# A complex spectrum based SAR image resampling method with restricted target sidelobes and statistics preservation

TL;DR: A resampling scheme for SAR images is presented that preserves spatial resolution and produces statistically accurate images at the same time and is completely faithful to the underlying signal.

Abstract: The aim of this work is to present a resampling scheme for SAR images that preserves spatial resolution and produces statistically accurate images at the same time. Indeed, SAR images are, for reasons due to their acquisition process, well sampled signals according to the Shannon sampling theory. In the presence of strong responses, that we will refer to as targets, a sinc-like function centered at the target is smeared over the entire image and is particularly visible in the range of tens of pixels surrounding the target. To mitigate this phenomenon, the usual solution is to apply an apodization window in the Fourier domain so as to change the cardinal sine impulse response into a much rapidly decaying one. This approach has two major drawbacks. It reduces the resolution of the image and introduces inaccurate statistical dependency between pixels. We propose to resample the image in an adaptive and robust way so that the target smear is canceled and the new sampled image is completely faithful to the underlying signal.

## Summary (2 min read)

### 1. INTRODUCTION

- SAR images are provided by complex signal processing being at the heart of the SAR technique (range compression, SAR synthesis).
- The raw data received by the antenna before these operations are usually not provided by space agencies.
- The provided Single Look Complex data (SLC) are affected by two important factors that can be seen in the complex Fourier spectrum of the image: over-sampling and weighting of the azimuth and range spectrum [1].
- These factors can change depending on the data provider even for similar resolutions of the SLC images.
- Section 2 introduces the notations and gives a method to cancel apodization when the weighting function is unknown.

### 2.1. Pseudo-raw image and pseudo-raw spectrum

- Besides, it happens that the non-zero part of the Fourier spectrum is in fact apodized, which means that it resulted from a multiplication in the Fourier domain by a frequency attenuating function.
- This function results from the weighting affecting the antenna pattern and the weighting applied to the data [1] which depends on the data provider.

### 2.2. Practical estimation of the pseudo-raw spectrum

- Now, let us focus on the inversion of (2), that is, on the computation of the pseudo-raw spectrum û0.
- When the subfrequency domain ω̂ and the frequency attenuating function γ are known (for instance provided by the spatial agency who generated the image) the relation (2) can be easily inverted and the authors get ∀(α, β) ∈ ω̂, û0(α, β) = û(α, β) γ(α, β) .

### 3.1. Model

- An interpretation of this phenomenon is that the target is sufficiently narrow to be transformed, by the acquisition process, to the impulse response, yielding the cardinal sine function.
- The obvious solution to this problem is to resample the image on a grid such that the coordinates of the target are integers, thus suppressing the side lobes contributions.
- The authors see that, contrary to u0, the resampled signal v0 is not polluted anymore by the oscillations of the cardinal sine.
- Since in practice, there may and will be numerous targets in a single image, a global translation will not be sufficient to accommodate all the targets of the image.
- Indeed, contrary to [4, 5], the authors made the choice to not explicitly detect targets to keep the process as robust as possible.

### 3.2. Local displacement vector field

- The idea is that, when sampled on the appropriate grid, the discrete total variation of a target-induced cardinal sine is minimal, whereas it is always higher for all non integer displacements of the grid (the red dashed curve in Fig. 3 is more oscillatory than the blue dotted curve and exhibits a higher discrete total variation).
- Since their numerical expriments revealed that the third choice led to the most satisfying results, it was systematically used in all the experimental results displayed below.
- The computation of the resampled image v0 defined by (9) from the pseudo-raw image u0 is summarized in Algorithm 1, and some experimental results are displayed and commented in Fig. 4 and Fig.

### 3.3. Statistical properties of the resampled image

- The authors investigate the statistical properties of the resulting image and they show that, under a reasonable assumption, their sampling scheme produces a signal that is completely faithful to the underlying signal.
- This means that the correlation between samples distant by an integer value is zero.
- Thus, provided that their estimated tx equals to δ the final discrete result of their resampling will be, according to (8), U∗0 (k + δ) except at pixel k = k0 (the target appears here) which are integer distant samples from the underlying fully-developed speckle and hence i.i.d Gaussian variables.

### 4. REFERENCES

- J. Tsao and B. Steinberg, “Reduction of Sidelobe and Speckle Artifacts in Microwave Imaging: the CLEAN technique,” IEEE Trans. on Antennas and Propagation, vol. 36, no. 4, 1988. [6].
- C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, SciTech Publishing, 2004.

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17 citations

### Cites background or methods from "A complex spectrum based SAR image ..."

...Whitening the spectrum [17], [22], [23] or downsampling the image are possible strategies [18]....

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...Algorithms developed using speckle generated under Goodmans fully developed speckle model generally assume an absence of spatial correlations [16], which is not the case in actual SAR images synthetized by space agencies [17], [21]....

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...At a later stage, we feed the network with real acquisitions, allowing learning of the spatial correlation introduced by the SAR processing steps, namely spectral windowing and oversampling [17] [18]....

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15 citations

### Cites background or methods from "A complex spectrum based SAR image ..."

...We refer the reader to [3], [28] for more details about the computation of the pseudo-raw image u0 from u (in particular in the case when the frequency attenuating function γ is unknown)....

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...In this section, we complete with more details and experimental results our previous work presented in [28], and we discuss the strengths and weaknesses of the proposed irregular resampling scheme....

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...Contributions: this paper extends the recent conference paper [28] and introduces:...

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...Algorithm 1: Irregular resampling scheme proposed in [28]...

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9 citations

### Cites background from "A complex spectrum based SAR image ..."

...As a downside, these operations introduce spatial correlations in the speckle [24]....

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4 citations

### Cites methods from "A complex spectrum based SAR image ..."

...As explained in [2, 3], computing the pseudo-raw image, such as that displayed in Fig....

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...Besides, we explained in [2] how the apodization function γ could be estimated (if unknown), so that we can invert (6) and compute the pseudo-raw image u0....

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##### References

1,814 citations

407 citations

### Additional excerpts

...Indeed, contrary to [4, 5], we made the choice to not explicitly detect targets to keep the process as robust as possible....

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316 citations

### "A complex spectrum based SAR image ..." refers background in this paper

...These processing have a strong impact on the appearance of the images (spreading of the strong targets) and induce a correlation between neighboring pixels, which can affect further processing like physical parameter estimation [2]....

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175 citations

### "A complex spectrum based SAR image ..." refers background in this paper

...This function results from the weighting affecting the antenna pattern and the weighting applied to the data [1] which depends on the data provider....

[...]

...The provided Single Look Complex data (SLC) are affected by two important factors that can be seen in the complex Fourier spectrum of the image: over-sampling and weighting of the azimuth and range spectrum [1]....

[...]

...3 that the bright targets observed on the pseudoraw image can be very well approached by a two-dimensional cardinal sine function defined by (as given by the SAR processing [1]):...

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41 citations

### Additional excerpts

...Indeed, contrary to [4, 5], we made the choice to not explicitly detect targets to keep the process as robust as possible....

[...]