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Purpose: Task-based assessment of image quality in undersampled magnetic resonance imaging provides a way of evaluating the impact of regularization on task performance. In this work, we evaluated the effect of total variation (TV) and wavelet regularization on human detection of signals with a varying background and validated a model observer in predicting human performance. Approach: Human observer studies used two-alternative forced choice (2-AFC) trials with a small signal known exactly task but with varying backgrounds for fluid-attenuated inversion recovery images reconstructed from undersampled multi-coil data. We used a 3.48 undersampling factor with TV and a wavelet sparsity constraints. The sparse difference-of-Gaussians (S-DOG) observer with internal noise was used to model human observer detection. Results: We observed a trend that the human observer detection performance remained fairly constant for a broad range of values in the regularization parameter before decreasing at large values. A similar result was found for the normalized ensemble root mean squared error. Without changing the internal noise, the model observer tracked the performance of the human observers as the regularization was increased but overestimated the PC for large amounts of regularization for TV and wavelet sparsity, as well as the combination of both parameters. Conclusions: For the task we studied, the S-DOG observer was able to reasonably predict human performance with both TV and wavelet sparsity regularizers over a broad range of regularization parameters. We observed a trend that task performance remained fairly constant for a range of regularization parameters before decreasing for large amounts of regularization.