Repository for 'Random Mesh Projectors for Inverse Problems' (paper)
We intend to make it as simple as possible to reproduce our results. If something is missing or you would like some guidance, please reach out to us.
- In this work, we come up with a learning scheme to regularize ill-posed inverse problems. Instead of learning to go from measurements to the model directly, we learn to estimate certain random projections of the model. Specifically, we estimate projections on many random Delaunay triangulations of the model. Later, we combine them using regularized iterative schemes.
- As an example of very ill-posed inverse problem we choose the traveltime tomography problem, where a few sensors are placed on the image domain and line-integrals are calculated along lines connecting pairs of sensors.
- We want to be able to work in environments where data is extremely scarce. Hence, we train on standard datasets (LSUN, celebA etc.) and test on arbitrary image patches from BP2004 velocity dataset. We show that our method does not try to paint dataset-specific structures, instead it captures the global shapes much more faithfully. We use the signal-to-noise ratio as a metric to quantify the quality of our reconstructions.
We use neural network architectures inspired by U-nets. All networks are given a non-negative least squares reconstuction input from measurements. This is to warm-start the network as the network need not learn the mapping from measurement domain back to image domain.
- ProjNets: These networks are trained to estimate projections on one random subspace from given input.
- SubNet: This network estimates projections on mutliple random subspaces while still maintaining robustness inherent in ProjNets.
- To reconstruct, once we know the estimated projections in these random low-dimensional subspaces we can build a linear system by concatenating the basis vectors of these low-dimensional subpsaces and solving the linear system.
We compare our results with a system that is trained to map the same input to the model directly. We show that our method achieves:
- Robustness to noise and arbitrary deviations from noise model
- Robustness against dataset used for training
To see all the results, the scheme and the rationale explained in detail, please look at our paper.
- Prepare training data. 20,000 training images and four geo images are available here: https://uofi.app.box.com/v/deepmesh-data .
- Use python3 to generate meshes from utils/mesh_code.py . By default this code generates ten 128 x 128 meshes with 50 triangles. In our paper we used 130 meshes to train 130 ProjNets and 350 meshes for SubNet.
Training multiple ProjNets
To train multiple ProjNets, use
The arguments are as follows:
usage: train_projnets.py [-h] [--imgs IMGS] [--val VAL] [--orig ORIG] [--input INPUT] [--nets NETS] [--path PATH] optional arguments: -h, --help show this help message and exit --imgs IMGS, --images IMGS Number of images to load --val VAL, --validation VAL Number of images for validation --orig ORIG, --originals ORIG Path to original images .npy --input INPUT, --input INPUT Path to ProjNet input images .npy --nets NETS, --networks NETS Number of ProjNets to train --path PATH, --projnetspath PATH Directory to store ProjNets. Ensure this directory exists.
For example, to train 50 ProjNets with 10,000 training images, 100 validation images and save them in
my_nets (ensure this directory exists), we can run the following:
cd projnet/ python3 train_projnets.py --nets=50 --imgs=10000 --val=10 --path='my_nets' --orig=../originals20k.npy --input=../custom25_infdb.npy
Reconstruct from trained ProjNets
To reconstruct from trained Projnets, use
projnet/reconstruct_from_projnets.py. Total variation regularization is used for the reconstruction.
The arguments are as follows:
usage: reconstruct_from_projnets.py [-h] [--orig ORIG] [--input INPUT] [--nets NETS] [--lam LAM] [--nc] [--projnets PROJNETS] [--b B] [--c C] [--path PATH] optional arguments: -h, --help show this help message and exit --orig ORIG, --originals ORIG Path to original images .npy --input INPUT, --input INPUT Path to ProjNet input images .npy --nets NETS, --networks NETS Number of ProjNets to use --lam LAM, --lambda LAM TV regularization parameter --nc, --nocoefs Use if already calculated coefficients --projnets PROJNETS, --projnetspath PROJNETS ProjNets directory. --b B, --basisstacked B Path to save stacked basis functions .npy --c C, --coefsstacked C Path to save stacked coefficients .npy --path PATH, --reconpath PATH Reconstruction directory. Ensure this directory exists.
For example, to reconstruct from 40 networks in
my_nets with a regularization parameter of 0.003 and store the reconstructions in
reconstructions, we can run the following:
cd projnet/ python3 reconstruct_from_projnets.py --nets=40 --projnets=my_nets --lam=0.003 --path=reconstructions --b=basis_40nets.npy --c=coefs_40nets.npy
The above command stores the stacked basis functions and stacked coefficients in
It is possible that you may wish to try a different regularization parameter for reconstruction.
