README.md

 # 3D Shape Analysis using Persistent Homology
 
+This is the official code accompanying our paper **Extracting Mitochondrial Cristae Characteristics from 3D Focused Ion Beam Scanning Electron Microscopy Data**. 
 
 
-## Getting started
+## Persistent Homology Toolset
 
-To make it easy for you to get started with GitLab, here's a list of recommended next steps.
+The persistent homology toolset provided here can be directly used for the shape analysis of other object types, and is not restricted to the mitochondria and their cristae.
 
-Already a pro? Just edit this README.md and make it your own. Want to make it easy? [Use the template at the bottom](#editing-this-readme)!
+Open "3D Shape Analysis using Persistent Homology.ipynb" to get started.
 
-## Add your files
+If you would like to run our real world example locally on your machine:
+- please download the [data](https://erda.dk) ("cristae_volume.npy", "intracristal_volume.npy", and "mito_volume.npy")
+- and unzip them into the "/binary_segmentation_masks" subfolder
 
-- [ ] [Create](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#create-a-file) or [upload](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#upload-a-file) files
-- [ ] [Add files using the command line](https://docs.gitlab.com/ee/gitlab-basics/add-file.html#add-a-file-using-the-command-line) or push an existing Git repository with the following command:
 
-```
-cd existing_repo
-git remote add origin https://lab.compute.dtu.dk/QIM/tools/3d-shape-analysis-using-persistent-homology.git
-git branch -M main
-git push -uf origin main
-```
 
-## Integrate with your tools
+## Segmentation Model
 
-- [ ] [Set up project integrations](https://lab.compute.dtu.dk/QIM/tools/3d-shape-analysis-using-persistent-homology/-/settings/integrations)
+For completeness, we also included our multiplanar segmentation model for mitochondria and cristae in the "/other/segmentation_model" folder. 
 
-## Collaborate with your team
+You won't need this part of the code, if your project is not related to mitochondria and cristae, or if you have your own segmentations already. 
 
-- [ ] [Invite team members and collaborators](https://docs.gitlab.com/ee/user/project/members/)
-- [ ] [Create a new merge request](https://docs.gitlab.com/ee/user/project/merge_requests/creating_merge_requests.html)
-- [ ] [Automatically close issues from merge requests](https://docs.gitlab.com/ee/user/project/issues/managing_issues.html#closing-issues-automatically)
-- [ ] [Enable merge request approvals](https://docs.gitlab.com/ee/user/project/merge_requests/approvals/)
-- [ ] [Automatically merge when pipeline succeeds](https://docs.gitlab.com/ee/user/project/merge_requests/merge_when_pipeline_succeeds.html)
+If you do decide to use our segmentation code:
+- You need to first perform image registration to align the slices in your image volume.
+- Place the registered image volume in "/raw_data/test" subfolder as a tiff file.
+- Adjust the filename in the loading function in "segmentation_main".
 
-## Test and Deploy
+Please note that the raw output needs to be binarized and cleaned up. We do not provide the code for binarization and clean up, because the best procedure is likely to be specific to each image volume and should be simple enough to figure out. 
 
-Use the built-in continuous integration in GitLab.
+The segmentation can take several days depending on the size of your image volume and your PC's computing power. 
 
-- [ ] [Get started with GitLab CI/CD](https://docs.gitlab.com/ee/ci/quick_start/index.html)
-- [ ] [Analyze your code for known vulnerabilities with Static Application Security Testing(SAST)](https://docs.gitlab.com/ee/user/application_security/sast/)
-- [ ] [Deploy to Kubernetes, Amazon EC2, or Amazon ECS using Auto Deploy](https://docs.gitlab.com/ee/topics/autodevops/requirements.html)
-- [ ] [Use pull-based deployments for improved Kubernetes management](https://docs.gitlab.com/ee/user/clusters/agent/)
-- [ ] [Set up protected environments](https://docs.gitlab.com/ee/ci/environments/protected_environments.html)
-
-***
-
-# Editing this README
-
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-
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-
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-
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-## Project status
-If you have run out of energy or time for your project, put a note at the top of the README saying that development has slowed down or stopped completely. Someone may choose to fork your project or volunteer to step in as a maintainer or owner, allowing your project to keep going. You can also make an explicit request for maintainers.
+Our registration of the Lausanne dataset (https://www.epfl.ch/labs/cvlab/data/data-em/) can be downloaded [here](https://erda.dk), and this is the volume which we segmented for analysis.
\ No newline at end of file

