Fourier Transform applied to Imaging Flashcards
What are the wide range of applications for the Fourier Transform?
- Image reconstruction
- Image compression
- Image analysis
- Image filtering
How can 1-D signal be represented?
Sum of series of Fourier components
What do each Fourier component represent?
Sinusoidal oscillations
- At a specific frequency
- with a specific amplitude
- and a specific phase
What are examples of 1D signal?
- Audio
- Temperature data (climate/medical)
- Blood pressure
- Voltage (EEG or ECG)
- Intensity (ultrasound)
What are images a representation of?
How things vary with position in space
What is an image?
Description of how a quantity varies with spatial localisation
What is a line profile?
1 dimensional image
How can function of position be broken down into?
A series of harmonic component
What is spatial frequency?
The number of times a particular signal repeats over a particular distance
A measure of how often sinusoidal components of feature repeat per unit of distance
What is the spatial frequency v?
The number of whole cycles repeated per unit distance
What is the spatial frequency related to?
wavelength
v=1/wavelength
What do spatial frequency describe?
Sinusoidal variations
What is observed in MRI theory?
Wave number
K-space
What is wave number k related to?
- Spatial frequency
2. Wavelength
Wave number K
the number of complete cycles in radians per unit distance
How many radians in every cycle?
2 pie
What do any sinusoidal functions have?
- frequency
- phase off-set
- Amplitude
What do function of space have?
High resolution frequency
High spatial frequency
Large K
Lower spatial frequency
Smaller K
less repetition per unit distance
What is the equation of K (spatial frequency)?
2pie x no. of cycles per unit distance
How can any function or signal be expressed as?
Sum of series of sinusoids
What are the sinusoids in 2D?
Sinusoidal variations in brightness across image
What happens in 2D?
Specify the direction
e.g. left-right
In 2D, what is the spatial frequency K?
The frequency with which the brightness modulates with position
What does magnitude of sinusoid correspond to?
Contrast
Difference between lowest darkest value and lightest brightest value
What do the spatial frequency of sinusoid describe?
How many periods (cycles) there are per unit distance
What do the orientation of sinusoid represent?
Direction of wave
What do phase of sinusoid represent?
How the wave is shifted relative to the origin e.g. how much sinusoid is shifted left or right
What do 2D sinusoids have?
Values that are fixed in one direction, and vary sinusoidally in other direction characterised by:
- Amplitude (height)
- Period
- Phase shift
- Orientation
What is the equation for period?
2pie/spatial frequency
What coordinate system is used in Imaging?
Rectangular coordinate system (x + y)
How can a 2D spatial sinusoid be written mathematically?
F(x,y) = Asin (KxX + KyY)-phase shift)
What is the equation for orientation?
Inverse of tan (ky/kx)
What is the equation for resultant spatial frequency?
K= square root (kX^2+Ky^2)
Where is the phase off-set stored in?
Complex number
What does the array element specify?
Phase and amplitude of spatial sinusoid component with respective Kx+Ky values
What do each sinusoid component have?
Direction + Spatial frequency dependent on position in array
What is the implication of applying inverse 2D-FT?
Perfectly recover original image
What does every pixel have?
Same value Completely uniform Calculate 2D FT No spatial coordination image, 0 frequency Get signal right in the origin at 0
Zero frequency level
Average value of function
When does Fourier Transform rotate?
If sinusoids are up and down
What is by far the largest component of the Image?
Zero frequency component
How to squeeze the grey-scale?
Take logarithm of intensity of FT
What does 2D-FT inform us?
The various combinations of 2D spectral sinusoids that are required to create image
What are required to get original image?
Amplitude and phase
Where are most of the image information located?
Low spatial frequency
There is less in the high spatial frequency
What is the role of Fourier filtering?
Remove high + low spatial frequencies before inverse transformation
What is the consequence of low spatial frequencies?
Lost contrast information
Only see edges
Region with low spatial frequency content
Smooth regions
Region with high spatial frequency content
Edges, texture
What does the low spatial frequencies encode?
The slowly varying image properties - the contrast
What does the high spatial frequencies encode?
More rapidly varying image properties - detail, edges
What has more spread-out transforms?
Smaller objects
What has more compressed transform?
Larger objects
If one rotates the image
The transform rotates the same amount
