diff --git a/denoising-plots/denoised_via_kerreg.pdf b/denoising-plots/denoised_via_kerreg.pdf
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diff --git a/denoising-plots/examples.ipynb b/denoising-plots/examples.ipynb
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-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "id": "6db15641",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [],
-      "text/plain": [
-       "<IPython.core.display.HTML object>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "\u001b[36m(pid=3980984)\u001b[0m /usr/lib/python3/dist-packages/scipy/__init__.py:146: UserWarning: A NumPy version >=1.17.3 and <1.25.0 is required for this version of SciPy (detected version 1.26.4\n",
-      "\u001b[36m(pid=3980984)\u001b[0m   warnings.warn(f\"A NumPy version >={np_minversion} and <{np_maxversion}\"\n",
-      "2024-03-14 15:03:03,729\tWARNING tune.py:186 -- Stop signal received (e.g. via SIGINT/Ctrl+C), ending Ray Tune run. This will try to checkpoint the experiment state one last time. Press CTRL+C (or send SIGINT/SIGKILL/SIGTERM) to skip. \n",
-      "2024-03-14 15:03:09,319\tWARNING tune.py:1057 -- Experiment has been interrupted, but the most recent state was saved.\n",
-      "Resume experiment with: Tuner.restore(path=\"/home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04\", trainable=...)\n",
-      "2024-03-14 15:03:15,927\tWARNING experiment_analysis.py:193 -- Failed to fetch metrics for 17 trial(s):\n",
-      "- objective_STIS_99690_00970: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00970: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00970_970_beta1=100.0000,beta2=0.0100,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph__2024-03-14_15-02-55')\n",
-      "- objective_STIS_99690_00971: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00971: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00971_971_beta1=10.0000,beta2=0.0100,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a_2024-03-14_15-02-55')\n",
-      "- objective_STIS_99690_00972: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00972: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00972_972_beta1=1.0000,beta2=0.0100,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a5_2024-03-14_15-02-55')\n",
-      "- objective_STIS_99690_00974: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00974: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00974_974_beta1=0.0100,beta2=0.0100,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a5_2024-03-14_15-02-55')\n",
-      "- objective_STIS_99690_00976: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00976: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00976_976_beta1=10.0000,beta2=100.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_2024-03-14_15-03-00')\n",
-      "- objective_STIS_99690_00977: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00977: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00977_977_beta1=1.0000,beta2=100.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph__2024-03-14_15-03-00')\n",
-      "- objective_STIS_99690_00978: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00978: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00978_978_beta1=0.1000,beta2=100.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph__2024-03-14_15-03-00')\n",
-      "- objective_STIS_99690_00980: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00980: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00980_980_beta1=100.0000,beta2=10.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00982: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00982: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00982_982_beta1=1.0000,beta2=10.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00983: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00983: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00983_983_beta1=0.1000,beta2=10.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00984: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00984: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00984_984_beta1=0.0100,beta2=10.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00986: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00986: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00986_986_beta1=10.0000,beta2=1.