general_support_functions.py

+# -*- coding: utf-8 -*-
+"""
+Extracting Mitochondrial Cristae Characteristics from 3D Focused Ion Beam Scanning Electron Microscopy Data
+
+Chenhao Wang, Leif Østergaard, Stine Hasselholt, Jon Sporring
+
+https://doi.org/10.1101/2022.11.08.515664
+"""
+
+# Imports
+import os
+import warnings
+import matplotlib.pyplot as plt
+warnings.filterwarnings('ignore')
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+
+
+
+###########################################################################
+# folder paths 
+
+root_folder = os.getcwd()
+
+
+
+###########################################################################
+# file management functions
+
+def setpath(path):
+    '''
+    changes working directory, creates folder if path doesn't exist
+    '''
+    try:
+        os.chdir(path)
+    except OSError:
+        os.makedirs(path)
+        os.chdir(path)           
+
+
+def all_file_names(folder, file_format = None):
+    '''
+    list all image filenames of a given format in a folder,
+    if file_format == None, returns all files regardless of format.
+    '''
+    if file_format is None:
+        names = os.listdir(folder)
+    else:
+        names = os.listdir(folder)
+        for name in names:
+            if name[-len(file_format):] != file_format:
+                names.remove(name)
+    return names
+
+
+
+
+
+###########################################################################
+# image visualization
+    
+def visualize_jet(img, max_val = 0.0000005):
+    plt.figure(figsize = (15,15))
+    #plt.imshow(img, cmap='jet')
+    ax = plt.gca()
+    im = ax.imshow(img, cmap='jet')
+    im.set_clim(vmin = 0, vmax=max_val)
+    divider = make_axes_locatable(ax)
+    cax = divider.append_axes("right", size="5%", pad=0.05)
+    plt.colorbar(im, cax=cax)
+    ax.tick_params(labelsize=12)
+    plt.tick_params(labelsize=15)    
+    
+    
+def visualize(img):
+    plt.figure(figsize = (15,15))
+    plt.imshow(img, cmap='gray')
+
+
+

