (MRI)

1. 1970: Lauterbur and Mansfield discorvered how to use magnetic fields to scan human bodies

2. 1973: discovery of magnetic resonance imaging by Paul C. Lauterbur and Sir Peter Mansfield

3. Dec. 10, 2003: Nobel Prize for Medicine awarded to Paul C. Lauterbur and Peter Mansfield

4. for more details see http://www.nobel.se

Interpolation:

Medical imaging systems collect data typically in a slice-by-slice manner. Usually, the pixel size p of the scene within a slice is different from spacing between adjacent slices. In addition, often the spacing between slices may not be the same for all slices. For visualization, manipulation and analysis of such anisotropic data, they often need to be converted into data of isotropic discretization or of desired level of discrete level of discretization in any of the three (or higher) dimensions. Interpolation techniques can be divided into two categories: scene based and object based. In scene-based methods, interpolated scene intensity values are determined directly from the intensity values of the given scene. In object-based methods, some object information extracted from the given scene is used in guiding the interpolation process. I have interpolation of slices from a given set of images. The slice can be either at a fixed depth or it can consist of patches selected from multiple slice interpolated with increasing depth along x or y axes. Interpolation is implemented in the Matlab files ‘interpolation.m’, which in turn uses two other files ‘funct1.m’ and funct2.m’. On running the ‘interpolation.m’, user is required to enter: - Information about input image files. - choose the type of interpolation. - select the depths for slices.

Noise Removal: Noise removal is implemented in the file ‘final.m’. The other file ‘final.fig’ contains the appropriate information about the GUI. It provides a GUI to read in the images, add noise, apply filters in spatial domain, calculated and visualize FFT , to apply high pass, low pass, band pass filters in frequency domain using the FFT. The user can view the original image, the noisy image, the filtered images, FFT of different images all together in the four windows of the GUI. Difficulties in Medical Image Processing: One has to be careful during processing operations like filtering and noise removal. It may happen that the apparent noise or irregularity is the signal of abnormality in the body. Secondly, Image data of the same region acquired after a time period may differ in orientation and shape due to patient movement and time changes in the region. It makes comparison and processing of the images difficult. Filters:

Median Filter

Common Names: Median filtering, Rank filtering

The median filter is normally used to reduce noise in an image, somewhat like the mean filter. However, it often does a better job than the mean filter of preserving useful detail in the image.

How It Works Like the mean filter, the median filter considers each pixel in the image in turn and looks at its nearby neighbors to decide whether or not it is representative of its surroundings. Instead of simply replacing the pixel value with the mean of neighboring pixel values, it replaces it with the median of those values. The median is calculated by first sorting all the pixel values from the surrounding neighborhood into numerical order and then replacing the pixel being considered with the middle pixel value. (If the neighborhood under consideration contains an even number of pixels, the average of the two middle pixel values is used.) Figure 1 illustrates an example calculation.

Figure 1 Calculating the median value of a pixel neighborhood. As can be seen the central pixel value of 150 is rather unrepresentative of the surrounding pixels and is replaced with the median value: 124. A 3×3 square neighborhood is used here --- larger neighborhoods will produce more severe smoothing.

Mean Filter

Common Names: Mean filtering, Smoothing, Averaging, Box filtering

Brief Description Mean filtering is a simple, intuitive and easy to implement method of smoothing images, i.e. reducing the amount of intensity variation between one pixel and the next. It is often used to reduce noise in images.

How It Works The idea of mean filtering is simply to replace each pixel value in an image with the mean (`average') value of its neighbours, including itself. This has the effect of eliminating pixel values which are unrepresentative of their surroundings. Mean filtering is usually thought of as a convolution filter. Like other convolutions it is based around a kernel, which represents the shape and size of the neighbourhood to be sampled when calculating the mean. Often a 3×3 square kernel is used, as shown in Figure 1, although larger kernels (e.g. 5×5 squares) can be used for more severe smoothing. (Note that a small kernel can be applied more than once in order to produce a similar - but not identical - effect as a single pass with a large kernel.)

