import math import numpy as np import torch import torch.nn.functional as F from einops import repeat def timestep_embedding(timesteps, dim, max_period=10000, repeat_only=False): """ Create sinusoidal timestep embeddings. :param timesteps: a 1-D Tensor of N indices, one per batch element. These may be fractional. :param dim: the dimension of the output. :param max_period: controls the minimum frequency of the embeddings. :return: an [N x dim] Tensor of positional embeddings. """ if not repeat_only: half = dim // 2 freqs = torch.exp( -math.log(max_period) * torch.arange(start=0, end=half, dtype=torch.float32) / half ).to(device=timesteps.device) args = timesteps[:, None].float() * freqs[None] embedding = torch.cat([torch.cos(args), torch.sin(args)], dim=-1) if dim % 2: embedding = torch.cat([embedding, torch.zeros_like(embedding[:, :1])], dim=-1) else: embedding = repeat(timesteps, 'b -> b d', d=dim) return embedding def make_beta_schedule(schedule, n_timestep, linear_start=1e-4, linear_end=2e-2, cosine_s=8e-3): if schedule == "linear": betas = ( torch.linspace(linear_start ** 0.5, linear_end ** 0.5, n_timestep, dtype=torch.float64) ** 2 ) elif schedule == "cosine": timesteps = ( torch.arange(n_timestep + 1, dtype=torch.float64) / n_timestep + cosine_s ) alphas = timesteps / (1 + cosine_s) * np.pi / 2 alphas = torch.cos(alphas).pow(2) alphas = alphas / alphas[0] betas = 1 - alphas[1:] / alphas[:-1] betas = np.clip(betas, a_min=0, a_max=0.999) elif schedule == "sqrt_linear": betas = torch.linspace(linear_start, linear_end, n_timestep, dtype=torch.float64) elif schedule == "sqrt": betas = torch.linspace(linear_start, linear_end, n_timestep, dtype=torch.float64) ** 0.5 else: raise ValueError(f"schedule '{schedule}' unknown.") return betas.numpy() def make_ddim_timesteps(ddim_discr_method, num_ddim_timesteps, num_ddpm_timesteps, verbose=True): if ddim_discr_method == 'uniform': c = num_ddpm_timesteps // num_ddim_timesteps ddim_timesteps = np.asarray(list(range(0, num_ddpm_timesteps, c))) steps_out = ddim_timesteps + 1 elif ddim_discr_method == 'uniform_trailing': c = num_ddpm_timesteps / num_ddim_timesteps ddim_timesteps = np.flip(np.round(np.arange(num_ddpm_timesteps, 0, -c))).astype(np.int64) steps_out = ddim_timesteps - 1 elif ddim_discr_method == 'quad': ddim_timesteps = ((np.linspace(0, np.sqrt(num_ddpm_timesteps * .8), num_ddim_timesteps)) ** 2).astype(int) steps_out = ddim_timesteps + 1 else: raise NotImplementedError(f'There is no ddim discretization method called "{ddim_discr_method}"') # assert ddim_timesteps.shape[0] == num_ddim_timesteps # add one to get the final alpha values right (the ones from first scale to data during sampling) # steps_out = ddim_timesteps + 1 if verbose: print(f'Selected timesteps for ddim sampler: {steps_out}') return steps_out def make_ddim_sampling_parameters(alphacums, ddim_timesteps, eta, verbose=True): # select alphas for computing the variance schedule # print(f'ddim_timesteps={ddim_timesteps}, len_alphacums={len(alphacums)}') alphas = alphacums[ddim_timesteps] alphas_prev = np.asarray([alphacums[0]] + alphacums[ddim_timesteps[:-1]].tolist()) # according the the formula provided in https://arxiv.org/abs/2010.02502 sigmas = eta * np.sqrt((1 - alphas_prev) / (1 - alphas) * (1 - alphas / alphas_prev)) if verbose: print(f'Selected alphas for ddim sampler: a_t: {alphas}; a_(t-1): {alphas_prev}') print(f'For the chosen value of eta, which is {eta}, ' f'this results in the following sigma_t schedule for ddim sampler {sigmas}') return sigmas, alphas, alphas_prev def betas_for_alpha_bar(num_diffusion_timesteps, alpha_bar, max_beta=0.999): """ Create a beta schedule that discretizes the given alpha_t_bar function, which defines the cumulative product of (1-beta) over time from t = [0,1]. :param num_diffusion_timesteps: the number of betas to produce. :param alpha_bar: a lambda that takes an argument t from 0 to 1 and produces the cumulative product of (1-beta) up to that part of the diffusion process. :param max_beta: the maximum beta to use; use values lower than 1 to prevent singularities. """ betas = [] for i in range(num_diffusion_timesteps): t1 = i / num_diffusion_timesteps t2 = (i + 1) / num_diffusion_timesteps betas.append(min(1 - alpha_bar(t2) / alpha_bar(t1), max_beta)) return np.array(betas) def rescale_zero_terminal_snr(betas): """ Rescales betas to have zero terminal SNR Based on https://arxiv.org/pdf/2305.08891.pdf (Algorithm 1) Args: betas (`numpy.ndarray`): the betas that the scheduler is being initialized with. Returns: `numpy.ndarray`: rescaled betas with zero terminal SNR """ # Convert betas to alphas_bar_sqrt alphas = 1.0 - betas alphas_cumprod = np.cumprod(alphas, axis=0) alphas_bar_sqrt = np.sqrt(alphas_cumprod) # Store old values. alphas_bar_sqrt_0 = alphas_bar_sqrt[0].copy() alphas_bar_sqrt_T = alphas_bar_sqrt[-1].copy() # Shift so the last timestep is zero. alphas_bar_sqrt -= alphas_bar_sqrt_T # Scale so the first timestep is back to the old value. alphas_bar_sqrt *= alphas_bar_sqrt_0 / (alphas_bar_sqrt_0 - alphas_bar_sqrt_T) # Convert alphas_bar_sqrt to betas alphas_bar = alphas_bar_sqrt**2 # Revert sqrt alphas = alphas_bar[1:] / alphas_bar[:-1] # Revert cumprod alphas = np.concatenate([alphas_bar[0:1], alphas]) betas = 1 - alphas return betas def rescale_noise_cfg(noise_cfg, noise_pred_text, guidance_rescale=0.0): """ Rescale `noise_cfg` according to `guidance_rescale`. Based on findings of [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://arxiv.org/pdf/2305.08891.pdf). See Section 3.4 """ std_text = noise_pred_text.std(dim=list(range(1, noise_pred_text.ndim)), keepdim=True) std_cfg = noise_cfg.std(dim=list(range(1, noise_cfg.ndim)), keepdim=True) # rescale the results from guidance (fixes overexposure) noise_pred_rescaled = noise_cfg * (std_text / std_cfg) # mix with the original results from guidance by factor guidance_rescale to avoid "plain looking" images noise_cfg = guidance_rescale * noise_pred_rescaled + (1 - guidance_rescale) * noise_cfg return noise_cfg