Fourier Analysis in Erdas Imagine

What is Fourier Analysis

The most common way of implementing these enhancements is by using a Fourier transformation. An enhancement that requires a convolution operation in the spatial domain can be implemented as a simple multiplication in frequency space- a much faster calculation.

A Fourier transform is a linear transformation that calculates the coefficients necessary for the sine and cosine terms to adequately represent the image. This theory is used extensively in electronics and signal processing, where electrical signals are continuous and not discrete.

Fourier Magnitude

The raster image generated by the FFT (Fast Fourier Transformation) calculation is not an optimum image for viewing or editing. Each pixel of a fourier image is a complex number. For display as a single image, these components are combined in a root-sum of squares operation. Also, since the dynamic range of Fourier spectra vastly exceeds the range of a typical display device, the Fourier Magnitude calculation involves a logarithmic function.

Finally, a Fourier image is symmetric about the origin (u, v = 0, 0). If the origin is plotted at the upper left corner, the symmetry is more difficult to see than if the origin is at the center of the image. Therefore, in the Fourier magnitude image, the origin is shifted to the center of the raster array.

• Image A– is one band of a badly striped Landsat TM scene
• Image B– is the Fourier Magnitude image derived from the Landsat image.

The origin of Image A is at (x, y) = (0, 0) in the upper left corner. In Image B, the origin (u, v) = (0, 0) is in the center of the raster. The low frequencies are plotted near this origin while the higher frequencies are plotted further out. Generally, the majority of the information in an image is in the low frequencies. This is indicated by the bright area at the center (origin) of the Fourier image.

Fourier Analysis using Erdas Imagine

Perform Fourier Analysis using Erdas Imagine software.

Steps :

1. Select Raster tab > Resolution group > Radiometric button > Periodic noise removal. The periodic noise removal dialog opens.

2. In the periodic noise removal dialogue under input file select image (ex.- tm_1) from the required directory.

3. Enter the output file name (ex.- fourier_transform).

[Layer one of this file is badly striped. In this example, you work with only one layer to make the processing go faster. However the techniques you use are applicable to multiple layers]

4. Enter 1 in the Select Layers field.

5. Enter 4 in the Minimum Affected Frequency field.

6. Click OK button to complete this task.

7. Now launch a new viewer and open the output file. You will notice that the drive has been removed.

You can apply for fourier transformation manually from Raster tab > Scientific button

Applications

Fourier transformations are typically used for the removal of noise such as striping, spots, or vibration in imagery by identifying the areas of high spatial frequency. Fourier editing can be used to remove regular errors in data such as those caused by sensor anomalies (for example, striping). This analysis technique can also be used across bands as another form of pattern
or feature recognition.