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We present a method that takes as input a single dual-pixel image, and simultaneously estimates the image's defocus map---the amount of defocus blur at each pixel---and recovers an all-in-focus image. Our method is inspired from recent works that leverage the dual-pixel sensors available in many consumer cameras to assist with autofocus, and use them for recovery of defocus maps or all-in-focus images. These prior works have solved the two recovery problems independently of each other, and often require large labeled datasets for supervised training. By contrast, we show that it is beneficial to treat these two closely-connected problems simultaneously. To this end, we set up an optimization problem that, by carefully modeling the optics of dual-pixel images, jointly solves both problems. We use data captured with a consumer smartphone camera to demonstrate that, after a one-time calibration step, our approach improves upon prior works for both defocus map estimation and blur removal, despite being entirely unsupervised.