other/segmentation_model/general_support_functions.py

+# -*- coding: utf-8 -*-
+"""
+General Support Functions - version 1.0
+
+@author: Chenhao Wang
+@date: June 2021
+"""
+import os
+import cv2
+import numpy as np
+from tqdm import tqdm
+import warnings
+import matplotlib.pyplot as plt
+warnings.filterwarnings('ignore')
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import skimage.io as skio
+
+###########################################################################
+# folder paths 
+
+root_folder = os.getcwd()
+
+raw_folder = root_folder + '/raw_data'
+raw_folder_train_img = raw_folder + "/train" + "/images"
+raw_folder_train_mask = raw_folder + "/train" + "/masks"
+raw_folder_valid_img = raw_folder + "/valid" + "/images"
+raw_folder_valid_mask = raw_folder + "/valid" + "/masks"
+
+
+cross_val_train_img_folder = root_folder + "\\x_val" + "\\train\\images"
+cross_val_train_mask_folder = root_folder + "\\x_val" + "\\train\\masks"
+cross_val_valid_img_folder = root_folder + "\\x_val" + "\\valid\\images"
+cross_val_valid_mask_folder = root_folder + "\\x_val" + "\\valid\\masks"
+cross_val_valid_img_augmented_folder = raw_folder + "\\x_val" + "\\valid_augmented\\images"
+cross_val_valid_mask_augmented_folder  = raw_folder + "\\x_val" + "\\valid_augmented\\masks"
+
+cross_val_test_img_folder = root_folder + "\\x_val" + "\\test\\images"
+cross_val_test_mask_folder = root_folder + "\\x_val" + "\\test\\prediction"
+
+
+
+
+full_sized_folder = root_folder + "/full_sized_data"
+full_sized_folder_train_img = full_sized_folder + "/train" + "/images"
+full_sized_folder_train_mask = full_sized_folder + "/train" + "/masks"
+full_sized_folder_valid_img = full_sized_folder + "/valid" + "/images"
+full_sized_folder_valid_mask = full_sized_folder + "/valid" + "/masks"
+
+cristae_image_folder = root_folder + "/cristae_annotation" + "/images"
+cristae_mask_folder = root_folder + "/cristae_annotation" + "/masks"
+
+
+test_folder = raw_folder + '/test'
+results_folder = root_folder + '/results'
+
+
+
+
+###########################################################################
+# file management functions
+
+def setpath(path):
+    '''
+    changes working directory, creates folder if path doesn't exist
+    '''
+    try:
+        os.chdir(path)
+    except OSError:
+        os.makedirs(path)
+        os.chdir(path)           
+
+
+def all_file_names(folder, file_format = None):
+    '''
+    list all image filenames of a given format in a folder,
+    if file_format == None, returns all files regardless of format.
+    '''
+    if file_format is None:
+        names = os.listdir(folder)
+    else:
+        names = os.listdir(folder)
+        for name in names:
+            if name[-len(file_format):] != file_format:
+                names.remove(name)
+    return names
+
+
+
+###########################################################################
+# image data loaders
+
+def load_volume_data_slices(folder_path):
+    """
+    loads all the image slices in the specified folder of a single image volume. 
+    """    
+    setpath(folder_path)
+    file_names = all_file_names(folder_path)
+    output_collection = []
+    for name in tqdm(file_names):
+        img = cv2.imread(name, cv2.IMREAD_GRAYSCALE)
+        output_collection.append(img)
+    output_collection = np.array(output_collection)
+    return file_names, output_collection
+
+
+def load_volume_data_slices_color(folder_path):
+    """
+    loads all the image slices in the specified folder of a single image volume. 
+    """    
+    setpath(folder_path)
+    file_names = all_file_names(folder_path)
+    output_collection = []
+    for name in tqdm(file_names):
+        img = cv2.imread(name)
+        output_collection.append(img)
+    output_collection = np.array(output_collection)
+    return file_names, output_collection
+
+
+
+
+def load_volume_data_tif_stacks(input_path,
+                                file_name):
+    """
+    loads all the image slices in the specified folder of a single image volume. 
+    """    
+    setpath(input_path)
+    img_stack = skio.imread(file_name, plugin="tifffile")
+    
+    
+    return img_stack
+
+
+
+def save_img_as_slices(volume,
+                       output_path,
+                       initial_tag = None):
+    
+    setpath(output_path)
+
+    for i, img_slice in enumerate(tqdm(volume)):
+        if i < 10:
+            save_name = "0000" + str(i) + ".png"
+        elif i < 100:
+            save_name = "000" + str(i) + ".png"
+        elif i < 1000:
+            save_name = "00" + str(i) + ".png"
+        elif i < 10000:
+            save_name = "0" + str(i) + ".png"
+        else:
+            save_name = str(i) + ".png"
+    
+        if initial_tag != None:
+            save_name = initial_tag + save_name
+            
+        cv2.imwrite(save_name, img_slice)
+
+
+
+"""
+Image saving function
+"""
+
+
+
+###########################################################################
+# image visualization
+    
+def visualize_jet(img, max_val = 0.0000005):
+    plt.figure(figsize = (15,15))
+    #plt.imshow(img, cmap='jet')
+    ax = plt.gca()
+    im = ax.imshow(img, cmap='jet')
+    im.set_clim(vmin = 0, vmax=max_val)
+    divider = make_axes_locatable(ax)
+    cax = divider.append_axes("right", size="5%", pad=0.05)
+    plt.colorbar(im, cax=cax)
+    ax.tick_params(labelsize=12)
+    plt.tick_params(labelsize=15)    
+    
+    
+def visualize(img):
+    plt.figure(figsize = (15,15))
+    plt.imshow(img, cmap='gray')
+
+
+
+###########################################################################
+# image filtering 
+
+def multiple_slice_gaussian_denoise(image_array,
+                                    temporal_size = 3,