Figure 1 3×3 averaging kernel often used in mean filtering

Computing the straightforward convolution of an image with this kernel carries out the mean filtering process.

Fourier Transform

Common Names: Fourier Transform, Spectral Analysis, Frequency Analysis

Brief Description The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency space, while the input image is the real space equivalent. In the Fourier space image, each point represents a particular frequency contained in the real domain image.

The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.

How It Works As we are only concerned with digital images, we will restrict this discussion to the Discrete Fourier Transform (DFT).

The DFT is the sampled Fourier Transform and therefore does not contain all frequencies forming an image, but only a set of samples which is large enough to fully describe the real domain image. The number of frequencies corresponds to the number of pixels in the real domain image, i.e. the image in the real and Fourier space are of the same size.

For a square image of size N×N, the two-dimensional DFT is given by:

where f(i,j) is the image in the real space and the exponential term is the basis function corresponding to each point F(k,l) in the Fourier space. The equation can be interpreted as: the value of each point F(k,l) is obtained by multiplying the real image with the corresponding base function and summing the result.

The basis functions are sine and cosine waves with increasing frequencies, i.e. F(0,0) represents the DC-component of the image which corresponds to the average brightness and F(N-1,N-1) represents the highest frequency.

In a similar way, the Fourier image can be re-transformed to the real domain. The inverse Fourier transform is given by:

To obtain the result for the above equations, a double sum has to be calculated for each image point. However, because the Fourier Transform is separable, it can be written as

where

Using these two formulas, the real domain image is first transformed into an intermediate image using N one-dimensional Fourier Transforms. This intermediate image is then transformed into the final image, again using N one-dimensional Fourier Transforms.

Expressing the two-dimensional Fourier Transform in terms of a series of 2N one-dimensional transforms decreases the number of required computations.

Even with these computational savings, the ordinary one-dimensional DFT has

complexity. This can be reduced to if we employ the Fast Fourier Transform (FFT) to compute the one dimension DFTs. This is a significant improvement, in particular for large images. There are various forms of the FFT and most of them restrict

the size of the input image that may be transformed, often to . The mathematical details are well described in the literature.

The Fourier Transform produces a complex number valued output image which can be displayed with two real images, either with the real and imaginary part or with magnitude and phase. In image processing, often only the magnitude of the Fourier Transform is displayed - as it contains most of the information of the geometric structure of the real space image. However, if we want to re-transform the Fourier image into the correct real space after some processing in the frequency domain, we must make sure to preserve both magnitude and phase of the Fourier image.

The Fourier domain image has a much greater range than the image in the real space. Hence, to be sufficiently accurate, its values are usually calculated and stored in float values.

Results: Interpolation:

slice 5 slice 5.6 (Interpolated) slice 6

Interpolation for varying depth along x axis: (a) for N = 10 (b) for N = 5

Interpolation for varying depth along y axis: (a) for N = 10 (b) for N = 5

Filtering GUI:

Further Improvements:

1. Methods can be developed to generate 3D surface plot of the image data.

2. The other feature can be the generation of image slice along any plane cutting through the 3D model.

3. Registration of different modality images followed by useful operations like interpolation, change detection, segmentation etc. can be done.

Acknowledgement: I express my acknowledgement to Dr. Uma Shankar Tiwari, under whose guidance I carried out the project development. I am thankful for his support and the valuable ideas he shared with me.

References:

Books: 3D Imaging in Medicine Edited by: Jayaram K. Udupa, Ph.D.

University of Pennsylvania Philadelphia, PA

Gabor T. Herman, Ph.D. University of Pennsylvania Philadelphia, PA

Digital Image Processing (2nd Edition) Authors: Rafael C. GonzalezRichard E. Woods

Web Resources:

GE Medical Systems http://www.gemedicalsystems.com Image Processing Fundamentals http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip.html Brain Imaging http://faculty.washington.edu/chudler/image.html

The Whole Brain Atlas http://www.med.harvard.edu/AANLIB/home.html

Papers: Survey: Interpolation Methods in Medical Image Processing

Thomas M. Lehmann,* Member, IEEE, Claudia G¨onner, and Klaus Spitzer