Edge-adaptive $ \ell_2 $ regularization image reconstruction from non-uniform Fourier data

Figure 1.  Non-uniform sampling $ { λ}_{\bf{k}} $ as in (5)

Figure 13.  $ 257\times257 $ pixel Shepp-Logan phantom

Figure 2.  $ f_1(x) $ (top) and $ f_2(x) $ (bottom) reconstructed via Algorithm 1 using $ m = 1 $ from $ 257 $ Fourier modes on $ 257 $ grid points. The red reconstructions indicate recovery from noise-free data as in (1), while yellow reconstructions are recovered from Fourier coefficients with zero-mean complex Gaussian noise added as in (3). Here we use a signal to noise ratio (SNR) of $ 20 $ dB. For $ f_1(x) $, we used parameters $ \rho = 1 $, $ \ell_{max} = 25 $, and $ \epsilon = 1.9 $. For $ f_2(x) $, we used parameters $ \rho = 1 $, $ \ell_{max} = 25 $, and $ \epsilon = 2.9 $

Figure 3.  (Left) $ 257\times257 $ pixel function $ f_3(x,y) $; (Middle) reconstruction via Algorithm 2 with $ m = 2 $ from $ 257\times257 $ jittered Fourier coefficients on $ 257\times257 $ grid points; (Right) pointwise error plotted on a logarithmic scale. The algorithm parameters used are $ \rho = .01 $, $ \epsilon = .9 $, and $ \ell_{max} = 5 $

Figure 4.  (Left) Ideal weighting matrix for the $ y $-direction edges computed from (19) using the exact $ 257\times257 $ two-dimensional function $ f_3(x,y) $; (Right) final weighting matrix for the $ y $-direction edges produced by Algorithm 2. The minimum weight of $ 0 $ is indicated by black while maximum weight of $ \frac{1}{\epsilon}\approx 1.11 $ is indicated by white, while gray indicates a weight in between $ 0 $ and $ \frac1\epsilon $

Figure 5.  (Left) $ f_1(x) $ and $ f_2(x) $; (Middle) Jump function approximations using (37); (Right) Jump function approximations with additive noise starting from (3) with SNR $ = 20 $ dB. Here we use $ 257 $ reconstruction points, $ 257 $ jittered Fourier modes, and regularization parameter $ \mu = 1 $

Figure 6.  Edge detection (center and right) in the $ x $ and $ y $ directions using (38) for $ f_3(x,y) $ (left). Here we use $ 257\times257 $ jittered Fourier modes, $ 257\times257 $ reconstruction points, and regularization parameter $ \mu = 1 $

Figure 17.  (Top) Reconstruction comparison of SAR vehicle data using (left) equation (44) and (right) Algorithm 6. (Bottom) A close up of the lower right tire

Figure 7.  (Left) Comparison of Algorithms 1 and 5 on $ f_1(x) $ using PA order $ m = 1 $ given $ 257 $ jittered Fourier samples reconstructed on $ 257 $ grid points; (Right) corresponding pointwise errors. For parameters, we use $ \rho = 1 $, $ \mu = 1 $, $ \lambda = 1 $, $ \epsilon = 1.9 $, $ \ell_{max} = 25 $, and threshold $ \tau = 1/257 $

Figure 8.  Comparison of Algorithms 2 (left) and 6 (right) for reconstructing $ f_3(x,y) $ from $ 257\times257 $ noise-free Fourier modes on $ 257\times257 $ grid points. For parameters, we use PA order $ m = 2 $ due to the piecewise quadratic nature of the function, $ \rho = .01 $, $ \epsilon = .9 $, and $ \ell_{max} = 5 $, $ \mu = .1 $, $ \tau = .025 $, and $ \lambda = 1 $

Figure 9.  Comparison of errors using Algorithms 2 (left) and 6 (right) for reconstructing $ f_3(x,y) $, using the same parameters as Figure 8