+                                    filter_strength = 0.5):
+    
+    """denoise gaussian white noise using Non-local Means Denoising algorithm:
+       http://www.ipol.im/pub/art/2011/bcm_nlm/
+       
+       Multiframe version.
+    """
+    
+    image_array_denoised = []
+   
+    margin = int(temporal_size/2)
+    
+    for i in tqdm(range(margin, len(image_array) - margin)):
+        # performs multiple frame denoising
+        noise_filtered = cv2.fastNlMeansDenoisingMulti(image_array,
+                                                       imgToDenoiseIndex = i,
+                                                       temporalWindowSize = temporal_size,
+                                                       h = filter_strength)
+        image_array_denoised.append(noise_filtered)
+        
+    return np.array(image_array_denoised)
+
+
+
+
+def adaptive_thresholding_gaussian(image_volume,
+                                   win_size = 41,
+                                   c = 2,
+                                   max_intensity_value = 255):
+
+    output = []
+    for image in tqdm(image_volume):
+        threshold = cv2.adaptiveThreshold(image,
+                                          max_intensity_value,
+                                          cv2.ADAPTIVE_THRESH_GAUSSIAN_C,\
+                                          cv2.THRESH_BINARY,
+                                          win_size,
+                                          c)
+        threshold = 1 - threshold/threshold.max()
+        output.append(threshold)
+    return np.array(output)
+
+
+
+
+###########################################################################
+# image filtering 
+
+def cut_images_into_patches_2D(input_path,
+                               mask_path,
+                               output_path,
+                               patch_size,
+                               patch_step_ratio_v = 2,
+                               patch_step_ratio_h = 2):
+    
+    """
+    cuts images into patches:
+        patch_step_size = patch_size / patch_step_ratio
+        
+    """
+    
+    input_names, input_volume = load_volume_data_slices(input_path)
+    mask_names, mask_volume = load_volume_data_slices(mask_path)
+    
+    patch_step_size_v = int((patch_size[0]/patch_step_ratio_v))
+    patch_step_size_h = int((patch_size[1]/patch_step_ratio_h))
+    
+    img_slice_shape = input_volume[0].shape
+    
+    v_starting_coors = np.arange(int((img_slice_shape[0] - patch_size[0])
+                                     /patch_step_size_v) + 1)*patch_step_size_v
+    
+    h_starting_coors = np.arange(int((img_slice_shape[1] - patch_size[1])
+                                     /patch_step_size_h) + 1)*patch_step_size_h
+    
+    
+    setpath(output_path)
+    for index in tqdm(range(len(input_volume))):
+        raw_img = input_volume[index]
+        raw_mask = mask_volume[index]
+        raw_name = input_names[index]
+        
+        for i in v_starting_coors: 
+            for j in h_starting_coors:  
+                    img_patch = raw_img[i : i + patch_size[0],
+                                        j : j + patch_size[1]]
+                 
+                    mask_patch = raw_mask[i : i + patch_size[0],
+                                          j : j + patch_size[1]]
+                    
+                    if np.sum(mask_patch) > 0:
+                        patch_save_name = raw_name.split(".")[0] + "_" + str(i) + "_" + str(j) + "." + raw_name.split(".")[1]
+                        cv2.imwrite(patch_save_name, img_patch)
+    
+    
+    
+    
+def cut_images_into_patches_2D_stack(input_path,
+                                     mask_path,
+                                     img_file_name,
+                                     mask_file_name,
+                                     output_path,
+                                     patch_size,
+                                     patch_step_ratio_v = 2,
+                                     patch_step_ratio_h = 2,
+                                     intial_tag = "train_img_"):
+    
+    """
+    cuts images into patches:
+        patch_step_size = patch_size / patch_step_ratio
+        
+    """
+
+    input_volume = load_volume_data_tif_stacks(input_path,
+                                               img_file_name)
+    
+    mask_volume = load_volume_data_tif_stacks(mask_path,
+                                              mask_file_name)
+    
+    patch_step_size_v = int((patch_size[0]/patch_step_ratio_v))
+    patch_step_size_h = int((patch_size[1]/patch_step_ratio_h))
+    
+    img_slice_shape = input_volume[0].shape
+    
+    v_starting_coors = np.arange(int((img_slice_shape[0] - patch_size[0])
+                                     /patch_step_size_v) + 1)*patch_step_size_v
+    
+    h_starting_coors = np.arange(int((img_slice_shape[1] - patch_size[1])
+                                     /patch_step_size_h) + 1)*patch_step_size_h
+    
+    
+    setpath(output_path)
+    for index in tqdm(range(len(input_volume))):
+        raw_img = input_volume[index]
+        raw_mask = mask_volume[index]
+        
+        for i in v_starting_coors: 
+            for j in h_starting_coors:  
+                    img_patch = raw_img[i : i + patch_size[0],
+                                        j : j + patch_size[1]]
+                 
+                    mask_patch = raw_mask[i : i + patch_size[0],
+                                          j : j + patch_size[1]]
+                    
+                    if np.sum(mask_patch) > 0:
+                        patch_save_name = intial_tag + str(index) + "_"  + str(i) + "_" + str(j) + ".png"
+                        cv2.imwrite(patch_save_name, img_patch)
+
+
+
+
+
+
+
+

