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README.md

@@ -1,13 +1,381 @@
----
-title: TCD
-emoji: 🔥
-colorFrom: green
-colorTo: blue
-sdk: gradio
-sdk_version: 4.19.2
-app_file: app.py
-pinned: false
-license: mit
----
-
-Check out the configuration reference at https://huggingface.co/docs/hub/spaces-config-reference
+# Trajectory Consistency Distillation
+
+[![Arxiv](https://img.shields.io/badge/arXiv-2211.15744-b31b1b)]()
+[![Project page](https://img.shields.io/badge/Web-Project%20Page-green)](https://mhh0318.github.io/tcd)
+[![Hugging Face Model](https://img.shields.io/badge/%F0%9F%A4%97HuggingFace-Model-purple)](https://huggingface.co/h1t/TCD-SDXL-LoRA)
+[![Hugging Face Space](https://img.shields.io/badge/%F0%9F%A4%97HuggingFace-Space-blue)](https://huggingface.co/spaces/h1t/TCD-SDXL-LoRA)
+
+Official Repository of the paper: [Trajectory Consistency Distillation]()
+
+![](./assets/teaser_fig.png)
+
+## 📣 News
+- (🔥New) 2024/2/29 We provided a demo of TCD on 🤗 Hugging Face Space. Try it out [here](https://huggingface.co/spaces/h1t/TCD-SDXL-LoRA).
+- (🔥New) 2024/2/29 We released our model [TCD-SDXL-Lora](https://huggingface.co/h1t/TCD-SDXL-LoRA) in 🤗 Hugging Face.
+- (🔥New) 2024/2/29 TCD is now integrated into the 🧨 Diffusers library. Please refer to the [Usage](#usage-anchor) for more information.
+
+## Introduction
+
+TCD, inspired by [Consistency Models](https://arxiv.org/abs/2303.01469), is a novel distillation technology that enables the distillation of knowledge from pre-trained diffusion models into a few-step sampler. In this repository, we release the inference code and our model named TCD-SDXL, which is distilled from [SDXL Base 1.0](https://huggingface.co/stabilityai/stable-diffusion-xl-base-1.0). We provide the LoRA checkpoint in this 🔥[repository](https://huggingface.co/h1t/TCD-SDXL-LoRA).
+
+⭐ TCD has following advantages:
+
+- `High-Quality with Few-Step`: TCD significantly surpasses the previous state-of-the-art few-step text-to-image model [LCM](https://github.com/luosiallen/latent-consistency-model/tree/main) in terms of image quality. Notably, LCM experiences a notable decline in quality at high NFEs. In contrast, _**TCD maintains superior generative quality at high NFEs, even exceeding the performance of DPM-Solver++(2S) with origin SDXL**_. 
+![](./assets/teaser.jpeg)
+<!-- We observed that the images generated with 8 steps by TCD-SDXL are already highly impressive, even outperforming the original SDXL 50-steps generation results. -->
+- `Versatility`: Integrated with LoRA technology, TCD can be directly applied to various models (including the custom Community Models, styled LoRA, ControlNet, IP-Adapter) that share the same backbone, as demonstrated in the [Usage](#usage-anchor).
+![](./assets/versatility.png)
+- `Avoiding Mode Collapse`: TCD achieves few-step generation without the need for adversarial training, thus circumventing mode collapse caused by the GAN objective. 
+In contrast to the concurrent work [SDXL-Lightning](https://huggingface.co/ByteDance/SDXL-Lightning), which relies on Adversarial Diffusion Distillation, TCD can synthesize results that are more realistic and slightly more diverse, without the presence of "Janus" artifacts.
+![](./assets/compare_sdxl_lightning.png)
+
+For more information, please refer to our paper [Trajectory Consistency Distillation]().
+
+<a id="usage-anchor"></a>
+
+## Usage
+To run the model yourself, you can leverage the 🧨 Diffusers library.
+```bash
+pip install diffusers transformers accelerate peft
+```
+And then we clone the repo.
+```bash
+git clone https://github.com/jabir-zheng/TCD.git
+cd TCD
+```
+Here, we demonstrate the applicability of our TCD LoRA to various models, including [SDXL](https://huggingface.co/stabilityai/stable-diffusion-xl-base-1.0), [SDXL Inpainting](https://huggingface.co/diffusers/stable-diffusion-xl-1.0-inpainting-0.1), a community model named [Animagine XL](https://huggingface.co/cagliostrolab/animagine-xl-3.0), a styled LoRA [Papercut](https://huggingface.co/TheLastBen/Papercut_SDXL), pretrained [Depth Controlnet](https://huggingface.co/diffusers/controlnet-depth-sdxl-1.0), [Canny Controlnet](https://huggingface.co/diffusers/controlnet-canny-sdxl-1.0) and [IP-Adapter](https://github.com/tencent-ailab/IP-Adapter) to accelerate image generation with high quality in few steps.
+
+### Text-to-Image generation
+```py
+import torch
+from diffusers import StableDiffusionXLPipeline
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+base_model_id = "stabilityai/stable-diffusion-xl-base-1.0"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+pipe = StableDiffusionXLPipeline.from_pretrained(base_model_id, torch_dtype=torch.float16, variant="fp16").to(device)
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+prompt = "Beautiful woman, bubblegum pink, lemon yellow, minty blue, futuristic, high-detail, epic composition, watercolor."
+
+image = pipe(
+    prompt=prompt,
+    num_inference_steps=4,
+    guidance_scale=0,
+    # Eta (referred to as `gamma` in the paper) is used to control the stochasticity in every step.
+    # A value of 0.3 often yields good results.
+    # We recommend using a higher eta when increasing the number of inference steps.
+    eta=0.3, 
+    generator=torch.Generator(device=device).manual_seed(0),
+).images[0]
+```
+![](./assets/t2i_tcd.png)
+
+### Inpainting
+```py
+import torch
+from diffusers import AutoPipelineForInpainting
+from diffusers.utils import load_image, make_image_grid
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+base_model_id = "diffusers/stable-diffusion-xl-1.0-inpainting-0.1"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+pipe = AutoPipelineForInpainting.from_pretrained(base_model_id, torch_dtype=torch.float16, variant="fp16").to(device)
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+img_url = "https://raw.githubusercontent.com/CompVis/latent-diffusion/main/data/inpainting_examples/overture-creations-5sI6fQgYIuo.png"
+mask_url = "https://raw.githubusercontent.com/CompVis/latent-diffusion/main/data/inpainting_examples/overture-creations-5sI6fQgYIuo_mask.png"
+
+init_image = load_image(img_url).resize((1024, 1024))
+mask_image = load_image(mask_url).resize((1024, 1024))
+
+prompt = "a tiger sitting on a park bench"
+
+image = pipe(
+  prompt=prompt,
+  image=init_image,
+  mask_image=mask_image,
+  num_inference_steps=8,
+  guidance_scale=0,
+  eta=0.3, # Eta (referred to as `gamma` in the paper) is used to control the stochasticity in every step. A value of 0.3 often yields good results.
