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P1 Introductory science and methods > Fourier Transform applied to Imaging > Flashcards

Fourier Transform applied to Imaging Flashcards

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1
Q

What are the wide range of applications for the Fourier Transform?

A
  1. Image reconstruction
  2. Image compression
  3. Image analysis
  4. Image filtering
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2
Q

How can 1-D signal be represented?

A

Sum of series of Fourier components

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3
Q

What do each Fourier component represent?

A

Sinusoidal oscillations

  1. At a specific frequency
  2. with a specific amplitude
  3. and a specific phase
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4
Q

What are examples of 1D signal?

A
  1. Audio
  2. Temperature data (climate/medical)
  3. Blood pressure
  4. Voltage (EEG or ECG)
  5. Intensity (ultrasound)
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5
Q

What are images a representation of?

A

How things vary with position in space

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6
Q

What is an image?

A

Description of how a quantity varies with spatial localisation

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7
Q

What is a line profile?

A

1 dimensional image

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8
Q

How can function of position be broken down into?

A

A series of harmonic component

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9
Q

What is spatial frequency?

A

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

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10
Q

What is the spatial frequency v?

A

The number of whole cycles repeated per unit distance

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11
Q

What is the spatial frequency related to?

A

wavelength

v=1/wavelength

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12
Q

What do spatial frequency describe?

A

Sinusoidal variations

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13
Q

What is observed in MRI theory?

A

Wave number

K-space

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14
Q

What is wave number k related to?

A
  1. Spatial frequency

2. Wavelength

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15
Q

Wave number K

A

the number of complete cycles in radians per unit distance

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16
Q

How many radians in every cycle?

A

2 pie

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17
Q

What do any sinusoidal functions have?

A
  1. frequency
  2. phase off-set
  3. Amplitude
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18
Q

What do function of space have?

A

High resolution frequency

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19
Q

High spatial frequency

A

Large K

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20
Q

Lower spatial frequency

A

Smaller K

less repetition per unit distance

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21
Q

What is the equation of K (spatial frequency)?

A

2pie x no. of cycles per unit distance

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22
Q

How can any function or signal be expressed as?

A

Sum of series of sinusoids

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23
Q

What are the sinusoids in 2D?

A

Sinusoidal variations in brightness across image

24
Q

What happens in 2D?

A

Specify the direction

e.g. left-right

25
Q

In 2D, what is the spatial frequency K?

A

The frequency with which the brightness modulates with position

26
Q

What does magnitude of sinusoid correspond to?

A

Contrast

Difference between lowest darkest value and lightest brightest value

27
Q

What do the spatial frequency of sinusoid describe?

A

How many periods (cycles) there are per unit distance

28
Q

What do the orientation of sinusoid represent?

A

Direction of wave

29
Q

What do phase of sinusoid represent?

A

How the wave is shifted relative to the origin e.g. how much sinusoid is shifted left or right

30
Q

What do 2D sinusoids have?

A

Values that are fixed in one direction, and vary sinusoidally in other direction characterised by:

  1. Amplitude (height)
  2. Period
  3. Phase shift
  4. Orientation
31
Q

What is the equation for period?

A

2pie/spatial frequency

32
Q

What coordinate system is used in Imaging?

A

Rectangular coordinate system (x + y)

33
Q

How can a 2D spatial sinusoid be written mathematically?

A

F(x,y) = Asin (KxX + KyY)-phase shift)

34
Q

What is the equation for orientation?

A

Inverse of tan (ky/kx)

35
Q

What is the equation for resultant spatial frequency?

A

K= square root (kX^2+Ky^2)

36
Q

Where is the phase off-set stored in?

A

Complex number

37
Q

What does the array element specify?

A

Phase and amplitude of spatial sinusoid component with respective Kx+Ky values

38
Q

What do each sinusoid component have?

A

Direction + Spatial frequency dependent on position in array

39
Q

What is the implication of applying inverse 2D-FT?

A

Perfectly recover original image

40
Q

What does every pixel have?

A 
Same value 
Completely uniform 
Calculate 2D FT 
No spatial coordination image, 0 frequency 
Get signal right in the origin at 0
41
Q

Zero frequency level

A

Average value of function

42
Q

When does Fourier Transform rotate?

A

If sinusoids are up and down

43
Q

What is by far the largest component of the Image?

A

Zero frequency component

44
Q

How to squeeze the grey-scale?

A

Take logarithm of intensity of FT

45
Q

What does 2D-FT inform us?

A

The various combinations of 2D spectral sinusoids that are required to create image

46
Q

What are required to get original image?

A

Amplitude and phase

47
Q

Where are most of the image information located?

A

Low spatial frequency

There is less in the high spatial frequency

48
Q

What is the role of Fourier filtering?

A

Remove high + low spatial frequencies before inverse transformation

49
Q

What is the consequence of low spatial frequencies?

A

Lost contrast information

Only see edges

50
Q

Region with low spatial frequency content

A

Smooth regions

51
Q

Region with high spatial frequency content

A

Edges, texture

52
Q

What does the low spatial frequencies encode?

A

The slowly varying image properties - the contrast

53
Q

What does the high spatial frequencies encode?

A

More rapidly varying image properties - detail, edges

54
Q

What has more spread-out transforms?

A

Smaller objects

55
Q

What has more compressed transform?

A

Larger objects

56
Q

If one rotates the image

A

The transform rotates the same amount

P1 Introductory science and methods (14 decks)

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