other/segmentation_model/segmentation_main.py

+# -*- coding: utf-8 -*-
+"""
+@author: CWANG
+"""
+
+import os
+os.environ['TF_CPP_MIN_LOG_LEVEL'] = '3'
+from deep_learning_support import *
+import skimage.io as skio
+
+
+# todos
+if_segment_mito = True
+if_segment_cristae = True
+
+
+
+# params
+batch_size = 2
+epoch = 3000
+patch_size = (256, 256, 256, 3)
+patch_step_ratio_v = 2
+patch_step_ratio_h = 2
+
+
+model_pretrained_name = "unet_2D_mito.hdf5"
+model_save_name = "unet_2D_cristae_multiclass.hdf5"
+
+
+
+
+# multiplanar segmentation todos
+overlap_factor_normal = 3
+overlap_factor_small = 2
+
+
+
+    
+if if_segment_mito == True:
+
+    volume_names = all_file_names(test_folder, file_format = None)
+
+    setpath(test_folder)
+
+    test_volume = skio.imread(volume_names[0], plugin="tifffile")
+
+
+    perform_memory_efficient_multiplanar_segmentation_large(test_volume,
+                                                            model_pretrained_name,
+                                                            sub_vol_start_index = 0,
+                                                            sub_vol_end_index = None,
+                                                            custom_loss = 'IOU_loss',
+                                                            overlap_factor = 2,
+                                                            patch_size = (256, 256, 256, 1),
+                                                            results_folder = '/mito_subvolume_result')
+    
+
+    merge_sub_volumes(test_volume,
+                      patch_size = (256, 256, 256, 1),
+                      input_folder = '/mito_subvolume_result',
+                      results_folder = '/mito_result')
+    
+    
+    
+    
+
+
+
+if if_segment_cristae == True:
+
+    volume_names = all_file_names(test_folder, file_format = None)
+
+    setpath(test_folder)
+
+    test_volume = skio.imread(volume_names[0], plugin="tifffile")
+
+
+
+    perform_memory_efficient_multiplanar_segmentation_large(test_volume,
+                                                            model_save_name,
+                                                            sub_vol_start_index = 0,
+                                                            sub_vol_end_index = None,
+                                                            custom_loss = 'multiclass_IOU_loss',
+                                                            overlap_factor = 2,
+                                                            patch_size = (256, 256, 256, 3),
+                                                            results_folder = '/cristae_subvolume_result')
+    
+
+    merge_sub_volumes(test_volume,
+                      patch_size = (256, 256, 256, 3),
+                      input_folder = '/cristae_subvolume_result',
+                      results_folder = '/cristae_result')
+    
+    
+    
+    
+
+    
\ No newline at end of file

