Medical Imaging Instrumentation & Image Analysis

1. 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, 07102 Dhawan@adm.njit.edu

2. 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.

3. 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.

5. A Multidisciplinary Paradigm

6. Electromagnetic Radiation Spectrum

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

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

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

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

11. 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

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

13. ROC Curve FPF=1-TNF TPF a b c

14. 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

15. Example True Positive Fraction (TPF) = 8/10 = 0.8 False Negative Fraction (FNF) = 2/10 = 0.2 False Positive Fraction (FPF) = 5/90 = 0.0556 True Negative Fraction (TNF) = 85/90 = 0.9444

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

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

18. y b Radiating Object f(a,b,g) Image g(x,y,z) Image Formation System h g z Image Domain Object Domain x a Image Formation

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

20. b y Radiating Object Image Image Formation System h Selected Cross-Section g z Image Domain Object Domain x a Radiation Source Reconstructed Cross-Sectional Image Image Formation: External Source

21. b y Image Radiating Object Image Formation System h Selected Cross-Section g z Image Domain Object Domain x a Reconstructed Cross-Sectional Image Image Formation: Internal Source

22. Fourier Transform

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

24. FT Properties Translation Convolution Cross-Correlation

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

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

27. Projection p1 A Reconstruction Space B Projection p3 Projection p2 Radon Transform

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

29. 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.

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

31. 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.

32. (a) wy wymax Fa(wx, wy) wx wxmax -wxmax (c) -wymax (b) Nyquist (Optimal) Sampling

33. 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.

34. 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.

35. 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.

36. 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 y(t) and zero mean: • This function now has changing time-frequency tiles because of scaling. • a<1: y(a,b) will be short and of high frequency • a>1: y(a,b) will be long and of low frequency

37. Wavelet Decomposition

38. 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

39. Wavelet Transform with Filters • The mother wavelet can be constructed using a scaling function f(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.

40. H H 2 H 2 Data G 2 G 2 G 2 Decomposition

41. Wavelet Decomposition Space

42. s u b - s a m p l e h - h h s u b - s a m p l e h - g h g X h g - h g g - g g h o r i z o n t a l l y v e r t i c a l l y L e v e l 0 L e v e l 1 Image Decomposition Image

43. Wavelet and Scaling Functions

44. Image Processing and Enhancement