Medical Imaging Instrumentation & Image Analysis Atam P. Dhawan, Ph.D. Dept. of Electrical & Computer Engineering Dept. of Biomedical Engineering New Jersey Institute of Technology Newark, NJ,

Imaging in Medical Sciences Imaging is an essential aspect of medical sciences for visualization of anatomical structures and functional or metabolic information of the human body. Structural and functional imaging of human body is important for understanding the human body anatomy, physiological processes, function of organs, and behavior of whole or a part of organ under the influence of abnormal physiological conditions or a disease.

Medical Imaging Radiological sciences in the last two decades have witnessed a revolutionary progress in medical imaging and computerized medical image processing. Advances in multi-dimensional medical imaging modalities X-ray Mammography X-ray Computed Tomography (CT) Single Photon Computed Tomography (SPECT) Positron Emission Tomography (PET) Ultrasound Magnetic Resonance Imaging (MRI) functional Magnetic Resonance Imaging (fMRI) Important radiological tools in diagnosis and treatment evaluation and intervention of critical diseases for significant improvement in health care.

Medical Imaging and Image Analysis The development of imaging instrumentation has inspired the evolution of new computerized image reconstruction, processing and analysis methods for better understanding and interpretation of medical images. The image processing and analysis methods have been used to help physicians to make important medical decision through physician-computer interaction. Recently, intelligent or model-based quantitative image analysis approaches have been explored for computer-aided diagnosis to improve the sensitivity and specificity of radiological tests involving medical images.

A Multidisciplinary Field Medical imaging in diagnostic radiology has evolved as a result of the significant contributions of a number of different disciplines from basic sciences, engineering, and medicine. Computerized image reconstruction, processing and analysis methods have been developed for medical imaging applications The application-domain knowledge has been used in developing models for accurate analysis and interpretation.

A Multidisciplinary Paradigm

Electromagnetic Radiation Spectrum

Medical Imaging Information Anatomical X-Ray Radiography X-Ray CT MRI Ultrasound Optical Functional/Metabolic SPECT PET fMRI, pMRI Ultrasound Optical Fluorescence Electrical Impedance

Medical Imaging Modalities Medical Imaging Methods Internal Combination: Internal & External External Using Energy Source SPECT PET MRI Fluorescence EI X-Ray Ultrasound Optical

Medical Imaging Thru Transmission Basic Principle: Radiation is attenuated when passed through the body.

ROC: Performance Measure Ntp = Notp + Nofn and Ntn = Nofp + Notn

Fractions True Positive Fraction (TPF): ratio of the number of positive observations to the number of positive true-condition cases. TPF = Notp/Ntp False Negative Fraction (FNF): ratio of the number of negative observations to the number of positive true-condition cases. FNF = Nofn/Ntp \ False Positive Fraction (FPF): ratio of the number of positive observations to the number of negative true-condition cases. FPF = Nofp/Ntn True Negative Fraction (TNF): ratio of the number of negative observations to the number of negative true-condition cases. TNF = Notn/Ntn TPF + FNF = 1 and TNF + FPF = 1

Measures Sensitivity is TPF Specificity is TNF Accuracy = (TPF+TNF)/Ntotal

ROC Curve FPF=1-TNF TPF b a c

A Case Study Total number of patients = Ntot=100 Total number of patients with biopsy proven cancer (true condition of object present) = Ntp=10 Total number of patients with biopsy proven normal tissue (true condition of object NOT present) = Ntn=90 Out of the patients with cancer Ntp, the number of patients diagnosed by the physician as having cancer = Number of True Positive cases = Notp=8 Out of the patients with cancer Ntp, the number of patients diagnosed by the physician as normal = Number of False Negative cases = Nofn=2 Out of the normal patients Ntn, the number of patients rated by the physician as normal = Number of True Negative cases = Notn=85 Out of the normal patients Ntn, the number of patients rated by the physician as having cancer = Number of False Positive cases = Nofp=5

Example True Positive Fraction (TPF) = 8/10 = 0.8 False Negative Fraction (FNF) = 2/10 = 0.2 False Positive Fraction (FPF) = 5/90 = True Negative Fraction (TNF) = 85/90 =

Linear System A system is said to be linear if it follows two properties: scaling and superposition.