+  strength=0.99,  # make sure to use `strength` below 1.0
+  generator=torch.Generator(device=device).manual_seed(0),
+).images[0]
+
+grid_image = make_image_grid([init_image, mask_image, image], rows=1, cols=3)
+```
+![](./assets/inpainting_tcd.png)
+
+### Versatile for Community Models
+```py
+import torch
+from diffusers import StableDiffusionXLPipeline
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+base_model_id = "cagliostrolab/animagine-xl-3.0"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+pipe = StableDiffusionXLPipeline.from_pretrained(base_model_id, torch_dtype=torch.float16, variant="fp16").to(device)
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+prompt = "A man, clad in a meticulously tailored military uniform, stands with unwavering resolve. The uniform boasts intricate details, and his eyes gleam with determination. Strands of vibrant, windswept hair peek out from beneath the brim of his cap."
+
+image = pipe(
+    prompt=prompt,
+    num_inference_steps=8,
+    guidance_scale=0,
+    # Eta (referred to as `gamma` in the paper) is used to control the stochasticity in every step.
+    # A value of 0.3 often yields good results.
+    # We recommend using a higher eta when increasing the number of inference steps.
+    eta=0.3, 
+    generator=torch.Generator(device=device).manual_seed(0),
+).images[0]
+```
+![](./assets/animagine_xl.png)
+
+### Combine with styled LoRA
+```py
+import torch
+from diffusers import StableDiffusionXLPipeline
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+base_model_id = "stabilityai/stable-diffusion-xl-base-1.0"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+styled_lora_id = "TheLastBen/Papercut_SDXL"
+
+pipe = StableDiffusionXLPipeline.from_pretrained(base_model_id, torch_dtype=torch.float16, variant="fp16").to(device)
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id, adapter_name="tcd")
+pipe.load_lora_weights(styled_lora_id, adapter_name="style")
+pipe.set_adapters(["tcd", "style"], adapter_weights=[1.0, 1.0])
+
+prompt = "papercut of a winter mountain, snow"
+
+image = pipe(
+    prompt=prompt,
+    num_inference_steps=4,
+    guidance_scale=0,
+    # Eta (referred to as `gamma` in the paper) is used to control the stochasticity in every step.
+    # A value of 0.3 often yields good results.
+    # We recommend using a higher eta when increasing the number of inference steps.
+    eta=0.3, 
+    generator=torch.Generator(device=device).manual_seed(0),
+).images[0]
+```
+![](./assets/styled_lora.png)
+
+### Compatibility with ControlNet
+#### Depth ControlNet
+```py
+import torch
+import numpy as np
+from PIL import Image
+from transformers import DPTFeatureExtractor, DPTForDepthEstimation
+from diffusers import ControlNetModel, StableDiffusionXLControlNetPipeline
+from diffusers.utils import load_image, make_image_grid
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+depth_estimator = DPTForDepthEstimation.from_pretrained("Intel/dpt-hybrid-midas").to(device)
+feature_extractor = DPTFeatureExtractor.from_pretrained("Intel/dpt-hybrid-midas")
+
+def get_depth_map(image):
+    image = feature_extractor(images=image, return_tensors="pt").pixel_values.to(device)
+    with torch.no_grad(), torch.autocast(device):
+        depth_map = depth_estimator(image).predicted_depth
+
+    depth_map = torch.nn.functional.interpolate(
+        depth_map.unsqueeze(1),
+        size=(1024, 1024),
+        mode="bicubic",
+        align_corners=False,
+    )
+    depth_min = torch.amin(depth_map, dim=[1, 2, 3], keepdim=True)
+    depth_max = torch.amax(depth_map, dim=[1, 2, 3], keepdim=True)
+    depth_map = (depth_map - depth_min) / (depth_max - depth_min)
+    image = torch.cat([depth_map] * 3, dim=1)
+
+    image = image.permute(0, 2, 3, 1).cpu().numpy()[0]
+    image = Image.fromarray((image * 255.0).clip(0, 255).astype(np.uint8))
+    return image
+
+base_model_id = "stabilityai/stable-diffusion-xl-base-1.0"
+controlnet_id = "diffusers/controlnet-depth-sdxl-1.0"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+controlnet = ControlNetModel.from_pretrained(
+    controlnet_id,
+    torch_dtype=torch.float16,
+    variant="fp16",
+).to(device)
+pipe = StableDiffusionXLControlNetPipeline.from_pretrained(
+    base_model_id,
+    controlnet=controlnet,
+    torch_dtype=torch.float16,
+    variant="fp16",
+).to(device)
+pipe.enable_model_cpu_offload()
+
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+prompt = "stormtrooper lecture, photorealistic"
+
+image = load_image("https://huggingface.co/lllyasviel/sd-controlnet-depth/resolve/main/images/stormtrooper.png")
+depth_image = get_depth_map(image)
+
+controlnet_conditioning_scale = 0.5  # recommended for good generalization
+
+image = pipe(
+    prompt, 
+    image=depth_image, 
+    num_inference_steps=4, 
+    guidance_scale=0,
+    eta=0.3, # A parameter (referred to as `gamma` in the paper) is used to control the stochasticity in every step. A value of 0.3 often yields good results.
