Relationship Between a SAR Image and its Spectrum
You can learn to see in Fourier: decode images like a radar pro!
Introduction: understanding SAR Images and their spectrum
SAR systems send out pulses of radio waves and measure the echoes reflected from the ground to construct detailed landscape images.
An important aspect of SAR imagery is the duality between its spatial and spectral representations. In the spatial domain, a SAR image consists of pixels that correspond to specific points on the Earth’s surface. However, this image can also be transformed into the frequency domain via the Fourier transform, where it is represented as a spectrum.
The spectral representation of an image describes how different spatial frequencies contribute to the overall image, with low frequencies capturing general shapes and high frequencies detailing fine structures.
⚠️Caution: The spectrum of a real image is not equivalent to that of a complex image, even if the phase of the complex image seems random! ⚠️
Understanding the relationship between a SAR image and its spectrum is key to optimizing image processing techniques. The spectral domain provides insight into the image’s spatial resolution, noise characteristics, and potential artifacts. Additionally, many SAR operations rely on spectral manipulations, such as filtering, reconstruction, and resampling.
In this blog, you’ll find an accompanying Python notebook that walks through basic manipulations of the spectrum. The notebook includes step-by-step examples to help illustrate the fundamental concepts of Fourier transforms and their application to radar (SAR) images, allowing you to explore the relationship between spatial and frequency domains hands-on:
Dual physical quantities: spatial domain and frequency domain
In the spatial domain, the SAR image is characterized by two key parameters: spatial resolution and image support.
Spatial resolution refers to the smallest distinguishable feature within the image, determining how fine the details are. Image support defines the extent of the scene captured or the physical size of the area represented in the image.
When we shift to the frequency domain through the Fourier transform, the image is described in terms of its Fourier spectrum. The Fourier spectrum represents the image in terms of its frequency components, where each point in the spectrum corresponds to a spatial frequency, indicating how rapidly image intensity changes across the scene. Lower frequencies correspond to smooth, broad patterns (such as the general shape of a landscape), and higher frequencies correspond to finer details (such as edges and sharp transitions in the scene).
In the case of real images, such as those we usually encounter in photography or visible imaging, the relationship between the image in the spatial domain and its spectrum is well-known and intuitive. SAR images, on the other hand, stand out for their complex nature. The phase component in SAR images is crucial, as it contains information linked to the precise distance between the radar and the objects observed, and plays a key role in image reconstruction and the analysis of specific targets. Consequently, as we move into the frequency domain, this phase makes a significant contribution to the spectrum. If we consider only the amplitude of the complex image (i.e. ∣Z∣, where Z is the complex value), and apply the Fourier transform on this amplitude alone, the results obtained will be fundamentally different from those obtained by taking into account the entire complex image.
Link Between the Image Support and the Sampling Rate in the Dual Domain
Classically, two dual variables in the Fourier sense have supports that are inversely proportional to the sampling step of the other. Dual variables are pairs of quantities where one is defined in the spatial domain (such as position or time) and the other in the frequency domain (such as spatial frequency or temporal frequency).
For example, in the case of a 1D signal, the spatial coordinate xxx and its corresponding frequency kx are dual variables. The relationship between them follows from the Fourier transform: the wider the range (support) of x, the narrower the support of kx, and vice versa. This is because the sampling step dx in the spatial domain determines the range of frequencies covered in the Fourier domain, which is proportional to 1/dx. Similarly, the sampling step dkxdk_xdkx in the frequency domain depends on the total support of x.
Thus, when you increase the spatial resolution (reduce dx), you broaden the frequency support, and when you increase the frequency resolution (reduce dkx), you require a larger spatial support. This inverse relationship is a fundamental property of Fourier pairs.
A SAR image is not acquired in the “Image” domain, but in a space that is close to that of the spectrum, without being strictly equivalent to it. We can call kx the dual variable of the x-space variable (or short-time axis or distance axis), and ky the dual variable of the y-space variable (or long-time axis, or azimut axis).