Image Formation: Object f, Image g Non-negativity Superposition Linear Response Function Image Formation Linear Image Formation

Image Formation  z Image Formation System h Object Domain Image Domain   x y Radiating Object f(  ) Image g(x,y,z)

Simple Case: LSI Image Formation g=h**f

Image Formation: External Source Reconstructed Cross-Sectional Image Radiation Source  z Image Formation System h Object Domain Image Domain Selected Cross-Section  x y Radiating Object  Image

Image Formation: Internal Source   z Reconstructed Cross-Sectional Image Image Formation System h Object Domain Image Domain Selected Cross-Section  x y Radiating Object Image

Fourier Transform

Properties of FT Scaling: It provides a proportional scaling. FT {ag(x,y)+bh(x,y)}= aFT{g(x,y)+ bFT{h(x,y) Linearity: Fourier transform, FT, is a linear transform.

FT Properties Translation Convolution Cross-Correlation

FT Properties…. Auto-Correlation Parseval’s Theorem Separability

Radon Transform x y q  p  p f(x,y) P(p,  ) Line integral projection P(p,  ) of the two-dimensional Radon transform.

Radon Transform Projection p 1 Projection p 2 Projection p 3 Reconstruction Space A B

Backprojection Reconstruction Method where L is the total number of projections acquired during the imaging process at viewing angles

Sampling Theorem The sampling theorem provides the mathematical foundation of Nyquist criterion to determine the optimal sampling rate for discretization of an analog signal without the loss of any frequency information. The Nyquist criterion states that to avoid any loss of information or aliasing artifact, an analog signal must be sampled with a sampling frequency that is at least twice the maximum frequency present in the original signal.

Sampling The sampled version of the image, fd[x,y] is obtained from sampling the analog version as

Sampling Effect In Fourier domain the spectrum overlapping has to be avoided by proper sampling of the image in spatial domain. Sampling in spatial domain produces a convolution in the frequency domain.

Nyquist (Optimal) Sampling xx yy  ymax  xmax  xmax  ymax F a (  x,  y ) (a) (b ) (c )

Wavelet Transform Fourier Transform only provides frequency information. Windowed Fourier Transform can provide time-frequency localization limited by the window size. Wavelet Transform is a method for complete time-frequency localization for signal analysis and characterization.

Wavelet Transform.. Wavelet Transform : works like a microscope focusing on finer time resolution as the scale becomes small to see how the impulse gets better localized at higher frequency permitting a local characterization Provides Orthonormal bases while STFT does not. Provides a multi-resolution signal analysis approach.

Wavelet Transform… Using scales and shifts of a prototype wavelet, a linear expansion of a signal is obtained. Lower frequencies, where the bandwidth is narrow (corresponding to a longer basis function) are sampled with a large time step. Higher frequencies corresponding to a short basis function are sampled with a smaller time step.

Continuous Wavelet Transform Shifting and scaling of a prototype wavelet function can provide both time and frequency localization. Let us define a real bandpass filter with impulse response  (t) and zero mean: This function now has changing time-frequency tiles because of scaling. a<1:  (a,b) will be short and of high frequency a>1:  (a,b) will be long and of low frequency

Wavelet Decomposition

Wavelet Coefficients Using orthonormal property of the basis functions, wavelet coefficients of a signal f(x) can be computed as The signal can be reconstructed from the coefficients as

Wavelet Transform with Filters The mother wavelet can be constructed using a scaling function  (x) which satisfies the two-scale equation Coefficients h(k) have to meet several conditions for the set of basis functions to be unique, orthonormal and have a certain degree of regularity. For filtering operations, h(k) and g(k) coefficients can be used as the impulse responses correspond to the low and high pass operations.

Decomposition H H G H G G Data

Wavelet Decomposition Space

Image Decomposition h g sub-sample Level 0Level 1 h-h h-g g-h g-g horizontallyvertically sub-sample g g h h X Image

Wavelet and Scaling Functions

Image Processing and Enhancement