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00987: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00987: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00987_987_beta1=1.0000,beta2=1.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a5_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00988: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00988: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00988_988_beta1=0.1000,beta2=1.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a5_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00989: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00989: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00989_989_beta1=0.0100,beta2=1.0000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a5_2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00990: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00990: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00990_990_beta1=100.0000,beta2=0.1000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph__2024-03-14_15-03-01')\n",
-      "- objective_STIS_99690_00991: FileNotFoundError('Could not fetch metrics for objective_STIS_99690_00991: both result.json and progress.csv were not found at /home/ypilavci/ray_results/objective_STIS_2024-03-14_14-56-04/objective_STIS_99690_00991_991_beta1=10.0000,beta2=0.1000,lambdaS=0.0000,lambdax=0.0000,sigma2=0.0400,t=ref_ph_8d1a09f1,x=ref_ph_a_2024-03-14_15-03-03')\n"
-     ]
-    }
-   ],
-   "source": [
-    "import numpy as np\n",
-    "import scipy\n",
-    "import matplotlib.pyplot as plt\n",
-    "import quaternion  # load the quaternion module\n",
-    "import bispy as bsp\n",
-    "import torch\n",
-    "from utils import STIS,optimize_loop,snr_bivariate,param_search,objective_STIS,objective_KReSP,KReSP,plot_on_sphere\n",
-    "from ray import  tune,init\n",
-    "import pickle\n",
-    "\n",
-    "\n",
-    "init(num_cpus=16)\n",
-    "\n",
-    "## PLOT AN EXAMPLE \n",
-    "\n",
-    "np.random.seed(5)\n",
-    "N = 1024 # length of the signal\n",
-    "t = np.linspace(0, 2*np.pi/4, N) # time vector\n",
-    "dt = t[1]-t[0]\n",
-    "\n",
-    "# ellipse parameters - AM-FM-PM polarized \n",
-    "theta1 = np.pi/4 - 2*t\n",
-    "chi1 = np.pi/16 - t\n",
-    "phi1 = 0 \n",
-    "f0 = 25/N/dt \n",
-    "S0 = bsp.utils.windows.hanning(N)\n",
-    "\n",
-    "x_quad = bsp.signals.bivariateAMFM(S0, theta1, chi1, 2*np.pi*f0*t+ phi1)\n",
-    "x = quaternion.as_float_array(x_quad)[:,:2]\n",
-    "\n",
-    "\n",
-    "bsp.utils.visual.plot3D(t,x_quad)\n",
-    "plt.savefig(\"clean_sig.pdf\")\n",
-    "plt.close()\n",
-    "\n",
-    "sigma = 0.2\n",
-    "n = np.zeros([N,4])\n",
-    "noise_complex = np.random.randn(N,2)\n",
-    "y = x +sigma*noise_complex\n",
-    "\n",
-    "uH = np.imag(scipy.signal.hilbert(noise_complex[:,0]))\n",
-    "vH = np.imag(scipy.signal.hilbert(noise_complex[:,1]))\n",
-    "n[:,0] = noise_complex[:,0]\n",
-    "n[:,1] = noise_complex[:,1]\n",
-    "n[:,2] = uH\n",
-    "n[:,3] = vH\n",
-    "n = quaternion.from_float_array(n)\n",
-    "y_quad = sigma*n + x_quad # Noisy signal\n",
-    "\n",
-    "ax = plt.figure().add_subplot(projection='3d')\n",
-    "# ax.view_init(15,-135,0)\n",
-    "plt.title(\"Normalized Stokes Parameters\")\n",
-    "plt.tight_layout()\n",
-    "plot_on_sphere(y_quad,ax,label=\"noisy signal\",t=t,scatter=True)\n",
-    "plot_on_sphere(x_quad,ax,label=\"original signal\",scatter=False)\n",
-    "\n",
-    "plt.legend()\n",
-    "plt.tight_layout()\n",
-    "plt.savefig(\"noisy_sig_sphere.pdf\")\n",
-    "plt.close()\n",
-    "\n",
-    "bsp.utils.visual.plot3D(t,y_quad)\n",
-    "plt.savefig(\"noisy_sig.pdf\")\n",
-    "\n",
-    "\n",
-    "print(\"sigma: \" + str(sigma) + \" Noise SNR: \"+  str(snr_bivariate(x,y) ) )\n",
-    "search_space = {\"x\":tune.grid_search([x]),\"t\":tune.grid_search([t]),\"y\":tune.grid_search([y]),\"lambdax\": tune.grid_search((0.1)**np.linspace(5,15,7)), \"lambdaS\":  tune.grid_search((0.1)**np.linspace(5,10,7)) , \"beta1\":tune.grid_search((0.10)**np.linspace(-2,2,5)),\"beta2\":tune.grid_search((0.10)**np.linspace(-2,2,5)),\"sigma2\":tune.grid_search([sigma**2])}\n",
