A synthetic model was designed by our lab where the number of particles, diffusion coefficient, and noise can be defined for each movie. MSD was used to characterize particle motions. MSD is defined by the following equations:

In two dimensions,

The term D in Eq. 3 is the diffusion coefficient of the particle, and the exponent α is a unitless parameter that characterizes the type of diffusion; α = 1 for simple diffusion, and α = ½ for a stretch of beads in a long-chain polymer (Osmanovic´ and Rabin, 2017 blue right-pointing triangle). In a Brownian motion model, the MSD is dependent on the size of the moving object as well as the mechanical and physical properties of the medium, as described in Eq. 4 (the Stokes–Einstein equation):

Here η is the viscosity of the medium, T is the temperature, R is the particle radius, and k is the Boltzmann constant. To match experimental observations, the pixel size of the synthetic data was set as 80 nm. The particle was simulated as a 2D Gaussian function with a radius of 60 nm, and the intensity level was scaled from 0 to 1 with a Poisson distribution. To avoid merging of multiple foci, the D value was set to 3.1 nm2/s. Movies including 49 particles lasting for 300 frames were simulated with a temporal interval of 1 s. The noise level of synthetic images was simulated as Gaussian noise, where the standard derivation (σ) varies as σ = 0, 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6. Synthetic movies without noise (σ = 0) were used as ground truth, and the coordinates of each particle were also used as ground truth for MSD curves. For restored images, the motion of the particle was tracked by a home-written algorithm (details below). A cutoff distance of 150 nm was used to define matching particles in two corresponding movies.

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