subpixel_morphology.py

+# -*- coding: utf-8 -*-
+"""
+Extracting Mitochondrial Cristae Characteristics from 3D Focused Ion Beam Scanning Electron Microscopy Data
+
+Chenhao Wang, Leif Østergaard, Stine Hasselholt, Jon Sporring
+
+https://doi.org/10.1101/2022.11.08.515664
+"""
+
+# Imports
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+
+##################################################
+# basic functions
+
+def U_ip1(U):
+    U_new = np.copy(U)
+    U_new[:-1] = U[1:]
+    return U_new
+
+def U_im1(U):
+    U_new = np.copy(U)
+    U_new[1:] = U[:-1]
+    return U_new
+
+def U_jp1(U):
+    U_new = np.copy(U)
+    U_new[:, :-1] = U[:, 1:]
+    return U_new
+
+def U_jm1(U):
+    U_new = np.copy(U)
+    U_new[:, 1:] = U[:, :-1]
+    return U_new
+
+def U_kp1(U):
+    U_new = np.copy(U)
+    U_new[:, :, :-1] = U[:, :, 1:]
+    return U_new
+
+def U_km1(U):
+    U_new = np.copy(U)
+    U_new[:, :, 1:] = U[:, :, :-1]
+    return U_new
+
+
+
+##################################################
+# Subpixel Morphology
+
+def subpixel_dilation_2D(I, 
+                         t, 
+                         Lambda):
+    
+    U = I.astype(np.float32)
+    max_iter = int(np.ceil(t/Lambda))
+    Lambda = t/max_iter
+    
+    
+    for i in range(max_iter):
+        U = U + Lambda * np.sqrt(np.square(np.clip(U_ip1(U) - U, a_min = 0, a_max = None)) +
+                                 np.square(np.clip(U_im1(U) - U, a_min = 0, a_max = None)) +
+                                 
+                                 np.square(np.clip(U_jp1(U) - U, a_min = 0, a_max = None)) + 
+                                 np.square(np.clip(U_jm1(U) - U, a_min = 0, a_max = None)))
+        
+        # regularization steps
+        U[U > 1] = 1
+        U[U < 0] = 0
+        
+        # enforces float32
+        U = U.astype(np.float32)
+    return U
+
+
+
+def subpixel_erosion_2D(I, 
+                        t, 
+                        Lambda):
+    
+    U = I.astype(np.float32)
+    max_iter = int(np.ceil(t/Lambda))
+    Lambda = t/max_iter
+    
+    
+    for i in range(max_iter):
+        U = U - Lambda * np.sqrt(np.square(np.clip(U - U_ip1(U), a_min = 0, a_max = None)) +
+                                 np.square(np.clip(U - U_im1(U), a_min = 0, a_max = None)) +
+                                 
+                                 np.square(np.clip(U - U_jp1(U), a_min = 0, a_max = None)) + 
+                                 np.square(np.clip(U - U_jm1(U), a_min = 0, a_max = None)))
+        
+        # regularization steps
+        U[U > 1] = 1
+        U[U < 0] = 0
+    
+        # enforces float32
+        U = U.astype(np.float32)
+        
+    return U
+
+
+
+def subpixel_dilation_3D(I, 
+                         t, 
+                         Lambda):
+    
+    U = I.astype(np.float32)
+    max_iter = int(np.ceil(t/Lambda))
+    Lambda = t/max_iter
+    
+    
+    for i in range(max_iter):
+        U = U + Lambda * np.sqrt(np.square(np.clip(U_ip1(U) - U, a_min = 0, a_max = None)) +
+                                 np.square(np.clip(U_im1(U) - U, a_min = 0, a_max = None)) +
+                                 
+                                 np.square(np.clip(U_jp1(U) - U, a_min = 0, a_max = None)) + 
+                                 np.square(np.clip(U_jm1(U) - U, a_min = 0, a_max = None)) +
+                                     
+                                 np.square(np.clip(U_kp1(U) - U, a_min = 0, a_max = None)) + 
+                                 np.square(np.clip(U_km1(U) - U, a_min = 0, a_max = None)))
+        
+        # regularization steps
+        U[U > 1] = 1
+        U[U < 0] = 0
+        
+        # enforces float32
+        U = U.astype(np.float32)
+    return U
+
+
+
+def subpixel_erosion_3D(I, 
+                        t, 
+                        Lambda):
+    
+    U = I.astype(np.float32)
+    max_iter = int(np.ceil(t/Lambda))
+    Lambda = t/max_iter
+    
+    
+    for i in range(max_iter):
+        U = U - Lambda * np.sqrt(np.square(np.clip(U - U_ip1(U), a_min = 0, a_max = None)) +
+                                 np.square(np.clip(U - U_im1(U), a_min = 0, a_max = None)) +
+                                 
+                                 np.square(np.clip(U - U_jp1(U), a_min = 0, a_max = None)) + 
+                                 np.square(np.clip(U - U_jm1(U), a_min = 0, a_max = None)) +
+                                     
+                                 np.square(np.clip(U - U_kp1(U), a_min = 0, a_max = None)) + 
+                                 np.square(np.clip(U - U_km1(U), a_min = 0, a_max = None)))
+        
+        # regularization steps
+        U[U > 1] = 1
+        U[U < 0] = 0
+    
+        # enforces float32
+        U = U.astype(np.float32)
+        
+    return U
+
+
+