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

## 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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##### Citations

57 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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24 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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14 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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7 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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2 citations

##### References

1,881 citations

442 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....

[...]

378 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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192 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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48 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....

[...]