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**Review of 1-D Fourier Theory:**

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(+i·2·uo·x)

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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) = Π(x /4) – Λ(x /2) + .5Λ(x) Using the Fourier transforms of Π and Λ 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) = Π(x /4) – 0.5((Π(x /3) * Π(x)) * – – Using the Fourier transforms of Π and Λ and the linearity and scaling and convolution properties , F(u) = 4sinc(4u) – 1.5sinc(3u)sinc(u)

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** The period; the distance between successive maxima of the waves**

Plane waves Let’s 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 x L Thus, similar triangles exist.**

ABC ~ ADB. Taking a ratio, A B C D q 1/u x q c = 90 - q q 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(2*p*x) cos(2*p*x)

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**sin(10*p*x) sin(10*p*x +4*pi*y)**

Plane waves: sine waves in the complex plane. sin(10*p*x) sin(10*p*x +4*pi*y)

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

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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**Separating the integrals,**

Separable Functions Separating the integrals,

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**F(u,v) = 1/2 [d(u+5,0) + d(u-5,0)]**

f(x,y) = cos(10px)*1 Fourier Transform F(u,v) = 1/2 [d(u+5,0) + d(u-5,0)] Real [F(u,v)] Imaginary [F(u,v)] v v v u u

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**F(u,v) = i/2 [d(u+5,0) - d(u-5,0)]**

f(x,y) = sin(10px) Fourier Transform F(u,v) = i/2 [d(u+5,0) - d(u-5,0)] Real [F(u,v)] Imaginary [F(u,v)] v v v u u

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**F(u,v) = i/2 [d(u+20,0) - d(u-20,0)]**

f(x,y) = sin(40px) Fourier Transform F(u,v) = i/2 [d(u+20,0) - d(u-20,0)] Real [F(u,v)] Imaginary [F(u,v)] v v v u u

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**F(u,v) = i/2 [d(u+10,v+5) - d(u-10,v-5)]**

f(x,y) = sin(20px + 10py) Fourier Transform F(u,v) = i/2 [d(u+10,v+5) - d(u-10,v-5)] Real [F(u,v)] Imaginary [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: a·f(x,y) + b·g(x,y) ↔ a·F(u,v) + b·G(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 Log10(|F(u,v)|)**

Real{F(u,v)}= 256 sinc2(16u)sinc2(16v) Imag{F(u,v)}= 0 Phase is 0 since Imaginary channel is 0 and F(u,v) > = 0 always Log10(|F(u,v)|)

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**Shifted one pixel right**

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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**Shifted seven pixels right**

L((x-7)/16)L(y/16) Shifted seven pixels right Log10(|F(u,v)|) Angle(F(u,v))

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**Shifted seven pixels right, 2 pixels up**

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 u v

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Image Fourier Space (log magnitude) u v Detail Contrast

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5 % 10 % 20 % 50 %

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**2D Fourier Transform problem: comb function.**

In one dimension, … … y 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 FT’s of Delta Functions: Good Things to Remember**

(“bed of nails” function) Terminology ok?

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**Note the 2D transforms of the 1D delta functions:**

y v (v) (x) x u y v (y) (u) u x

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**Example problem: Answer.**

Find the Fourier transform of f(x) = Π(x /4) – Λ(x /2) + .5Λ(x) Using the Fourier transforms of Π and Λ and the linearity and scaling properties, F(u) = 4sinc(4u) - 2sinc(2u) + .5sinc(u)

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