Medical Image Analysis Image Formation Figures come from

Medical Image Analysis Image Formation Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Color representation ◦ RGB: red, green, blue ◦ ISH: intensity, saturation, hue Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

From www. wikipedia. org Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Image Coordinate System Object: Image: Translation Rotation Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Image Coordinate System y b Radiating Object f(a, b, g) Image g(x, y, z) Image-Formation System h g z Image Domain Object Domain a x Figure 2. 1. The object and image domain coordinates systems. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

2 -D image rotation ◦ Original ( , ) ◦ ( , ) coordinate system rotated by Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Linear Systems and Impulse Response Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Point Source and Impulse Functions Sifting property From www. wikipedia. org

Probability and Random Variable Functions Gaussian density function Independent random variables

Orthogonal Jacobian random variables transformation

Image Formation Linear spatially invariant (LSI) system Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

b y Radiating Object Image-Formation System h Selected Cross-Section g Object Domain Image Domain z x a Radiation Source Reconstructed Cross-Sectional Image Figure 2. 2. A schematic block diagram of a general 3 -D image formation system requiring an external radiation source (such as light) for photographic imaging or x-ray for transmission imaging. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

b y Radiating Object Image-Formation System h Selected Cross-Section g Object Domain Image Domain z x a Reconstructed Cross-Sectional Image Figure 2. 3. A schematic block diagram of an emission based image formation system commonly used for fluorescence or nuclear medicine imaging modalities. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

PSF and spatial resolution ◦ Full width at half maximum (FWHM) ◦ 1 -D Gaussian distribution, FWHM= From www. wikipedia. org

Line spread function (LSF) Modulation transfer function (MTF) Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

From www. wikipedia. org

Signal-to-noise ratio Contrast-to-noise ratio Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Pin-hole Imaging y x z f(x 1, y 1) Pin-hole g(x 2, y 2) z 1 Object Plane z 2 Focal Plane Image Plane Figure 2. 4. A schematic diagram of a pin-hole imaging camera. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Fourier Transform Linearity Scaling Translation Convolution Cross-correlation Auto-correlation Parseval’s theorem Separability Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Figure 2. 5. (a) A vertical stripe image generated from a sinusoidal waveform of a period of 8 pixels and (b) the logarithmic magnitude image of its Fourier transform Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Figure 2. 6. (a) A rotated stripe image and (b) the logarithmic magnitude image of its Fourier transform. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Figure 2. 7. (a) An image with a square region at the center and (b) the logarithmic magnitude image of its Fourier transform. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Radon Transform y q p f(x, y) q x p P(p, q) q Figure 2. 8. Line integral projection P(p, q) of the two-dimensional Radon transform. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Projection p 1 A Reconstruction Space B Projection p 3 Projection p 2 Figure 2. 9. A schematic diagram for reconstructing images from projections. Three projections are back-projected to reconstruct objects A and B. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Sampling Nyquist rate Figure 2. 10. (a) A 2 -D distribution of Gaussian impulses in the spatial domain and (b) its representation in the Fourier domain. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

wy wymax Fa(wx, wy) wxmax -wxmax wx -wymax (a) Figure 2. 11. (a) The Fouier spectrum of a bandlimited two-dimensional signal; (b) Sampling of the bandlimited signal with a sampling frequency higher than the Nyquist rate; (c). Undersampling of the bandlimited signal with a sampling frequency lower than the Nyquist rate. Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

(b) Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

(c) Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Figure 2. 12. (a) A vertical stripe image generated from a sinusoidal waveform of a period of 1. 8 pixels, (b) the logarithmic magnitude image of its Fourier transform, (c) the central cross section of the logarithmic magnitude of the Fourier transform, along the horizontal direction, of the signal shown in Figure 2. 5 (a), which is sampled satisfying the Nyquist criterion, and (d) the central cross section along the horizontal direction of the logarithmic magnitude of the Fourier transform of the aliasing pattern shown in Figure 2. 12 (a). Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Over-sampled Log |F(wx, 0)| wx Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Aliasing Log |F(wx, 0)| wx Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.

Discrete Fourier Transform Figures come from the textbook: Medical Image Analysis, Second Edition, by Atam P. Dhawan, IEEE Press, 2011.