Lecture 5: Imaging Theory (3/6): Plane Waves and the Two-Dimensional Fourier Transform.

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Lecture 5 Imaging Theory (3/6) Plane Waves and
the Two-Dimensional Fourier Transform.

• Review of 1-D Fourier Theory
• Fourier Transform x ? u
• F(u) describes the magnitude and phase of the
  exponentials used to build f(x).

Consider uo, a specific value of u. The
integral sifts out the portion of f(x) that
consists of exp(i2?uox)
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Review 1-D Fourier Theorems / Properties

• If f(x) ? F(u) and h(x) ? H(u) ,
• Performing the Fourier transform twice on a
  function f(x) yields f(-x).
• Linearity af(x) bh(x) ? aF(u) bH(u)
• Scaling f(ax) ?
• Shift f(x-xo) ?

Duality multiplying by a complex exponential in
the space domain results in a shift in the
spatial frequency domain.
Convolution f(x)h(x) ? F(u)H(u)
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Can you explain this movie via the convolution
theorem?
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Example problem

• Find the Fourier transform of

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Example problem Answer.

• Find the Fourier transform of

f(x) ?(x /4) ?(x /2) .5?(x)
Using the Fourier transforms of ? and ? and the
linearity and scaling properties, F(u)
4sinc(4u) - 2sinc2(2u) .5sinc2(u)
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Example problem Alternative Answer.

• Find the Fourier transform of

f(x) ?(x /4) 0.5((?(x /3) ?(x))

1 -.5 0 .5 1
Using the Fourier transforms of ? and ? and the
linearity and scaling and convolution properties
, F(u) 4sinc(4u) 1.5sinc(3u)sinc(u)
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Plane waves

• Lets get an intuitive feel for the plane wave

? The period the distance between successive
maxima of the waves

• defines the direction
• of the undulation.

? Lines of constant phase undulation in the
complex plane
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Plane waves, continued.
y

• Thus, similar triangles exist.
• ABC ADB.
• Taking a ratio,

q
1/u
x
L
1/v
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Plane waves, continued (2).
As u and v increase, L decreases.
y
Frequency of the plane wave
1/u
q
x
(cycles/mm)
L

• Each set of u and v defines a complex plane wave
  with a different L and ?.

1/v
q

• gives the direction of the undulation,
• and can be found by

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Plane waves sine and cosine waves
sin(2px)
cos(2px)
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Plane waves sine waves in the complex plane.
sin(10px)
sin(10px 4piy)
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Two-Dimensional Fourier Transform
Two-Dimensional Fourier Transform

• Where in f(x,y), x and y are real, not complex
  variables.
• Two-Dimensional Inverse Fourier Transform

? ? amplitude basis functions and phase of
required basis functions
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Separable Functions
Two-Dimensional Fourier Transform

• What if f(x,y) were separable? That is,
• f(x,y) f1(x) f2(y)

Breaking up the exponential,
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Separable Functions
Separating the integrals,
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f(x,y) cos(10px)1
Fourier Transform
F(u,v) 1/2 d(u5,0) d(u-5,0)
Imaginary F(u,v)
Real F(u,v)
v
v
v
u
u
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f(x,y) sin(10px)
Fourier Transform
F(u,v) i/2 d(u5,0) - d(u-5,0)
Imaginary F(u,v)
Real F(u,v)
v
v
v
u
u
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f(x,y) sin(40px)
Fourier Transform
F(u,v) i/2 d(u20,0) - d(u-20,0)
Imaginary F(u,v)
Real F(u,v)
v
v
v
u
u
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f(x,y) sin(20px 10py)
Fourier Transform
F(u,v) i/2 d(u10,v5) - d(u-10,v-5)
Imaginary F(u,v)
Real F(u,v)
v
v
v
u
u
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Properties of the 2-D Fourier Transform

• Let f(x,y) ? F(u,v) and g(x,y) ? G(u,v)

Linearity af(x,y) bg(x,y) ? aF(u,v)
bG(u,v) Scaling g(ax,by) ?
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Log display often more helpful
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Properties of the 2-D Fourier Transform

• Let G(x,y) ? G(u,v)

Shift g(x a ,y b) ?
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L(x/16)L(y/16) Real and even
RealF(u,v) 256 sinc2(16u)sinc2(16v)
ImagF(u,v) 0
Phase is 0 since Imaginary channel is 0
and F(u,v) gt 0 always
Log10(F(u,v))
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Shift g(x a ,y b) ?
L((x-1)/16) L(y/16) Shifted one pixel
right
Log10(F(u,v))
Angle(F(u,v))
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L((x-7)/16)L(y/16) Shifted seven pixels
right
Log10(F(u,v))
Angle(F(u,v))
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L((x-7)/16)L((y-2)/16) Shifted seven
pixels right, 2 pixels up
Log10(F(u,v))
Angle(F(u,v))
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Properties of the 2-D Fourier Transform

• Let g(x,y) ? G(u,v) and h(x,y) ? H(u,v)

Convolution
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Image Fourier Space
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Image Fourier Space
(log magnitude)
Detail
Contrast
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10
5
20
50
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2D Fourier Transform problem comb function.

• In one dimension,

y
-2 -1 0 1 2
In two dimensions,
y
x
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2D Fourier Transform problem comb function,
continued.

• Since the function does not describes how comb(y)
  varies in x, we can assume that by definition
  comb(y) does not vary in x.
• We can consider comb(y) as a separable function,
• where g(x,y)gX(x)gY(y)
• Here, gX(x) 1
• Recall, if g(x,y) gX(x)gY(y), then its
  transform is
• gX(x)gY(y) ? GX(u)GY(v)

y
x
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2D Fourier Transform problem comb function,
continued (2).

• gX(x)gY(y) ? GX(u)GY(v)
• So, in two dimensions,

y
x
g(x,y)
G(u,v)
v
u
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2D FTs of Delta Functions Good Things to
Remember

• (bed of nails function)

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Note the 2D transforms of the 1D delta functions
y
v
?(v)
?(x)
u
x
y
v
?(u)
?(y)
u
x
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Example problem Answer.

• Find the Fourier transform of

f(x) ?(x /4) ?(x /2) .5?(x)
Using the Fourier transforms of ? and ? and the
linearity and scaling properties, F(u)
4sinc(4u) - 2sinc(2u) .5sinc(u)