+    controlnet_conditioning_scale=controlnet_conditioning_scale,
+    generator=torch.Generator(device=device).manual_seed(0),
+).images[0]
+
+grid_image = make_image_grid([depth_image, image], rows=1, cols=2)
+```
+![](./assets/controlnet_depth_tcd.png)
+
+#### Canny ControlNet
+```py
+import torch
+from diffusers import ControlNetModel, StableDiffusionXLControlNetPipeline
+from diffusers.utils import load_image, make_image_grid
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+base_model_id = "stabilityai/stable-diffusion-xl-base-1.0"
+controlnet_id = "diffusers/controlnet-canny-sdxl-1.0"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+controlnet = ControlNetModel.from_pretrained(
+    controlnet_id,
+    torch_dtype=torch.float16,
+    variant="fp16",
+).to(device)
+pipe = StableDiffusionXLControlNetPipeline.from_pretrained(
+    base_model_id,
+    controlnet=controlnet,
+    torch_dtype=torch.float16,
+    variant="fp16",
+).to(device)
+pipe.enable_model_cpu_offload()
+
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+prompt = "ultrarealistic shot of a furry blue bird"
+
+canny_image = load_image("https://huggingface.co/datasets/hf-internal-testing/diffusers-images/resolve/main/sd_controlnet/bird_canny.png")
+
+controlnet_conditioning_scale = 0.5  # recommended for good generalization
+
+image = pipe(
+    prompt, 
+    image=canny_image, 
+    num_inference_steps=4, 
+    guidance_scale=0,
+    eta=0.3, # A parameter (referred to as `gamma` in the paper) is used to control the stochasticity in every step. A value of 0.3 often yields good results.
+    controlnet_conditioning_scale=controlnet_conditioning_scale,
+    generator=torch.Generator(device=device).manual_seed(0),
+).images[0]
+
+grid_image = make_image_grid([canny_image, image], rows=1, cols=2)
+```
+
+![](./assets/controlnet_canny_tcd.png)
+
+### Compatibility with IP-Adapter
+⚠️ Please refer to the official [repository](https://github.com/tencent-ailab/IP-Adapter/tree/main) for instructions on installing dependencies for IP-Adapter.
+```py
+import torch
+from diffusers import StableDiffusionXLPipeline
+from diffusers.utils import load_image, make_image_grid
+
+from ip_adapter import IPAdapterXL
+from scheduling_tcd import TCDScheduler 
+
+device = "cuda"
+base_model_path = "stabilityai/stable-diffusion-xl-base-1.0"
+image_encoder_path = "sdxl_models/image_encoder"
+ip_ckpt = "sdxl_models/ip-adapter_sdxl.bin"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+pipe = StableDiffusionXLPipeline.from_pretrained(
+    base_model_path, 
+    torch_dtype=torch.float16, 
+    variant="fp16"
+)
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+ip_model = IPAdapterXL(pipe, image_encoder_path, ip_ckpt, device)
+
+ref_image = load_image("https://raw.githubusercontent.com/tencent-ailab/IP-Adapter/main/assets/images/woman.png").resize((512, 512))
+
+prompt = "best quality, high quality, wearing sunglasses"
+
+image = ip_model.generate(
+    pil_image=ref_image, 
+    prompt=prompt,
+    scale=0.5,
+    num_samples=1, 
+    num_inference_steps=4, 
+    guidance_scale=0,
+    eta=0.3, # A parameter (referred to as `gamma` in the paper) is used to control the stochasticity in every step. A value of 0.3 often yields good results.
+    seed=0,
+)[0]
+
+grid_image = make_image_grid([ref_image, image], rows=1, cols=2)
+```
+![](./assets/ip_adapter.png)
+
+### Local Gradio Demo
+Install the `gradio` library first,
+```bash
+pip install gradio==3.50.2
+```
+then local gradio demo can be launched by:
+```py
+python gradio_app.py
+```
+![](./assets/gradio_demo.png)
+
+## Citation
+```bibtex
+@article{zheng2024trajectory,
+  title = {Trajectory Consistency Distillation},
+  author = {Zheng, Jianbin and Hu, Minghui and Fan, Zhongyi and Wang, Chaoyue and Ding, Changxing and Tao, Dacheng and Cham, Tat-Jen},
+  journal = {arXiv},
+  year = {2024},
+}
+```
+
+## Acknowledgments
+This codebase heavily relies on the 🤗[Diffusers](https://github.com/huggingface/diffusers) library and [LCM](https://github.com/luosiallen/latent-consistency-model).

gradio_app.py

@@ -0,0 +1,117 @@
+import random
+
+import gradio as gr
+import torch
+from diffusers import StableDiffusionXLPipeline
+
+from scheduling_tcd import TCDScheduler
+
+device = "cuda"
+base_model_id = "stabilityai/stable-diffusion-xl-base-1.0"
+tcd_lora_id = "h1t/TCD-SDXL-LoRA"
+
+pipe = StableDiffusionXLPipeline.from_pretrained(
+    base_model_id,
+    torch_dtype=torch.float16,
+    variant="fp16"
+).to(device)
+pipe.scheduler = TCDScheduler.from_config(pipe.scheduler.config)
+
+pipe.load_lora_weights(tcd_lora_id)
+pipe.fuse_lora()
+
+
+def inference(prompt, num_inference_steps=4, seed=-1, eta=0.3):
+    if seed is None or seed == '' or seed == -1:
+        seed = int(random.randrange(4294967294))
+    generator = torch.Generator(device=device).manual_seed(int(seed))
+    image = pipe(
+        prompt=prompt,
+        num_inference_steps=num_inference_steps,
+        guidance_scale=0,
+        eta=eta,
+        generator=generator,
+    ).images[0]
+    return image
+
+
+# Define style
+title = "<h1 style='text-align: center'>Trajectory Consistency Distillation</h1>"
+description = "Official 🤗 Gradio demo for Trajectory Consistency Distillation"
+article = "<p style='text-align: center'><a href='https://arxiv.org/abs/' target='_blank'>Trajectory Consistency Distillation</a> | <a href='https://github.com/jabir-zheng/TCD' target='_blank'>Github Repo</a></p>"
+
+
+default_prompt = "Painting of the orange cat Otto von Garfield, Count of Bismarck-Schönhausen, Duke of Lauenburg, Minister-President of Prussia. Depicted wearing a Prussian Pickelhaube and eating his favorite meal - lasagna."