Figure 10.  Pointwise error plot comparisons between Algorithm 1 and Algorithm 5 for $ f_1(x) $ given $ 257 $ jittered Fourier samples reconstructed on $ 257 $ grid points. (Left) Parameters $ m = 1 $, $ \ell_{max} = 25 $, $ \epsilon = 1.9 $, $ \tau = 1/257 $, $ \lambda = 1 $, and vary $ \rho = .01,.1,1,10,100 $; (Right) Parameters $ m = 1 $, $ \rho = 1 $, $ \ell_{max} = 25 $, $ \epsilon = 1.9 $, $ \mu = 1 $, $ \tau = 1/257 $, and vary $ \lambda = .01,.1,1,10,100 $

Figure 11.  Comparison of Algorithms 1 and 5 on $ f_2(x) $ using (top) $ m = 1 $ (middle) $ m = 2 $ and (bottom) $ m = 3 $ given $ 257 $ jittered Fourier samples reconstructed on $ 257 $ grid points. Left are the reconstructions, right are the corresponding pointwise errors. For parameters, we use $ \rho = 1 $, $ \ell_{max} = 25 $, $ \epsilon = 2.9 $, $ \mu = 1 $, $ \tau = 1/257 $, and $ \lambda = 1 $

Figure 12.  (Left) Comparisons of Algorithms 1 and 5 on $ f_1(x) $ and $ f_2(x) $ given $ 257 $ noisy jittered Fourier samples. $ SNR = 15 $ for $ f_1(x) $ and SNR $ = 20 $ for $ f_2(x) $. In both cases we use PA order $ m = 1 $ and reconstruct on $ 257 $ grid points; (Right) corresponding pointwise errors. For $ f_1(x) $ we use parameters $ \rho = 1 $, $ \epsilon = 1.9 $, $ \ell_{max} = 25 $, $ \mu = 1 $, $ \tau = 1/257 $, and $ \lambda = 1 $. For $ f_2 $ we use parameters $ \rho = 1 $, $ \epsilon = 2.9 $, $ \ell_{max} = 25 $, $ \mu = 1 $, $ \tau = 1/257 $, and $ \lambda = 1 $

Figure 14.  (Top) Reconstruction on $ 257\times 257 $ grid points of the Shepp-Logan phantom using Algorithms 2 (left) and 6 (right) with PA order $ m = 1 $ from $ 129^2 $ Fourier coefficients randomly chosen from a grid of $ 257\times257 $. (Bottom) respective pointwise errors. For parameters, we use $ \rho = .01 $, $ \epsilon = .9 $, $ \ell_{max} = 5 $, $ \mu = .01 $, $ \tau = 0.1 $, $ \lambda = .1 $. The relative error using Algorithm 2 was $ RE = .7160 $, while Algorithm 6 yielded $ RE = .4873 $

Figure 15.  (Top) Reconstruction on $ 257\times 257 $ grid points of the Shepp-Logan phantom using Algorithms 2 (left) and 6 (right) with PA order $ m = 1 $ from $ 181^2 $ Fourier coefficients randomly chosen from a grid of $ 257\times257 $. (Bottom) respective pointwise errors. For parameters, we use $ \rho = .01 $, $ \epsilon = .9 $, $ \ell_{max} = 5 $, $ \mu = .01 $, $ \tau = 0.1 $, $ \lambda = .1 $. The relative error using Algorithm 2 was $ RE = .5180 $, while Algorithm 6 yielded $ RE = 0.3259 $

Figure 16.  (Top) Reconstruction on $ 257\times 257 $ grid points of the Shepp-Logan phantom using Algorithms 2 (left) and 6 (right) with PA order $ m = 1 $ from $ 225^2 $ Fourier coefficients randomly chosen from a grid of $ 257\times257 $. (Bottom) respective pointwise errors. For parameters, we use $ \rho = .01 $, $ \epsilon = .9 $, $ \ell_{max} = 5 $, $ \mu = .01 $, $ \tau = 0.1 $, $ \lambda = .1 $. The relative error using Algorithm 2 was $ RE = .3458 $, while Algorithm 6 yielded $ RE = .2930 $

Figure 18.  Comparison of edge-adaptive $ \ell_1 $ and edge-adaptive $ \ell_2 $ methods on $ f_2(x) $ using 257 Fourier coefficients. (left) error plot from reconstructions with noise-free data. (right) error plot from reconstruction with 20 dB zero-mean Gaussian noise added to the data