2. Theoretical Basis

Sirius has willingly been developped by CS as an Open Source software. As so, we believe this documentation shall not only explain how to use it, but also explain the algorithms behind it. It should be noticed the used of the word ‘algorithms’ instead of ‘theory’ here. We believe that the theory can merely be found on the internet with high quality courses and documentations, and so will only sparsely point at it. However, we thought to begin with, and now know for sure, that the hardest part is rather knowing how to comprehend its discretization and deal with its outcomes. Hence this is what we will be focusing on in this documentation:

• 2.1. Upsampling
  • 2.1.1. "Fourier Zero Padding"
    • 2.1.1.1. Sinus cardinal interpolation
    • 2.1.1.2. Simple example
    • 2.1.1.3. Summary
  • 2.1.2. Tile processing of the large image
    • 2.1.2.1. The implicit edges discontinuities and their associated artifacts
    • 2.1.2.2. The possible solutions
      • 2.1.2.2.1. Margin the original data
      • 2.1.2.2.2. Mirroring the original data
      • 2.1.2.2.3. The periodic + smooth decomposition
    • 2.1.2.3. Conclusion : Sirius uses the p+s decomposition
  • 2.1.3. The discontinuous signals
    • 2.1.3.1. Illustrating the issue
    • 2.1.3.2. The windowing solution
  • 2.1.4. User kernel interpolator
    • 2.1.4.1. One can give Sirius his own low-pass filter
    • 2.1.4.2. The Sirius Filter must lie on the spatial domain
    • 2.1.4.3. The Sirius Filter must be sampled like the upsampled (targeted) signal
    • 2.1.4.4. The Sirius Filter is odd and a GDAL supported image format
    • 2.1.4.5. Sirius auto-shifts the Filter before its FT is computed
    • 2.1.4.6. Sirius can normalize the Filter
    • 2.1.4.7. When Sirius uses a Filter to upsample, the spectrum is periodized instead of zero padded
  • 2.1.5. Convolution near the data edges
    • 2.1.5.1. The real edges
    • 2.1.5.2. The virtual edges
    • 2.1.5.3. What Sirius proposes
  • 2.1.6. Float upsampling factor
• 2.2. Downsampling
  • 2.2.1. Anti-Aliasing process
  • 2.2.2. Non integer downsampling factor
• 2.3. References
• 2.4. Associated Python Code
  • 2.4.1. :mod:`conf`
    • 2.4.1.1. Module Contents
      • setupapp
  • 2.4.2. :mod:`simple_signal_proc`
    • 2.4.2.1. Module Contents
      • create_1D_rectn
      • sincx, dtype = None
      • create_1D_sincn, step = 1
      • create_1D_lanczosn, a = 3, step = 1
      • create_1D_miren
      • create_1D_triangle_signaln = 4, step = 1
      • create_1D_cosinen = 100, w = 1
      • create_1D_rampn = 100, step = 1
      • show_1Dsignal, title = "Figure"
      • mirrors, margin_factor = 2
      • zero_pads, margin_factor = 2, dtype = None
      • periodizes, margin_factor = 2, dtype = None
      • fft1D_zero_padtf, zoom_factor = 2
      • fft1D_periodizationtf, zoom_factor = 2
      • zoom_freq_zpdsignal, zoom_factor = 2
      • zoom_freq_img_periodizationsignal, zoom_factor = 2, user_kernel = None
      • interp1dsignal, zoom_factor
      • zoom_freq_img_perAndSmoothsignal, zoom_factor