+examples = [
+    [
+        "Beautiful woman, bubblegum pink, lemon yellow, minty blue, futuristic, high-detail, epic composition, watercolor.",
+        4
+    ],
+    [
+        "Beautiful man, bubblegum pink, lemon yellow, minty blue, futuristic, high-detail, epic composition, watercolor.",
+        8
+    ],
+    [
+        "Painting of the orange cat Otto von Garfield, Count of Bismarck-Schönhausen, Duke of Lauenburg, Minister-President of Prussia. Depicted wearing a Prussian Pickelhaube and eating his favorite meal - lasagna.",
+        16
+    ],
+    [
+        "closeup portrait of 1 Persian princess, royal clothing, makeup, jewelry, wind-blown long hair, symmetric, desert, sands, dusty and foggy, sand storm, winds bokeh, depth of field, centered.",
+        16
+    ],
+]
+
+outputs = gr.Label(label='Generated Images')
+
+with gr.Blocks() as demo:
+    gr.Markdown(f'# {title}\n### {description}')
+    
+    with gr.Row():
+        with gr.Column():
+            prompt = gr.Textbox(label='Prompt', value=default_prompt)
+            num_inference_steps = gr.Slider(
+                label='Inference steps',
+                minimum=4,
+                maximum=16,
+                value=4,
+                step=1,
+            )
+            
+            with gr.Accordion("Advanced Options", visible=False):
+                with gr.Row():
+                    with gr.Column():
+                        seed = gr.Number(label="Random Seed", value=-1)
+                    with gr.Column():
+                        eta = gr.Slider(
+                                label='Gamma',
+                                minimum=0.,
+                                maximum=1.,
+                                value=0.3,
+                                step=0.1,
+                            )
+
+            with gr.Row():
+                clear = gr.ClearButton(
+                    components=[prompt, num_inference_steps, seed, eta])
+                submit = gr.Button(value='Submit')
+
+            examples = gr.Examples(
+                label="Quick Examples",
+                examples=examples,
+                inputs=[prompt, num_inference_steps, 0, 0.3],
+                outputs="outputs",  # 适当调整此处
+                cache_examples=False
+            )
+
+        with gr.Column():
+            outputs = gr.Image(label='Generated Images')
+
+    gr.Markdown(f'{article}')
+
+    submit.click(
+        fn=inference,
+        inputs=[prompt, num_inference_steps, seed, eta],
+        outputs=outputs,
+    )
+
+demo.launch()

scheduling_tcd.py

@@ -0,0 +1,657 @@
+# Copyright 2023 Stanford University Team and The HuggingFace Team. All rights reserved.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+# DISCLAIMER: This code is strongly influenced by https://github.com/pesser/pytorch_diffusion
+# and https://github.com/hojonathanho/diffusion
+
+import math
+from dataclasses import dataclass
+from typing import List, Optional, Tuple, Union
+
+import numpy as np
+import torch
+
+from diffusers.configuration_utils import ConfigMixin, register_to_config
+from diffusers.utils import BaseOutput, logging
+from diffusers.utils.torch_utils import randn_tensor
+from diffusers.schedulers.scheduling_utils import SchedulerMixin
+
+
+logger = logging.get_logger(__name__)  # pylint: disable=invalid-name
+
+
+@dataclass
+class TCDSchedulerOutput(BaseOutput):
+    """
+    Output class for the scheduler's `step` function output.
+
+    Args:
+        prev_sample (`torch.FloatTensor` of shape `(batch_size, num_channels, height, width)` for images):
+            Computed sample `(x_{t-1})` of previous timestep. `prev_sample` should be used as next model input in the
+            denoising loop.
+        pred_noised_sample (`torch.FloatTensor` of shape `(batch_size, num_channels, height, width)` for images):
+            The predicted noised sample `(x_{s})` based on the model output from the current timestep.
+    """
+
+    prev_sample: torch.FloatTensor
+    pred_noised_sample: Optional[torch.FloatTensor] = None
+
+
+# Copied from diffusers.schedulers.scheduling_ddpm.betas_for_alpha_bar
+def betas_for_alpha_bar(
+    num_diffusion_timesteps,
+    max_beta=0.999,
+    alpha_transform_type="cosine",
+):
+    """
+    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].
+
+    Contains a function alpha_bar that takes an argument t and transforms it to the cumulative product of (1-beta) up
+    to that part of the diffusion process.
+
+
+    Args:
+        num_diffusion_timesteps (`int`): the number of betas to produce.
+        max_beta (`float`): the maximum beta to use; use values lower than 1 to
+                     prevent singularities.
+        alpha_transform_type (`str`, *optional*, default to `cosine`): the type of noise schedule for alpha_bar.
+                     Choose from `cosine` or `exp`
+
+    Returns:
+        betas (`np.ndarray`): the betas used by the scheduler to step the model outputs
+    """
+    if alpha_transform_type == "cosine":
+
+        def alpha_bar_fn(t):
+            return math.cos((t + 0.008) / 1.008 * math.pi / 2) ** 2
+
+    elif alpha_transform_type == "exp":
+
+        def alpha_bar_fn(t):
+            return math.exp(t * -12.0)
+
+    else:
+        raise ValueError(f"Unsupported alpha_tranform_type: {alpha_transform_type}")
+
+    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_fn(t2) / alpha_bar_fn(t1), max_beta))
+    return torch.tensor(betas, dtype=torch.float32)
+
+
+# Copied from diffusers.schedulers.scheduling_ddim.rescale_zero_terminal_snr
+def rescale_zero_terminal_snr(betas: torch.FloatTensor) -> torch.FloatTensor:
+    """
+    Rescales betas to have zero terminal SNR Based on https://arxiv.org/pdf/2305.08891.pdf (Algorithm 1)
+
+
+    Args:
+        betas (`torch.FloatTensor`):
+            the betas that the scheduler is being initialized with.
+
+    Returns:
+        `torch.FloatTensor`: rescaled betas with zero terminal SNR
+    """
+    # Convert betas to alphas_bar_sqrt
+    alphas = 1.0 - betas
+    alphas_cumprod = torch.cumprod(alphas, dim=0)
+    alphas_bar_sqrt = alphas_cumprod.sqrt()
+
+    # Store old values.
+    alphas_bar_sqrt_0 = alphas_bar_sqrt[0].clone()
+    alphas_bar_sqrt_T = alphas_bar_sqrt[-1].clone()
+
+    # 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 = torch.cat([alphas_bar[0:1], alphas])
+    betas = 1 - alphas
+
+    return betas
+
+
+class TCDScheduler(SchedulerMixin, ConfigMixin):
+    """
+    `TCDScheduler` incorporates the `Strategic Stochastic Sampling` introduced by the paper `Trajectory Consistency Distillation`, 
+    extending the original Multistep Consistency Sampling to enable unrestricted trajectory traversal.
