Topic 7 - Fourier Transforms

DIGITAL IMAGE PROCESSING

Course 3624

Department of Physics and Astronomy

Professor Bob Warwick

7.1 Review of The Fourier Transform

Outline of this Topic

7.1 Introduction to the Fourier Transform (FT)

7.2 The Discrete Fourier Transform (DFT)

7.3 The Properties of the DFT

7.4 Computation of the DFT (via the FFT)

7.1 Introduction to the FT

In 1-d, assuming continuous variables: Fourier Transform Pair

may often be real, whereas F(u)

is generally complex, ie

• Many DIP techniques rely on the application of an image transform of which the Fourier Transform is the most popular.

• The Fourier domain provides very important insight into the information content of the data.

Spatial Frequency

The spatial frequency of this signal would be:

with units of either:

cycles per m (physical space)

cycles per pixel (image space)

F(u) is a two-side function of u

(i.e. extends to +ve and –ve frequencies)

If f(x) is real:

Fourier Transform of a Rectangular Function

Fourier Transform of a Gaussian

Notes:•The Inverse Relationship – narrow function in the spatial domain results in a wide function in the Fourier Domain.•In this case F(u) is not complex ie

The Impulse or Delta Function

A Table of FT Pairs

RECTANGULAR FUNCTION

GAUSSIAN FUNCTION

DELTA FUNCTION

COSINE FUNCTION

SINE FUNCTION

SHAH FUNCTION

The 2-D Fourier Transform

x

y

u

v

Example: The "Box Car" Function

More on Convolution Integrals

The CONVOLUTION THEOREM: the Fourier Transform of the product of two functions equals the convolution of the Fourier transforms of the individual functions.

Example: Convolution of a Shah Function with a Rectangular Function

7.2 The Discrete Fourier Transform (DFT)Consider a 1-d sampled dataset of dimension N:

We need to evaluate F(u) on a grid of N points (in u space). A good choice for the spacing is Δu=1/X (=1 cycle across the full extent of the image)

The DFT continuedFinally we set , we redistribute the normalization term and drop the dashes:

EXAMPLE: Determine the DFT of a 4-point dataset with input values:

f0 = 1 f1 = 1 f2 = 0 f3 = 0

|Fu|

-4 -3 -2 -1 0 1 2 3 4 5 6 7 u

X

X

X

X

X

X

X

X

X

X

X

X

Why the periodicity?

7.3 Some Properties of the DFT(a) SAMPLING THEORY

Input data stream (continuous variables)

Represent sampling by a

comb of delta functions s(x) [a Shah Function]

The sampled version of f(x) can be represented by s(x)f(x)

Assume F(u) is zero outside range -w<u<w

S(u) is a Shah Function with spacing Δu=1/Δx

In the Fourier domain we have the convolution S(u)*F(u)

This explains the periodic nature of the DFT ie it repeats at a rate (in u) of 1/Δx

However, if the bandwidth of the input signal w is too high , such that w > 1/2Δx, then the result will be

The overlap of the repeating functions results in information loss

ALIASING

Some Properties of the DFT cont.(b) SHANNON’S SAMPLING THEOREMTo avoid information loss it is necessary to sample a signal at a rate

equivalent to at least twice the maximum frequency component present in the signal.

That is we need w < 1/2Δx, where 1/2Δx is known as the Nyquist frequency.

To avoid aliasing sample at a higher rate

(or alternatively filter the input data stream to remove signals above the Nyquist frequency)

✔

✖

The process to recover the original data stream!

Some Properties of the DFT cont.(c) CHARACTER OF THE DFTExample: 1-d 8-point transform.

(d) The 2-D DFT

(e) The Optical TransformA trick to shift the (u,v) origin from the top left corner to the centre of the Fuv image is to multiply

the input image by alternate 1’s and -1’s before computing the DFT i.e.

|Fu|fx

0 1 2 3 4 5 6 7 x

0 1 2 3 4 5 6 7 x

N=8N=8

The Optical Transform

7.4 Computation of the DFT In DIP applications we need to compute the 2-d DFT

The “separability” property of the 2-d transform leads to a simplification:

Hence, the 2-d DFT reduces to the computation of a series of 1-d transforms.

In total we require 2N 1-d transforms, each of which involves N x N

complex multiplies and adds ie N4 2N3 calculations

Step 1 involves N 1-d transforms along the rows of the original image (x N scaling)

Step 2 involves N 1-d transforms down the columns of the intermediate image

fxy Fxv Fuv 1 2

x

y

x

v

u

v

The FFT Algorithm

Implementation of the FFT Algorithm

A Fortran FFT Routine

Re-ordering

Successive doubling

Calculating the Inverse Fourier Transform

If we take the complex conjugate of the inverse transform and scale by 1/N:

To calculate the Inverse Transform with a software procedure that computes the 1-d “forward” transform:

(i)Convert Fu Fu*

(ii) Apply the forward transform

(iii) Scale the result by N

(iv) Convert fx* fx

Example of a 2-d DFT calculation

fxy (-1)x+y

|Fuv | log |Fuv |

And the Inverse ….

Ignoring the phase! Ignoring the amplitude!

(-1)x+y

Richard Alan Peters II

Example of a 2-d DFT

log |Fuv|

Magnitude-only Reconstruction

Phase-only Reconstruction