medical image analysis image formation figures come from the textbook: medical image analysis, by...

Medical Image AnalysisMedical Image AnalysisImage Formation

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Color representation◦RGB: red, green, blue◦ISH: intensity, saturation, hue

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

From www.wikipedia.org

Image Coordinate SystemImage Coordinate SystemObject: Image:TranslationRotation

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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Image Coordinate SystemImage Coordinate System

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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Image-Formation System h

Object Domain

Image Domain

x

y

Radiating Object f() Image g(x,y,z)

Figure 2.1. The object and image domain coordinates systems.

Linear Systems and Impulse Linear Systems and Impulse ResponseResponse

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Reconstructed Cross-Sectional Image

Radiation Source

z

Image-Formation Systemh

Object Domain

Image Domain

Selected Cross-Section

x

yRadiating Object

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, by Atam P. Dhawan, IEEE Press, 2003.

z

Reconstructed Cross-Sectional Image

Image-Formation Systemh

Object Domain

Image Domain

Selected Cross-Section

x

y

Radiating Object

Image

Figure 2.3. A schematic block diagram of an emission based image formation system commonly used for fluorescence or nuclear medicine imaging modalities.

Pin-hole ImagingPin-hole Imaging

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Pin-hole

Object Plane Focal Plane Image Plane

f(x1,y1)

g(x2,y2)z1z2

x

z

y

Figure 2.4. A schematic diagram of a pin-hole imaging camera.

Fourier TransformFourier TransformLinearityScalingTranslationConvolutionCross-correlationAuto-correlationParseval’s theoremSeparability

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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, by Atam P. Dhawan, IEEE Press, 2003.

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, by Atam P. Dhawan, IEEE Press, 2003.

Figure 2.7. (a) An image with a square region at the center and (b) the logarithmic magnitude image of its Fourier transform.

Radon TransformRadon Transform

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

x

y

q

p

p

f(x,y)

P(p,)

Figure 2.8. Line integral projection P(p,q) of the two-dimensional Radon transform.

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

Projection p1

Projection p2

Projection p3

Reconstruction Space

A

B

Figure 2.9. A schematic diagram for reconstructing images from projections. Three projections are back-projected to reconstruct objects A and B.

SamplingSamplingNyquist rate

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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, by Atam P. Dhawan, IEEE Press, 2003.

x

y

ymax

xmaxxmax

ymax

Fa(x,y)

(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, by Atam P. Dhawan, IEEE Press, 2003.

(b)

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

(c)

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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).

Over-sampled

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

x

Log |F(x, 0)|

Aliasing

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

x

Log |F(x, 0)|

Discrete Fourier Discrete Fourier TransformTransform

Figures come from the textbook: Medical Image Analysis, by Atam P. Dhawan, IEEE Press, 2003.

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