+
+    This model inherits from [`SchedulerMixin`] and [`ConfigMixin`]. [`~ConfigMixin`] takes care of storing all config
+    attributes that are passed in the scheduler's `__init__` function, such as `num_train_timesteps`. They can be
+    accessed via `scheduler.config.num_train_timesteps`. [`SchedulerMixin`] provides general loading and saving
+    functionality via the [`SchedulerMixin.save_pretrained`] and [`~SchedulerMixin.from_pretrained`] functions.
+
+    Args:
+        num_train_timesteps (`int`, defaults to 1000):
+            The number of diffusion steps to train the model.
+        beta_start (`float`, defaults to 0.0001):
+            The starting `beta` value of inference.
+        beta_end (`float`, defaults to 0.02):
+            The final `beta` value.
+        beta_schedule (`str`, defaults to `"linear"`):
+            The beta schedule, a mapping from a beta range to a sequence of betas for stepping the model. Choose from
+            `linear`, `scaled_linear`, or `squaredcos_cap_v2`.
+        trained_betas (`np.ndarray`, *optional*):
+            Pass an array of betas directly to the constructor to bypass `beta_start` and `beta_end`.
+        original_inference_steps (`int`, *optional*, defaults to 50):
+            The default number of inference steps used to generate a linearly-spaced timestep schedule, from which we
+            will ultimately take `num_inference_steps` evenly spaced timesteps to form the final timestep schedule.
+        clip_sample (`bool`, defaults to `True`):
+            Clip the predicted sample for numerical stability.
+        clip_sample_range (`float`, defaults to 1.0):
+            The maximum magnitude for sample clipping. Valid only when `clip_sample=True`.
+        set_alpha_to_one (`bool`, defaults to `True`):
+            Each diffusion step uses the alphas product value at that step and at the previous one. For the final step
+            there is no previous alpha. When this option is `True` the previous alpha product is fixed to `1`,
+            otherwise it uses the alpha value at step 0.
+        steps_offset (`int`, defaults to 0):
+            An offset added to the inference steps. You can use a combination of `offset=1` and
+            `set_alpha_to_one=False` to make the last step use step 0 for the previous alpha product like in Stable
+            Diffusion.
+        prediction_type (`str`, defaults to `epsilon`, *optional*):
+            Prediction type of the scheduler function; can be `epsilon` (predicts the noise of the diffusion process),
+            `sample` (directly predicts the noisy sample`) or `v_prediction` (see section 2.4 of [Imagen
+            Video](https://imagen.research.google/video/paper.pdf) paper).
+        thresholding (`bool`, defaults to `False`):
+            Whether to use the "dynamic thresholding" method. This is unsuitable for latent-space diffusion models such
+            as Stable Diffusion.
+        dynamic_thresholding_ratio (`float`, defaults to 0.995):
+            The ratio for the dynamic thresholding method. Valid only when `thresholding=True`.
+        sample_max_value (`float`, defaults to 1.0):
+            The threshold value for dynamic thresholding. Valid only when `thresholding=True`.
+        timestep_spacing (`str`, defaults to `"leading"`):
+            The way the timesteps should be scaled. Refer to Table 2 of the [Common Diffusion Noise Schedules and
+            Sample Steps are Flawed](https://huggingface.co/papers/2305.08891) for more information.
+        timestep_scaling (`float`, defaults to 10.0):
+            The factor the timesteps will be multiplied by when calculating the consistency model boundary conditions
+            `c_skip` and `c_out`. Increasing this will decrease the approximation error (although the approximation
+            error at the default of `10.0` is already pretty small).
+        rescale_betas_zero_snr (`bool`, defaults to `False`):
+            Whether to rescale the betas to have zero terminal SNR. This enables the model to generate very bright and
+            dark samples instead of limiting it to samples with medium brightness. Loosely related to
+            [`--offset_noise`](https://github.com/huggingface/diffusers/blob/74fd735eb073eb1d774b1ab4154a0876eb82f055/examples/dreambooth/train_dreambooth.py#L506).
+    """
+
+    order = 1
+
+    @register_to_config
+    def __init__(
+        self,
+        num_train_timesteps: int = 1000,
+        beta_start: float = 0.00085,
+        beta_end: float = 0.012,
+        beta_schedule: str = "scaled_linear",
+        trained_betas: Optional[Union[np.ndarray, List[float]]] = None,
+        original_inference_steps: int = 50,
+        clip_sample: bool = False,
+        clip_sample_range: float = 1.0,
+        set_alpha_to_one: bool = True,
+        steps_offset: int = 0,
+        prediction_type: str = "epsilon",
+        thresholding: bool = False,
+        dynamic_thresholding_ratio: float = 0.995,
+        sample_max_value: float = 1.0,
+        timestep_spacing: str = "leading",
+        timestep_scaling: float = 10.0,
+        rescale_betas_zero_snr: bool = False,
+    ):
+        if trained_betas is not None:
+            self.betas = torch.tensor(trained_betas, dtype=torch.float32)
+        elif beta_schedule == "linear":
+            self.betas = torch.linspace(beta_start, beta_end, num_train_timesteps, dtype=torch.float32)
+        elif beta_schedule == "scaled_linear":
+            # this schedule is very specific to the latent diffusion model.
+            self.betas = torch.linspace(beta_start**0.5, beta_end**0.5, num_train_timesteps, dtype=torch.float32) ** 2
+        elif beta_schedule == "squaredcos_cap_v2":
+            # Glide cosine schedule
+            self.betas = betas_for_alpha_bar(num_train_timesteps)
+        else:
+            raise NotImplementedError(f"{beta_schedule} does is not implemented for {self.__class__}")
+
+        # Rescale for zero SNR
+        if rescale_betas_zero_snr:
+            self.betas = rescale_zero_terminal_snr(self.betas)
+
+        self.alphas = 1.0 - self.betas
+        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)
+
+        # At every step in ddim, we are looking into the previous alphas_cumprod
+        # For the final step, there is no previous alphas_cumprod because we are already at 0
+        # `set_alpha_to_one` decides whether we set this parameter simply to one or
+        # whether we use the final alpha of the "non-previous" one.
