Referring to the second equation, we can easily sample image xt from a normal distribution as: x, = \\/1 , x,_1 + Ea ...(3) ;where 6 ~ N(0,I) 1. Here, the epsilon s is the "noise\" term that is randomly sampled from the standard gaussian distribution and is rst scaled and then added (scaled) X(t_1). 2. In this way, starting from x0, the original image is iteratively corrupted from t=1...T Mathematical Details Of The Forward Diffusion Process The distribution q in the forward diffusion process is defined as Markov Chain given by: T q ( x 1 , . . . , XT |xo) := q(x, 1x1-1) . .. (1) t=1 q ( x, | x1-1 ) : = N(x,; V1 - B,X1-1, B,I) ... (2) 1. We begin by taking an image from our dataset: Xo. Mathematically it's stated as sampling a data point from the original (but unknown) data distribution: Xo ~ q(Xo). 2. The PDF of the forward process is the product of individual distribution starting from timestep 1 - T. 3. The forward diffusion process is fixed and known. 4. All the intermediate noisy images starting from timestep 1 to T are also called "latents." The dimension of the latents is the same as the original image. 5. The PDF used to define the FDK is a "Normal/Gaussian distribution" (eqn. 2) . 6. At each timestep t, the parameters that define the distribution of image Xt are set as: . Mean: V1 - Btxt-1 . Covariance: B+I 7. The term B (beta) is known as the "diffusion rate" and is precalculated using a "variance scheduler". The term / is an identity matrix. Therefore, the distribution at each time step is called Isotropic Gaussian. 8. The original image is corrupted at each time step by adding a small amount of gaussian noise (&). The amount of noise added is regulated by the scheduler. 9. By choosing sufficiently large timesteps and defining a well-behaved schedule of B the repeated application of FDK gradually converts the data distribution to be nearly an isotropic gaussian distribution