# FFT regression and cross-noise reduction for comparing images in remote sensing

### Abstract:

In many remote sensing studies it is desired to quantify the functional relationship between images of a given target that were acquired by different sensors. Such comparisons are problematic because when the pixel values of one image are plotted versus the other, the 'cross-noise' is quite high. Typically, the correlation coefficient is quite low, even when the compared images look alike. Nevertheless, we can try to quantify the functional relationship between two images by a suitable regression model function**Y**=

**(**

*f***X**), while choosing one of them as 'the reference'

**Y**and using the other one as a 'predictor'

**X**. The underlying assumption of classical regression is that

**Y**is absolutely correct while

**X**is erroneous. Thus, the objective is to fit

**X**to

**Y**by choosing the parameters of

**Y**=

**(**

*f***X**), which minimize the 'residuals' (

**Ŷ**-

**Y**). When comparing images in remote sensing this objective is not valid because

**Y**itself is error prone. The alternative FFT regression method presented herein comprises a two-stage sensor fusion approach, whereby the initially low correlation between

**X**and

**Y**is increased and the residuals are dramatically decreased. First, pairwise image transforms are applied to

**X**and

**Y**whereby the correlation coefficient is increased, e.g. from roughly 0.4 to about 0.8-0.85. A predicted image

**Y**

**is then derived by least squares minimization between the amplitude matrices of**

*fft***X**and

**Y**, via the 2D FFT. In the second stage, there are two options: For one time predictions, the phase matrix of

**Y**is combined with the amplitude matrix of

**Y**, whereby an improved predicted image

*fft***Y**

**is formed. Usually, the residuals of**

*plock***Y**

**versus**

*plock***Y**are about half of the values of

**Y**versus

*fft***Y**. For long term predictions, the phase matrix of a 'field mask' is combined with the amplitude matrices of the reference image

**Y**and the predicted image

**Y**. The field mask is a binary image of a pre-selected region of interest in

*fft***X**and

**Y**. The resultant images

**Y**

**and**

*pref***Y**

**are modified versions of**

*pred***Y**and

**Y**respectively. The residuals of

*fft***Y**

**versus**

*pred***Y**

**are even lower than the residuals of**

*pref***Y**

**versus**

*plock***Y**. Images

**Y**

**and**

*pref***Y**

**represent a close consensus of two independent imaging methods which view the same target. The practical utility of FFT regression is demonstrated by examples wherein remotely sensed NDVI images**

*pred***X**are used for predicting yield distributions in agricultural fields. Reference yield maps

**Y**were derived by yield monitors which measure the flow rate of the crop while it is being harvested. The 2D FFT transforms, as well as other mathematical operations in this paper were performed in the 'MATLAB' environment.

**Document Type:** Research Article

**DOI:** http://dx.doi.org/10.1080/01431160110075910

Publication date: May 20, 2002

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