+        self.final_alpha_cumprod = torch.tensor(1.0) if set_alpha_to_one else self.alphas_cumprod[0]
+
+        # standard deviation of the initial noise distribution
+        self.init_noise_sigma = 1.0
+
+        # setable values
+        self.num_inference_steps = None
+        self.timesteps = torch.from_numpy(np.arange(0, num_train_timesteps)[::-1].copy().astype(np.int64))
+        self.custom_timesteps = False
+
+        self._step_index = None
+
+    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._init_step_index
+    def _init_step_index(self, timestep):
+        if isinstance(timestep, torch.Tensor):
+            timestep = timestep.to(self.timesteps.device)
+
+        index_candidates = (self.timesteps == timestep).nonzero()
+
+        # The sigma index that is taken for the **very** first `step`
+        # is always the second index (or the last index if there is only 1)
+        # This way we can ensure we don't accidentally skip a sigma in
+        # case we start in the middle of the denoising schedule (e.g. for image-to-image)
+        if len(index_candidates) > 1:
+            step_index = index_candidates[1]
+        else:
+            step_index = index_candidates[0]
+
+        self._step_index = step_index.item()
+
+    @property
+    def step_index(self):
+        return self._step_index
+
+    def scale_model_input(self, sample: torch.FloatTensor, timestep: Optional[int] = None) -> torch.FloatTensor:
+        """
+        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
+        current timestep.
+
+        Args:
+            sample (`torch.FloatTensor`):
+                The input sample.
+            timestep (`int`, *optional*):
+                The current timestep in the diffusion chain.
+        Returns:
+            `torch.FloatTensor`:
+                A scaled input sample.
+        """
+        return sample
+
+    def _get_variance(self, timestep, prev_timestep):
+        alpha_prod_t = self.alphas_cumprod[timestep]
+        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
+        beta_prod_t = 1 - alpha_prod_t
+        beta_prod_t_prev = 1 - alpha_prod_t_prev
+
+        variance = (beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)
+
+        return variance
+
+    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler._threshold_sample
+    def _threshold_sample(self, sample: torch.FloatTensor) -> torch.FloatTensor:
+        """
+        "Dynamic thresholding: At each sampling step we set s to a certain percentile absolute pixel value in xt0 (the
+        prediction of x_0 at timestep t), and if s > 1, then we threshold xt0 to the range [-s, s] and then divide by
+        s. Dynamic thresholding pushes saturated pixels (those near -1 and 1) inwards, thereby actively preventing
+        pixels from saturation at each step. We find that dynamic thresholding results in significantly better
+        photorealism as well as better image-text alignment, especially when using very large guidance weights."
+
+        https://arxiv.org/abs/2205.11487
+        """
+        dtype = sample.dtype
+        batch_size, channels, *remaining_dims = sample.shape
+
+        if dtype not in (torch.float32, torch.float64):
+            sample = sample.float()  # upcast for quantile calculation, and clamp not implemented for cpu half
+
+        # Flatten sample for doing quantile calculation along each image
+        sample = sample.reshape(batch_size, channels * np.prod(remaining_dims))
+
+        abs_sample = sample.abs()  # "a certain percentile absolute pixel value"
+
+        s = torch.quantile(abs_sample, self.config.dynamic_thresholding_ratio, dim=1)
+        s = torch.clamp(
+            s, min=1, max=self.config.sample_max_value
+        )  # When clamped to min=1, equivalent to standard clipping to [-1, 1]
+        s = s.unsqueeze(1)  # (batch_size, 1) because clamp will broadcast along dim=0
+        sample = torch.clamp(sample, -s, s) / s  # "we threshold xt0 to the range [-s, s] and then divide by s"
+
+        sample = sample.reshape(batch_size, channels, *remaining_dims)
+        sample = sample.to(dtype)
+
+        return sample
+
+    def set_timesteps(
+        self,
+        num_inference_steps: Optional[int] = None,
+        device: Union[str, torch.device] = None,
+        original_inference_steps: Optional[int] = None,
+        timesteps: Optional[List[int]] = None,
+        strength: int = 1.0,
+    ):
+        """
+        Sets the discrete timesteps used for the diffusion chain (to be run before inference).
+
+        Args:
+            num_inference_steps (`int`, *optional*):
+                The number of diffusion steps used when generating samples with a pre-trained model. If used,
+                `timesteps` must be `None`.
+            device (`str` or `torch.device`, *optional*):
+                The device to which the timesteps should be moved to. If `None`, the timesteps are not moved.
+            original_inference_steps (`int`, *optional*):
+                The original number of inference steps, which will be used to generate a linearly-spaced timestep
+                schedule (which is different from the standard `diffusers` implementation). We will then take
+                `num_inference_steps` timesteps from this schedule, evenly spaced in terms of indices, and use that as
+                our final timestep schedule. If not set, this will default to the `original_inference_steps` attribute.
+            timesteps (`List[int]`, *optional*):
+                Custom timesteps used to support arbitrary spacing between timesteps. If `None`, then the default
+                timestep spacing strategy of equal spacing between timesteps on the training/distillation timestep
+                schedule is used. If `timesteps` is passed, `num_inference_steps` must be `None`.
+        """
+        # 0. Check inputs
+        if num_inference_steps is None and timesteps is None:
+            raise ValueError("Must pass exactly one of `num_inference_steps` or `custom_timesteps`.")
+
+        if num_inference_steps is not None and timesteps is not None:
+            raise ValueError("Can only pass one of `num_inference_steps` or `custom_timesteps`.")
+
+        # 1. Calculate the TCD original training/distillation timestep schedule.
+        original_steps = (
+            original_inference_steps if original_inference_steps is not None else self.config.original_inference_steps
+        )
+        
+        if original_steps is not None:
+            if original_steps > self.config.num_train_timesteps:
+                raise ValueError(
+                    f"`original_steps`: {original_steps} cannot be larger than `self.config.train_timesteps`:"
+                    f" {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle"
+                    f" maximal {self.config.num_train_timesteps} timesteps."
+                )
+            # TCD Timesteps Setting
+            # The skipping step parameter k from the paper.
+            k = self.config.num_train_timesteps // original_steps
+            # TCD Training/Distillation Steps Schedule
+            tcd_origin_timesteps = np.asarray(list(range(1, int(original_steps * strength) + 1))) * k - 1
+        else:
+            tcd_origin_timesteps = np.asarray(list(range(0, int(self.config.num_train_timesteps * strength))))
+
+        # 2. Calculate the TCD inference timestep schedule.
+        if timesteps is not None:
+            # 2.1 Handle custom timestep schedules.
+            train_timesteps = set(tcd_origin_timesteps)
+            non_train_timesteps = []
+            for i in range(1, len(timesteps)):
+                if timesteps[i] >= timesteps[i - 1]:
+                    raise ValueError("`custom_timesteps` must be in descending order.")
