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This paper proposes a sparse image representation using deformable templates of simple geometric structures that are commonly observed in images of natural scenes. These deformable templates include active curve templates and active corner templates. An active curve template is a composition of Gabor wavelet elements placed with equal spacing on a straight line segment or a circular arc segment of constant curvature, where each Gabor wavelet element is allowed to locally shift its location and orientation, so that the original line and arc segment of the active curve template can be deformed to fit the observed image. An active corner or angle template is a composition of two active curve templates that share a common end point, and the active curve templates are allowed to vary their overall lengths and curvatures, so that the original corner template can deform to match the observed image. This paper then proposes a hierarchical computational architecture of summax maps that pursues a sparse representation of an image by selecting a small number of active curve and corner templates from a dictionary of all such templates. Experiments show that the proposed method is capable of finding sparse representations of natural images. It is also shown that object templates can be learned by selecting and composing active curve and corner templates.