Speckle contrast is a simple metric for characterizing tissue dynamics such as blood flow. In conventional laser speckle contrast imaging (LSCI), speckle patterns are captured by a camera and their contrast, spatial or temporal, is calculated as the ratio of the intensity standard deviation to the mean. In practice, the presence of detection noise leads to a bias in the measured speckle contrast that must be corrected. This correction requires a precise knowledge of camera gain and readout noise, which can vary across the camera sensor and be inaccurate in low-light conditions. We describe a method based on spatial covariance to quantify speckle dynamics in an unbiased manner without prior knowledge of detection noise. We further describe a method to optimally combine covariance measurements across different length scales to improve precision. We show that with slight oversampling, covariance-based measurements provide better signal-to-noise ratios than variance-based measurements alone. Our method is validated with simulations and applied to both in-vivo mouse brain imaging and low-light-level speckle plethysmography in humans.