+
+                if timesteps[i] not in train_timesteps:
+                    non_train_timesteps.append(timesteps[i])
+
+            if timesteps[0] >= self.config.num_train_timesteps:
+                raise ValueError(
+                    f"`timesteps` must start before `self.config.train_timesteps`:"
+                    f" {self.config.num_train_timesteps}."
+                )
+
+            # Raise warning if timestep schedule does not start with self.config.num_train_timesteps - 1
+            if strength == 1.0 and timesteps[0] != self.config.num_train_timesteps - 1:
+                logger.warning(
+                    f"The first timestep on the custom timestep schedule is {timesteps[0]}, not"
+                    f" `self.config.num_train_timesteps - 1`: {self.config.num_train_timesteps - 1}. You may get"
+                    f" unexpected results when using this timestep schedule."
+                )
+
+            # Raise warning if custom timestep schedule contains timesteps not on original timestep schedule
+            if non_train_timesteps:
+                logger.warning(
+                    f"The custom timestep schedule contains the following timesteps which are not on the original"
+                    f" training/distillation timestep schedule: {non_train_timesteps}. You may get unexpected results"
+                    f" when using this timestep schedule."
+                )
+
+            # Raise warning if custom timestep schedule is longer than original_steps
+            if original_steps is not None:
+                if len(timesteps) > original_steps:
+                    logger.warning(
+                        f"The number of timesteps in the custom timestep schedule is {len(timesteps)}, which exceeds the"
+                        f" the length of the timestep schedule used for training: {original_steps}. You may get some"
+                        f" unexpected results when using this timestep schedule."
+                    )
+            else:
+                if len(timesteps) > self.config.num_train_timesteps:
+                    logger.warning(
+                        f"The number of timesteps in the custom timestep schedule is {len(timesteps)}, which exceeds the"
+                        f" the length of the timestep schedule used for training: {self.config.num_train_timesteps}. You may get some"
+                        f" unexpected results when using this timestep schedule."
+                    )
+
+            timesteps = np.array(timesteps, dtype=np.int64)
+            self.num_inference_steps = len(timesteps)
+            self.custom_timesteps = True
+
+            # Apply strength (e.g. for img2img pipelines) (see StableDiffusionImg2ImgPipeline.get_timesteps)
+            init_timestep = min(int(self.num_inference_steps * strength), self.num_inference_steps)
+            t_start = max(self.num_inference_steps - init_timestep, 0)
+            timesteps = timesteps[t_start * self.order :]
+            # TODO: also reset self.num_inference_steps?
+        else:
+            # 2.2 Create the "standard" TCD inference timestep schedule.
+            if num_inference_steps > self.config.num_train_timesteps:
+                raise ValueError(
+                    f"`num_inference_steps`: {num_inference_steps} cannot be larger than `self.config.train_timesteps`:"
+                    f" {self.config.num_train_timesteps} as the unet model trained with this scheduler can only handle"
+                    f" maximal {self.config.num_train_timesteps} timesteps."
+                )
+
+            if original_steps is not None:
+                skipping_step = len(tcd_origin_timesteps) // num_inference_steps
+
+                if skipping_step < 1:
+                    raise ValueError(
+                        f"The combination of `original_steps x strength`: {original_steps} x {strength} is smaller than `num_inference_steps`: {num_inference_steps}. Make sure to either reduce `num_inference_steps` to a value smaller than {int(original_steps * strength)} or increase `strength` to a value higher than {float(num_inference_steps / original_steps)}."
+                    )
+
+            self.num_inference_steps = num_inference_steps
+
+            if original_steps is not None:
+                if num_inference_steps > original_steps:
+                    raise ValueError(
+                        f"`num_inference_steps`: {num_inference_steps} cannot be larger than `original_inference_steps`:"
+                        f" {original_steps} because the final timestep schedule will be a subset of the"
+                        f" `original_inference_steps`-sized initial timestep schedule."
+                    )
+            else:
+                if num_inference_steps > self.config.num_train_timesteps:
+                    raise ValueError(
+                        f"`num_inference_steps`: {num_inference_steps} cannot be larger than `num_train_timesteps`:"
+                        f" {self.config.num_train_timesteps} because the final timestep schedule will be a subset of the"
+                        f" `num_train_timesteps`-sized initial timestep schedule."
+                    )
+
+            # TCD Inference Steps Schedule
+            tcd_origin_timesteps = tcd_origin_timesteps[::-1].copy()
+            # Select (approximately) evenly spaced indices from tcd_origin_timesteps.
+            inference_indices = np.linspace(0, len(tcd_origin_timesteps), num=num_inference_steps, endpoint=False)
+            inference_indices = np.floor(inference_indices).astype(np.int64)
+            timesteps = tcd_origin_timesteps[inference_indices]
+
+        self.timesteps = torch.from_numpy(timesteps).to(device=device, dtype=torch.long)
+
+        self._step_index = None
+
+    def step(
+        self,
+        model_output: torch.FloatTensor,
+        timestep: int,
+        sample: torch.FloatTensor,
+        eta: float,
+        generator: Optional[torch.Generator] = None,
+        return_dict: bool = True,
+    ) -> Union[TCDSchedulerOutput, Tuple]:
+        """
+        Predict the sample from the previous timestep by reversing the SDE. This function propagates the diffusion
+        process from the learned model outputs (most often the predicted noise).
+
+        Args:
+            model_output (`torch.FloatTensor`):
+                The direct output from learned diffusion model.
+            timestep (`int`):
+                The current discrete timestep in the diffusion chain.
+            sample (`torch.FloatTensor`):
+                A current instance of a sample created by the diffusion process.
+            eta (`float`):
+                A stochastic parameter (referred to as `gamma` in the paper) used to control the stochasticity in every step. 
+                When eta = 0, it represents deterministic sampling, whereas eta = 1 indicates full stochastic sampling.
+            generator (`torch.Generator`, *optional*):
+                A random number generator.
+            return_dict (`bool`, *optional*, defaults to `True`):
+                Whether or not to return a [`~schedulers.scheduling_tcd.TCDSchedulerOutput`] or `tuple`.
+        Returns:
+            [`~schedulers.scheduling_utils.TCDSchedulerOutput`] or `tuple`:
+                If return_dict is `True`, [`~schedulers.scheduling_tcd.TCDSchedulerOutput`] is returned, otherwise a
+                tuple is returned where the first element is the sample tensor.