-    "config = param_search(objective_STIS,search_space)\n",
-    "model = STIS(t,y,lambdax=config[\"lambdax\"],lambdaS=config[\"lambdaS\"],beta1=config[\"beta1\"],beta2=config[\"beta2\"],sigma2=sigma**2,p=2)\n",
-    "optimizer = torch.optim.Adam(model.parameters(), lr=0.01)\n",
-    "model = optimize_loop(model,optimizer,numit=1000)\n",
-    "\n",
-    "x_stis = quaternion.from_float_array(model.Xquad.detach().numpy())\n",
-    "bsp.utils.visual.plot2D(t,x_stis)\n",
-    "plt.savefig(\"denoised_via_all_terms.pdf\")\n",
-    "\n",
-    "print(\"sigma: \" + str(sigma) + \" STIS SNR: \"+  str(snr_bivariate(x,model.X.detach().numpy())))\n",
-    "\n",
-    "\n",
-    "# NO STOKES REGULARIZATION  \n",
-    "search_space = {\"x\":tune.grid_search([x]),\"t\":tune.grid_search([t]),\"y\":tune.grid_search([y]),\"lambdax\": tune.grid_search((0.1)**np.linspace(5,15,7)), \"lambdaS\":tune.grid_search([0.0]) , \"beta1\":tune.grid_search([0.0]),\"beta2\":tune.grid_search([0.0]),\"sigma2\":tune.grid_search([sigma**2])}\n",
-    "config = param_search(objective_STIS,search_space)\n",
-    "model = STIS(t,y,lambdax=config[\"lambdax\"],lambdaS=config[\"lambdaS\"],beta1=config[\"beta1\"],beta2=config[\"beta2\"],sigma2=sigma**2,p=2)\n",
-    "optimizer = torch.optim.Adam(model.parameters(), lr=0.01)\n",
-    "model = optimize_loop(model,optimizer,numit=1000)\n",
-    "\n",
-    "print(\"sigma: \" + str(sigma) + \" STIS SNR: \"+  str((snr_bivariate(x,model.X.detach().numpy()))))\n",
-    "x_nostokes = quaternion.from_float_array(model.Xquad.detach().numpy())\n",
-    "bsp.utils.visual.plot2D(t,x_nostokes)\n",
-    "plt.savefig(\"denoised_via_no_stokes.pdf\")\n",
-    "\n",
-    "# ONLY SMOOTH STOKES\n",
-    "search_space = {\"x\":tune.grid_search([x]),\"t\":tune.grid_search([t]),\"y\":tune.grid_search([y]),\"lambdax\": tune.grid_search([0.0]), \"lambdaS\":  tune.grid_search((0.1)**np.linspace(5,10,7)) , \"beta1\":tune.grid_search((0.1)**np.linspace(-2,2,7)),\"beta2\":tune.grid_search((0.1)**np.linspace(-2,2,5)),\"sigma2\":tune.grid_search([sigma**2])}\n",
-    "config = param_search(objective_STIS,search_space)\n",
-    "model = STIS(t,y,lambdax=config[\"lambdax\"],lambdaS=config[\"lambdaS\"],beta1=config[\"beta1\"],beta2=config[\"beta2\"],sigma2=sigma**2,p=2)\n",
-    "optimizer = torch.optim.Adam(model.parameters(), lr=0.01)\n",
-    "model = optimize_loop(model,optimizer,numit=1000)\n",
-    "x_onlystokes = quaternion.from_float_array(model.Xquad.detach().numpy())\n",
-    "bsp.utils.visual.plot2D(t,x_onlystokes)\n",
-    "plt.savefig(\"denoised_via_no_signal_smoother.pdf\")\n",
-    "\n",
-    "print(\"sigma: \" + str(sigma) +  \" STIS SNR: \"+  str((snr_bivariate(x,model.X.detach().numpy()))))\n",
-    "\n",
-    "# Kernel regression on normalized \n",
-    "search_space = {\"x\":tune.grid_search([x]),\"t\":tune.grid_search([t]),\"y\":tune.grid_search([y]),\"alpha\": tune.grid_search((0.1)**np.linspace(5,15,5)),\"lambda_1\": tune.grid_search((0.1)**np.linspace(5,15,5)), \"lambda_s\":  tune.grid_search((0.1)**np.linspace(5,10,5)) , \"beta\":tune.grid_search((0.10)**np.linspace(-2,2,5)),\"gamma\":tune.grid_search((0.1)**np.linspace(0,1,5)),\"sigma2\":tune.grid_search([sigma**2])}\n",
-    "config = param_search(objective_KReSP,search_space)\n",
-    "model = KReSP(t,y,lambda_1=config[\"lambda_1\"],beta=config[\"beta\"],lambda_s=config[\"lambda_s\"],alpha=config[\"alpha\"],gamma=config[\"gamma\"],eps=10**-7,win_width=64,sigma2=sigma**2)\n",
-    "optimizer = torch.optim.Adam(model.parameters(), lr=0.01)\n",
-    "model = optimize_loop(model,optimizer,numit=300)\n",
-    "\n",
-    "\n",
-    "print(\"sigma: \" + str(sigma) + \" KReSP SNR: \"+  str(snr_bivariate(x,model.X.detach().numpy())))\n",
-    "x_kerreg = quaternion.from_float_array(model.Xquad.detach().numpy())\n",
-    "bsp.utils.visual.plot2D(t,x_kerreg)\n",
-    "plt.savefig(\"denoised_via_kerreg.pdf\")\n",
-    "\n",
-    "with open('example.pkl', 'wb') as f:\n",
-    "\tpickle.dump(x,f)\n",
-    "\tpickle.dump(y,f)\n",
-    "\tpickle.dump(x_stis,f)\n",
-    "\tpickle.dump(x_nostokes,f)\n",
-    "\tpickle.dump(x_onlystokes,f)\n",
-    "\tpickle.dump(x_kerreg,f)\n"
-   ]
-  }
- ],
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