As you have saved the stacked basis functions and coefficients, you do not need to calculate these again. You can use the
python3 reconstruct_from_projnets.py --lam=0.002 --path=reconstructions_new_lam --b=basis_40nets.npy --c=coefs_40nets.npy --nc
The subspace network (SubNet), takes the basis for the random projection as an input along with the measurements. One can use
subnet/subnet.py to train the subnet.
subnet.py allows for resuming training from a particular checkpoint and also, skipping training and moving directly to evaluation on required datasets. We have a common parser for
reconstruct_from_subnet.py as they share many same arguments. The parser arguments are as below::
usage: subnet.py [-h] [-ni NITER] [-dnpy DATA_ARRAY] [-mnpy MEASUREMENT_ARRAY] [-ntrain TRAINING_SAMPLES] [-t TRAIN] [-r_iter RESUME_FROM] [-n NAME] [-lr LEARNING_RATE] [-bs BATCH_SIZE] [-i_s IMG_SIZE] [-e EVAL] [-e_orig EVAL_ORIGINALS] [-e_meas EVAL_MEASUREMENTS] [-e_name EVAL_NAME] [-pdir PROJECTORS_DIR] [-nproj NUM_PROJECTORS_TO_USE] [-ntri DIM_RAND_SUBSPACE] [-r_orig RECON_ORIGINALS] [-r_coefs RECON_COEFFICIENTS] [-m MASK] optional arguments: -h, --help show this help message and exit -ni NITER, --niter NITER Number of iterations fot the training to run -dnpy DATA_ARRAY, --data_array DATA_ARRAY Numpy array of the data -mnpy MEASUREMENT_ARRAY, --measurement_array MEASUREMENT_ARRAY Numpy array of the measurements -ntrain TRAINING_SAMPLES, --training_samples TRAINING_SAMPLES Number of training samples -t TRAIN, --train TRAIN Training flag -r_iter RESUME_FROM, --resume_from RESUME_FROM resume training from r_iter checkpoint, if 0 start training afresh -n NAME, --name NAME Name of directory under results/ where the results of the current run will be stored -lr LEARNING_RATE, --learning_rate LEARNING_RATE learning rate to be used for training -bs BATCH_SIZE, --batch_size BATCH_SIZE mini-batch size -i_s IMG_SIZE, --img_size IMG_SIZE image size -e EVAL, --eval EVAL Evaluation (on other dataset) flag -e_orig EVAL_ORIGINALS, --eval_originals EVAL_ORIGINALS list of strings with names of original .npy arrays -e_meas EVAL_MEASUREMENTS, --eval_measurements EVAL_MEASUREMENTS list of strings with names of measurement .npy arrays -e_name EVAL_NAME, --eval_name EVAL_NAME Name to be given to the evaluation experiments -pdir PROJECTORS_DIR, --projectors_dir PROJECTORS_DIR directory where all projector matrices are stored -nproj NUM_PROJECTORS_TO_USE, --num_projectors_to_use NUM_PROJECTORS_TO_USE Number of projector matrices to use -ntri DIM_RAND_SUBSPACE, --dim_rand_subspace DIM_RAND_SUBSPACE Number of triangles per mesh -r_orig RECON_ORIGINALS, --recon_originals RECON_ORIGINALS npy array of originals to compare against -r_coefs RECON_COEFFICIENTS, --recon_coefficients RECON_COEFFICIENTS npy array of coefficients to use -m MASK, --mask MASK Mask to apply for convex hull of sensors
One must provide the
-mnpy arguments which correspond to the data and the measurements numpy arrays. Along with that, a directory which has all the basis vectors must be provided via
-pdir argument. Note that running the training does not require you to be in the subnet folder.
python3 subnet/subnet.py -niter 20000 -dnpy 'originals20k.npy' -mnpy 'custom25_10db.npy' -n test_subnet -e_orig [geo_originals.npy','geo_originals.npy'] -e_meas ['geo_pos_recon_10db.npy','geo_pos_recon_infdb.npy'] -e_name ['geo_tr0_t10','geo_tr0_tinf'] -pdir 'meshes/' -nproj 350 -ntri 50
To reconstruct from SubNet, one needs to run
reconstruct_from_subnet.py. This file takes the coefficients calculated and the projections and runs an iterative projected least squares. Note that since we only train one network for all subspaces we need not use any extra regularization (like TV) to reconstruct the model. An example usage is given below:
python3 subnet/subnet.py -lr 0.0005 -r_orig 'geo_originals.npy' -r_coef 'geo_tr10_tinf' -m 'mask.npy' -nproj 350 -ntri 50
The direct net uses the same parser as subnet. An example usage is given below for reference:
python3 subnet/directnet.py -niter 20000 -dnpy 'originals20k.npy' -mnpy 'custom25_0db.npy' -n test_subnet -e_orig [geo_originals.npy','geo_originals.npy'] -e_meas ['geo_pos_recon_10db.npy','geo_pos_recon_infdb.npy'] -e_name ['geo_tr0_t10','geo_tr0_tinf']