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Robust, computationally efficient and accurate estimation of 2D speckle pattern displacement is the key signal processing task in most digital speckle photography principles. State-of-the-art techniques are based on the evaluation of correlation functions with high computational complexity when sub-pixel accuracy is needed. Within this paper we describe a new algorithm based on the evaluation of 2D phase data of laser speckle patterns, capable of estimating 2D speckle pattern displacement with reduced computational effort but high resolution when compared to correlation-based processing methods. Using our new algorithm, the resolution and measurement rate of some digital speckle photography principles could be noticeably improved.