• ݕग़ɾൃݟͷޮԽɹྫɿಛྔநग़ʹΑΔલܠఱମʔഎܠ • ΦʔτϝʔγϣϯɹྫɿΩϡʔ؍ଌɾύΠϓϥΠϯॲཧɾσʔλϕʔε particular, the raw reconstructed images in Figure 6 clearly show that smooth edges in the ground-truth images, which are attributed to a smooth transition in the emissivity and opacity of the plasma in the accretion flow, are much better reconstructed with ℓ1 +TSV regularization. As a consequence of this, the TSV term comes reproduces a much clearer shadow feature in the reconstructed images. For the Free-fall model, the size of the black hole shadow is larger in the ℓ1 +TSV image that the isoTV term and gets closer to the ground truth than the isoTV term. For sub-Keplerian and Keplerian models, the black hole shadow is visible in the ℓ1 +TSV images but is mostly obscured (except for the darker funnel region) in the ℓ1 +isoTV methods. The appearance of the reconstructed images indicates that ℓ1 +TSV regularization is justified based on a more physically reasonable assumption and is therefore more suitable to image the objects seen in many astronomical observations. In the following subsections, we evaluate the images more quantita- tively with the image fidelity metrics described in Section 4. 5.3. NRMSE Analysis and Optimal Beam Sizes In Figure 7, we evaluate the NRMSE metric on the image domain and its gradient domain over various spatial scales, as in previous work (Chael et al. 2016; Akiyama et al. 2017a, 2017b). The black curves represent the ideal NRMSE curves between the original (unconvolved) ground-truth image and the ground-truth image after convolution with a Gaussian beam scaled to each resolution on the horizontal axis. These curves represent the highest fidelity available at a given resolution, as would be provided by an algorithm that reconstructs the image Figure 5. Ground-truth image (left-most) and images reconstructed with CS-CLEAN (second from left), ℓ1 +isoTV (second from right), and ℓ1 +TSV (right-most) regularization. All reconstructed images are convolved with elliptical Gaussian beams represented by the yellow ellipses, for which the size corresponds to the optimal resolution determined with the image-domain NRMSE curve in Figure 7 (see Section 5.3). The same transfer function is adopted for four images of each model (i.e., on each row). 9 The Astrophysical Journal, 858:56 (14pp), 2018 May 1 Kuramochi et al. the rank and cardinality of the low-rank and sparse matrices, respectively. For the low-rank matrix L, we check the distribution of singular values, applying SVD, as shown in the main panel of Figure 2. The matrix L is expressed as = L UDVT , where U and V are orthogonal matrices, and D is a diagonal one. Then, we set the rank of the low-rank matrix by setting zeros for the singular values at indices larger than the rank. Within the sparse matrix S, the transient events can be easily extracted because these events are innately sparse. The time variation of the sky background can be monitored by checking the noise matrix G. After the data matrix M has been decomposed into the three matrices L, S, and G, further data processing is necessary, because otherwise the data size would be three times larger than that of the original data. The low-rank matrix L is easily compressed into three small matrices, as shown in Figure 1. For the sparse matrix S, the frames that contain a transient event(s) must be preserved, movie data, due to the speed of computation and memory consumption. We have rewritten the GoDec code in C++, utilizing the OpenBLAS5 and LAPACK6 libraries. We use Quick Select, instead of full sorting, to select non-zero elements for a sparse matrix in the GoDec algorithm. 3. APPLICATION OF THE PROPOSED METHOD We used the movie data set of a CMOS sensor for 400 frames obtained with the Tomo-e PM in 2015 December, which contains some transient events lasting for a short duration (Ohsawa et al. 2016). Panels (a) and (e) of Figure 3 show the subarray images with 300×300 pixels in two different time frames, which contained a transient point source and a meteor, respectively. We applied the decomposition to the data by setting r=10 and = ´ k 1 108. Panels (b)–(d) and Figure 3. Example decomposition images for movie data of the Tomo-e Gozen from two frames (top and bottom rows). Original (denoted as the matrix M in the main text), low-rank (L), sparse (S), and noise (G) images are shown in the four columns from left to right, respectively. A transient point source appears near the center of the image at the time frame of the top row, as spotted in the original image (a), in contrast to (e), which was taken in a different frame (bottom row), and as clearly visible in the sparse one (c), in contrast to (g). On the other hand, a line, which is a light trail caused by a meteor, is seen in the second time frame (bottom row), as in the original image (e) and the sparse one (g). These transient events are not recognized in the low-rank images (b) and (f). The noise images (d) and (h) do not contain any noticeable patterns. The Astrophysical Journal, 835:1 (5pp), 2017 January 20 Morii et al. Galaxy Zoo: Classifying Galaxies with Crowdsourcing and Active Learning Low-rank + sparse decomposition (Morii+2017) Sparse modeling (Kuramochi+2018, ...)
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