AbstractIntroductionIntroduction to Positron Emission TomographyNoise Processes Associated With PETImage DenoisingWavelet Transform – Continuous and DiscreteContinuous Wavelet TransformDiscrete Wavelet TransformTheories of Wavelet Based Noise RemovalMaterials & MethodsImaging ProtocolPhantomMouseImage Processing – Wavelet Filter ImplementationResultsAnimalPhantomDiscussionConclusionReferencesThe Use of Wavelet Filters to De-noise µPETDataJoe GrudzinskiDepartment of Biomedical EngineeringDecember 15, 200623AbstractThe reconstruction scheme of filter back projection (FBP) yields very noisy reconstructed PET data because PET data is Poisson and not Gaussian distributed. This noisy data often makes the diagnostic task very difficult and overestimates tracer uptake values. To compensate for this amplified noise, it has recently been proposed to post-process PET data using wavelet filters in order to “denoise” or restore the PET data. To assess how the wavelet filters perform, the SNR and resolution of pre-processing is compared quantitatively to that of post processing. Furthermore, a qualitative assessmentof pre-processing and post-processing is made by a licensed radiologist. IntroductionIntroduction to Positron Emission TomographyThe use of Positron Emission Tomography (PET) in the clinic has drastically increased over the past couple decades. Because of its drastic improvements in hardware,the resolution and sensitivity has increased thus resulting in revolutionary uses in neurology and oncology. Its use has even crossed disciplines into pre-clinical research with the invention of the µPET system. Though its improvements in resolution and sensitivity have broadened its range of applications, the issues associated with acquiring and displaying PET data are still the same. In order to understand the acquisition of PET data, it is important to first understand the source of the signal. When fluorodeoxyglucose (18F-FDG), a radiolabelled glucose analog, is injected into the body of a patient it is selectively taken4up by cells that are rapidly metabolizing glucose such as tumors. The 18F that is attached decays into positrons that travel a short distance (positron range) before colliding with an electron and annihilating into two 511 KeV photons. These photons are approximately 180 degrees apart from each other and are detected by two coincident detectors within thePET detector ring. This event created what is known as a line of response. Throughout the scan, each event produces a line of response that is binned with respect to the coincident detectors that detected the event. At the end of a scan, there are often millions of binned lines of response. All of these lines of response map out the potential location of each event’s location. To produce an image from these data one must reconstruct theselines of response. Noise Processes Associated With PETThe positron emission process itself follows a Poisson distribution and possesses a variance equal to the number of counts. Due to the numerous corrections applied (ie. randoms, attenuation) and image reconstructions, the noise properties get altered. Each source of noise introduces its own noise distribution into the resulting PET data which makes removing noise difficult. If one better understands the sources of noise in the reconstruction, it may be easier to remove noise in a post-processing manner.The reconstruction method of filter back projection (FBP) has been the workhorseof nuclear medicine. Theoretically, if FBP is used on noiseless data then it should be ableto perfectly reconstruct the lines of response into an image that represents the true distribution of radioactivity. In reality though, FBP has the tendency to spread noise variance from high to low densities thus making the image have more of a uniform noise5distribution. To better assess how FBP contributes to noise, it is imperative to understandthe general process such as the implementation of filters.Starting with all of the data binned into a sinogram, it is possible to now acquire projection profiles at N projection angles as to obey the Nyquist criterion. Then, the one dimensional Fourier transform is taken of each profile. In Fourier space, each projection profile is then multiplied by the filter of choice to acquire a filtered Fourier transform of each profile. The inverse Fourier transform is then taken of each filtered projection and the image is finally made when the each filtered profile is then back projected onto imagespace.The filter type has proven to be the most crucial step reconstructing a noise free image. It is hypothesized that the Ramp filter is the ideal FBP filter in suppressing noise because it enhances high spatial frequencies which contain local details such as edges andsuppresses low frequencies which contain global details such as contrast. Two problems associated with this particular filter are that artifacts sometimes arise from the sharp cut-off at the Nyquist frequency and amplification of high spatial frequency information often leads to high frequency noise amplification. Due to the shortcomings of the Ramp filter, there has been development of other filters as well. The Shepp and Hann filters conversely roll-off at higher frequencies as to not amplify high frequencies like the Rampfilter does. In comparison, the Hann filter rounds off at the higher frequencies faster thandoes the Shepp filter. The results of these filters used in FBP will be discussed later in the paper.6Because of its speed and ease of implementation, FBP will not lose use to iterativemethods of reconstruction. But to improve image quality and fidelity, ways of removing noise are required. Figure 1: Frequency Response of Different FiltersImage DenoisingWithin nuclear medicine very high frequencies are dominated by noise and not signal. The common practice here is to apply a low pass filter in conjunction with a Ramp filter to improve SNR. However, the SNR improvement due to the low pass filter comes at the expense of degraded image resolution because high resolution features will also be smoothed (Sahiner and Yagle, 3579). Ideally, it would be desirable to reduce the noise in the reconstructed image over regions where high resolution features are not present by using spatially-varying filtering. Fourier techniques are only effective at denoising signals that are globally periodic and stationary due to the nature of
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