If we look at the spectrum of an SLC radar image, the shape of the support looks like this:
It depends on several acquisition parameters:
- The width of the spectrum approximated along the kx axis, Δkx, is inversely proportional to the distance resolution
- The width of the spectrum approximated along the ky axis, Δky is inversely proportional to azimuth resolution.
- Along this axis, the center is located at a coordinate corresponding to 2fc/c, where c is the celerity of light and fc is the center frequency of the transmitted signals.
- the width of the spectrum approximated along the ky axis is inversely proportional to the azimuth resolution.
The spectrum of a SAR image is not necessarily centered.
Particularly if the acquisition mode is squinted.
As it generally includes a few zeros, intended to achieve a final spatial sampling slightly finer than the resolution, it sometimes looks like several pieces. Here are some spectrum examples:
Equivalence Between Translation in One Domain and Multiplication in the Other
In signal processing, there exists a fundamental relationship between translation (or shifting) in the spatial domain and phase modulation in the spectral domain, as governed by the Fourier transform properties. This relationship plays a crucial role in understanding how SAR (Synthetic Aperture Radar) images behave when manipulated, and it can be described as follows:
If an image or a signal is shifted by a distance Δx in the spatial domain, its Fourier transform in the spectral domain is modified by a phase modulation. Specifically, if f(x) represents an image and F(k) its Fourier transform, then a shift of Δx in the spatial domain causes the following transformation in the frequency domain:
Centering the spectrum of an image can therefore be achieved either by a translation in the Fourier domain or, equivalently, by applying a multiplication in the spatial domain by a complex exponential:
These two approaches are related due to the properties of the Fourier transform, which links spatial and frequency domains.
The link between Convolution and multiplication
The link between convolution in the spatial domain and multiplication in the frequency domain is a key concept. According to the convolution theorem, convolution in one domain (spatial) corresponds to multiplication in the other (frequency). Mathematically, this can be expressed as:
Where ∗ denotes convolution in the spatial domain, and ⋅ represents multiplication in the frequency domain.
- SAR systems use convolution to perform operations such as image reconstruction or filtering, where the radar signal is convolved with a specific kernel or response function. This can be solved in the Fourier domain
- The impulse response function (IRF), which describes how a point target appears in a SAR image, is a key concept. The IRF is used to model the system’s effect on the received signal, and through convolution with the SAR data, we can focus the image and enhance resolution. In the frequency domain, this convolution becomes a simpler multiplication, enabling efficient computational techniques to process the data.
Resampling and Spectral Manipulations: Zero Padding and Spectrum Cropping
In the frequency domain, zero padding results in a finer sampling of the Fourier-transformed spectrum, which can provide a higher apparent resolution in the frequency domain. This means that the Fourier transform of the zero-padded image will contain more frequency components, allowing for a smoother representation of the spectral information. However, it is important to note that zero padding does not increase the true resolution of the image — it merely interpolates the existing information more finely.
✔️ Please note: to be carried out correctly, while preserving the significance of the spectrum center, the spectrum must be centered, zero-padded, and then off-center.
Spectrum cropping is the opposite approach to zero padding and involves removing or truncating high-frequency components from the Fourier-transformed image, thereby reducing the spectrum’s bandwidth. This manipulation has a direct effect on the image’s spatial characteristics when transformed back to the spatial domain.
For correct spectrum cropping, it is again important to select an area of the spectrum from its center.
✨Looking at the spectrum of a complex radar image, we can therefore see at a glance :
- whether the resolution varies greatly within the image (if the spectrum has a shape that deviates from the theoretical rectangular shape)
- whether or not the sampling step is finer than the physical resolution (if the spectrum contains a high proportion of zeros)
- if the acquisition is highly squinted (the spectrum has a non-symmetrical shape)
- if the antenna positions used to reconstitute the image are highly squinted (is the spectrum widely off-center)
The books that accompanied me were written in French:
Mohamed Tria’s thesis
https://theses.hal.science/tel-00011181/file/TheseMomo.pdf
The course handout by Pr. Jean-Marie Nicolas
https://perso.telecom-paristech.fr/tupin/JMN/DOCJMN/monradar.pdf