+        """
+        if self.num_inference_steps is None:
+            raise ValueError(
+                "Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"
+            )
+
+        if self.step_index is None:
+            self._init_step_index(timestep)
+
+        # 1. get previous step value
+        prev_step_index = self.step_index + 1
+        if prev_step_index < len(self.timesteps):
+            prev_timestep = self.timesteps[prev_step_index]
+        else:
+            prev_timestep = torch.tensor(0)
+
+        timestep_s = torch.floor((1 - eta) * prev_timestep).to(dtype=torch.long)
+        
+        # 2. compute alphas, betas
+        alpha_prod_t = self.alphas_cumprod[timestep]
+        beta_prod_t = 1 - alpha_prod_t
+
+        alpha_prod_t_prev = self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod
+        beta_prod_t_prev = 1 - alpha_prod_t_prev
+        
+        alpha_prod_s = self.alphas_cumprod[timestep_s] if timestep_s >= 0 else self.final_alpha_cumprod
+        beta_prod_s = 1 - alpha_prod_s
+
+        # 3. Compute the predicted noised sample x_s based on the model parameterization
+        if self.config.prediction_type == "epsilon":  # noise-prediction
+            pred_original_sample = (sample - beta_prod_t.sqrt() * model_output) / alpha_prod_t.sqrt()
+            pred_epsilon = model_output
+            pred_noised_sample = alpha_prod_s.sqrt() * pred_original_sample + beta_prod_s.sqrt() * pred_epsilon 
+        elif self.config.prediction_type == "sample":  # x-prediction
+            pred_original_sample = model_output
+            pred_epsilon = (sample - alpha_prod_t ** (0.5) * pred_original_sample) / beta_prod_t ** (0.5)
+            pred_noised_sample = alpha_prod_s.sqrt() * pred_original_sample + beta_prod_s.sqrt() * pred_epsilon
+        elif self.config.prediction_type == "v_prediction":  # v-prediction
+            pred_original_sample = (alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
+            pred_epsilon = (alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
+            pred_noised_sample = alpha_prod_s.sqrt() * pred_original_sample + beta_prod_s.sqrt() * pred_epsilon
+        else:
+            raise ValueError(
+                f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample` or"
+                " `v_prediction` for `TCDScheduler`."
+            )
+            
+        # 4. Sample and inject noise z ~ N(0, I) for MultiStep Inference
+        # Noise is not used on the final timestep of the timestep schedule.
+        # This also means that noise is not used for one-step sampling.
+        # Eta (referred to as "gamma" in the paper) was introduced to control the stochasticity in every step. 
+        # When eta = 0, it represents deterministic sampling, whereas eta = 1 indicates full stochastic sampling.
+        if eta > 0:
+            if self.step_index != self.num_inference_steps - 1:
+                noise = randn_tensor(
+                    model_output.shape, generator=generator, device=model_output.device, dtype=pred_noised_sample.dtype
+                )
+                prev_sample = (alpha_prod_t_prev / alpha_prod_s).sqrt() * pred_noised_sample + (1 - alpha_prod_t_prev / alpha_prod_s).sqrt() * noise
+            else:
+                prev_sample = pred_noised_sample
+        else:
+            prev_sample = pred_noised_sample
+
+        # upon completion increase step index by one
+        self._step_index += 1
+
+        if not return_dict:
+            return (prev_sample, pred_noised_sample)
+
+        return TCDSchedulerOutput(prev_sample=prev_sample, pred_noised_sample=pred_noised_sample)
+
+    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.add_noise
+    def add_noise(
+        self,
+        original_samples: torch.FloatTensor,
+        noise: torch.FloatTensor,
+        timesteps: torch.IntTensor,
+    ) -> torch.FloatTensor:
+        # Make sure alphas_cumprod and timestep have same device and dtype as original_samples
+        alphas_cumprod = self.alphas_cumprod.to(device=original_samples.device, dtype=original_samples.dtype)
+        timesteps = timesteps.to(original_samples.device)
+
+        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
+        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
+        while len(sqrt_alpha_prod.shape) < len(original_samples.shape):
+            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)
+
+        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
+        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
+        while len(sqrt_one_minus_alpha_prod.shape) < len(original_samples.shape):
+            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)
+
+        noisy_samples = sqrt_alpha_prod * original_samples + sqrt_one_minus_alpha_prod * noise
+        return noisy_samples
+
+    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.get_velocity
+    def get_velocity(
+        self, sample: torch.FloatTensor, noise: torch.FloatTensor, timesteps: torch.IntTensor
+    ) -> torch.FloatTensor:
+        # Make sure alphas_cumprod and timestep have same device and dtype as sample
+        alphas_cumprod = self.alphas_cumprod.to(device=sample.device, dtype=sample.dtype)
+        timesteps = timesteps.to(sample.device)
+
+        sqrt_alpha_prod = alphas_cumprod[timesteps] ** 0.5
+        sqrt_alpha_prod = sqrt_alpha_prod.flatten()
+        while len(sqrt_alpha_prod.shape) < len(sample.shape):
+            sqrt_alpha_prod = sqrt_alpha_prod.unsqueeze(-1)
+
+        sqrt_one_minus_alpha_prod = (1 - alphas_cumprod[timesteps]) ** 0.5
+        sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.flatten()
+        while len(sqrt_one_minus_alpha_prod.shape) < len(sample.shape):
+            sqrt_one_minus_alpha_prod = sqrt_one_minus_alpha_prod.unsqueeze(-1)
+
+        velocity = sqrt_alpha_prod * noise - sqrt_one_minus_alpha_prod * sample
+        return velocity
+
+    def __len__(self):
+        return self.config.num_train_timesteps
+
+    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler.previous_timestep
+    def previous_timestep(self, timestep):
+        if self.custom_timesteps:
+            index = (self.timesteps == timestep).nonzero(as_tuple=True)[0][0]
+            if index == self.timesteps.shape[0] - 1:
+                prev_t = torch.tensor(-1)
+            else:
+                prev_t = self.timesteps[index + 1]
+        else:
+            num_inference_steps = (
+                self.num_inference_steps if self.num_inference_steps else self.config.num_train_timesteps
+            )
+            prev_t = timestep - self.config.num_train_timesteps // num_inference_steps
+
+        return prev_t