User: SkySat L1B
Proposed
Figure 1. Examples of reconstruction on a real ground fan target from 35 low-resolution SkySat L1A frames. From left to right: reference low-resolution L1A frame, L1B product provided by Planet (×1.25), proposed method (×2).
vi =Σ1((u◦Ai)∗k)+ni, (1)
where k is the Point Spread Function (PSF) modeling jointly optical blur and pixel integration, the operator Σ1 is the bi- dimensional ideal sampling operator due to the sensor array, and ni models the image noise. Since the satellite moves dur- ing the acquisition, small exposure times are necessary to avoid blur, thus the images are noisy. The resulting digital image v is encoded in a linear intensity scale, as is frequent in remote sens- ing applications. In the specific case of affine transforms, as- suming that k is an isotropic kernel, A and k commute. Hence, it is possible to first estimate u ∗ k, then invert k on the high- resolution image; this property allows for an efficient super- resolution method.
Note that in absence of noise, if the kernel k introduces a cutoff below the critical sampling frequency, then the image v is said to be well sampled. Therefore, by the Nyquist–Shannon sampling theorem (Shannon, 1948) the continuous signal u ∗ k could be reconstructed from the samples in v. In this scenario, there would be no gain in applying multi-frame super-resolution on this system, except for denoising. The interesting case is
when the system is designed to have the frequency cutoff of k above the critical sampling frequency. The resulting image v is then said to be aliased. Aliasing is the phenomenon of replica- tion of higher frequencies onto lower frequencies. Supermode (Latry, Rouge ́, 2000) for SPOT5 used this information to pro- duce a higher resolution image from two images acquired sim- ultaneously with a half-pixel shift. The objective of multi-frame super-resolution is to collect samples from multiple different images to estimate a de-aliased u ∗ k.
Assistant: The text you've provided describes a method for enhancing the resolution of images captured by satellites, specifically focusing on the processing of low-resolution SkySat Level 1A (L1A) frames to generate a higher-resolution product, referred to as the Level 1B (L1B) product. Here's a summary and breakdown of the key concepts:

### Key Concepts:

1. **Reconstruction from Low-Resolution Frames:**
   - The method involves the use of multiple low-resolution images (L1A frames) to reconstruct a higher-resolution image (L1B).
   - An example is given where a ground-based target (a fan) is reconstructed using 35 low-resolution frames.

2. **Image Formation Model:**
   - The image formation is represented by the equation:
     \[
     v_i = \Sigma_1((u \circ A_i) * k) + n_i
     \]
     Where:
     - \( v_i \) represents the observed noisy image.
     - \( u \) represents the true underlying image.
     - \( A_i \) denotes the affine transformations applied (such as rotations or translations) to each low-resolution frame.
     - \( k \) is the Point Spread Function (PSF), which models both optical blur and pixel integration.
     - \( n_i \) is the noise associated with the image.

3. **Noise and Aliasing:**
   - Due to satellite motion during acquisitions, low exposure times are required, which results in noisy images.
   - If the PSF \( k \) introduces a cutoff frequency that is above the critical sampling frequency, the resulting image \( v \) is considered "aliased," meaning higher frequency information has been incorrectly mapped to lower frequencies.

4. **Super-Resolution Concept:**
   - Super-resolution is sought when the system causes aliasing. In this context, the goal is to utilize multiple frames to estimate a de-aliased version of the image, where the underlying image \( u \) can be better approximated.
   - The Nyquist-Shannon sampling theorem states that if the sampling frequency is adequately high, the original continuous signal can be reconstructed from its samples, indicating that under ideal conditions, multi-frame super-resolution primarily serves the purpose of denoising rather than resolution enhancement.

5. **Application Example:**
   - The text references a previous technique, "Supermode," used with SPOT5 satellites, which utilized a similar concept to combine two images taken with a half-pixel shift to create a higher resolution output. This illustrates the practical application of multi-frame super-resolution techniques in remote sensing.

### Conclusion:
The passage describes a sophisticated approach to image reconstruction in satellite imaging, emphasizing the challenges posed by noise and aliasing, as well as the theoretical underpinnings that justify the multi-frame super-resolution methods. By leveraging multiple low-resolution frames, the proposed method seeks to improve the clarity and detail of the reconstructed image, which is critical for accurate analysis in remote sensing applications.
User: How to determine cutoff frequency of an imaging system?
Assistant: