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We develop a new retrieval scheme for obtaining two-dimensional surface maps of exoplanets from scattered light curves. In our scheme, the combination of the L1-norm and total squared variation, which is one of the techniques used in sparse modeling, is adopted to find the optimal map. We apply the new method to simulated scattered light curves of the Earth, and find that the new method provides a better spatial resolution of the reconstructed map than those using Tikhonov regularization. We also apply the new method to observed scattered light curves of the Earth obtained during the two-year Deep Space Climate Observatory/Earth Polychromatic Imaging Camera observations presented by Fan et al. The method with Tikhonov regularization enables us to resolve North America, Africa, Eurasia, and Antarctica. In addition to that, the sparse modeling identifies South America and Australia, although it fails to find Antarctica, maybe due to low observational weights on the poles. Besides, the proposed method is capable of retrieving maps from noise-injected light curves of a hypothetical Earthlike exoplanet at 5 pc with a noise level expected from coronagraphic images from a 8 m space telescope. We find that the sparse modeling resolves Australia, Afro-Eurasia, North America, and South America using 2 yr observation with a time interval of one month. Our study shows that the combination of sparse modeling and multiepoch observation with 1 day or 5 days per month can be used to identify main features of an Earth analog in future direct-imaging missions such as the Large UV/Optical/IR Surveyor. [ABSTRACT FROM AUTHOR]