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Finding Strong Gravitational Lenses in the Kilo Degree Survey with Convolutional Neural Networks

C. E. Petrillo, C. Tortora, S. Chatterjee, G. Vernardos, L. V. E. Koopmans, G. Verdoes Kleijn, N. R. Napolitano, G. Covone, P. Schneider, A. Grado, J. McFarland

arXiv:1702.07675v2astro-ph.GA

TL;DR

Large, homogeneous lens samples can improve the effectiveness and reliability of gravitational-lensing studies. This paper applies a morphological CNN to KiDS and projects ∼100 LRG lens candidates conservatively, with an optimistic yield up to ∼2400.

  • Problem

    Large, homogeneous lens samples can improve the effectiveness and reliability of gravitational-lensing studies.

    Source: §1 · p. 2

  • Method

    A morphological CNN identifies strong-lens candidates in KiDS using simulated lensing features placed on observed colour-magnitude selected galaxies.

    Source: §1 · p. 2, §3.1 · p. 4

  • Results

    From 21,789 LRGs, the CNN retrieved 761 candidates and visual inspection retained 56; full KiDS is projected to yield ∼100 conservatively or up to ∼2400 optimistically.

    Source: §5 · p. 11

  • Takeaways & Limitations

    The project begins a systematic census of strong lenses in KiDS.

    Source: §1 · p. 2

    False positives currently outnumber true positives by ∼40, and spectroscopic validation is needed to confirm or reject candidate lensing systems.

    Source: §4.3.1 · p. 10, §5 · p. 11

Abstract

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The volume of data that will be produced by new-generation surveys requires automatic classification methods to select and analyze sources. Indeed, this is the case for the search for strong gravitational lenses, where the population of the detectable lensed sources is only a very small fraction of the full source population. We apply for the first time a morphological classification method based on a Convolutional Neural Network (CNN) for recognizing strong gravitational lenses in $255$ square degrees of the Kilo Degree Survey (KiDS), one of the current-generation optical wide surveys. The CNN is currently optimized to recognize lenses with Einstein radii $\gtrsim 1.4$ arcsec, about twice the $r$-band seeing in KiDS. In a sample of $21789$ colour-magnitude selected Luminous Red Galaxies (LRG), of which three are known lenses, the CNN retrieves 761 strong-lens candidates and correctly classifies two out of three of the known lenses. The misclassified lens has an Einstein radius below the range on which the algorithm is trained. We down-select the most reliable 56 candidates by a joint visual inspection. This final sample is presented and discussed. A conservative estimate based on our results shows that with our proposed method it should be possible to find $\sim100$ massive LRG-galaxy lenses at $z\lsim 0.4$ in KiDS when completed. In the most optimistic scenario this number can grow considerably (to maximally $\sim$2400 lenses), when widening the colour-magnitude selection and training the CNN to recognize smaller image-separation lens systems.

1 INTRODUCTION

Strong lensing constrains foreground-galaxy mass and supports studies of galaxy structure and cosmology, but visual searches do not scale to large surveys. The paper introduces a CNN-based morphological lens finder for KiDS as part of a broader effort to build large lens samples.

1 INTRODUCTION

  • Strong lensing tightly constrains projected lens-galaxy mass and, with complementary measurements, supports studies of dark matter, inner density profiles, stellar populations, and the Hubble constant.Lensing also magnifies otherwise inaccessible background objects.Source: §1 · pp. 1–2
  • A large, homogeneously selected lens sample could improve the precision and reliability of lensing studies, including measurements of mass-density slopes and dark-matter substructure.The largest homogeneous sample cited is SLACS, with almost 100 observed lenses.Source: §1 · p. 2
  • Ongoing optical wide surveys are expected to find about 10^3 lenses, while Euclid and LSST could raise discovery rates to about 10^5 systems.The passage also expects a similar order of magnitude from the Square Kilometer Array.Source: §1 · p. 2
  • Visual inspection remains effective for small samples but is impractical at survey scale, prompting automatic methods that identify arcs, subtract galaxy light, model lens probabilities, or use crowdsourcing.The authors say methods perform best on different lens types and data, and a common-dataset comparison is beyond this paper’s scope.Source: §1 · p. 2
  • CNNs automatically extract image features during training, and prior astronomical applications include galaxy morphology, photometric redshifts, and star–galaxy classification.This paper presents a CNN morphological lens finder and applies it to KiDS as part of the Lenses in KiDS project.Source: §1 · p. 2

2 THE KIDS SURVEY

KiDS combines wide optical coverage with sharp r-band imaging, and its DR3 data provide the photometry and redshifts used here. The study selects an expanded sample of LRGs for CNN training and lens searches.

2 THE KIDS SURVEY

  • KiDS covers 1,500 square degrees in four optical bands, with median r-band PSF FWHM of 0.65 arcsec and a 0.21-arcsec pixel scale.The survey was designed primarily for weak-lensing tomography, while its image quality and area also suit strong-lens searches.Source: §2 · p. 3

2.1 Data Release Three

  • KiDS DR3 supplies multi-band photometry and photometric redshifts for 292 tiles; the analysis uses an unmasked effective area of 255 square degrees.Photometric redshifts are estimated with BPZ, and sources are filtered using catalogue flags and masks.Source: §2.1 · p. 3

2.2 Luminous red galaxy sample

  • The selection targets massive LRGs and combines an adapted low-redshift colour-magnitude cut with an r-band source-size cut relative to each tile’s PSF.The sample is used both to train the CNN and to search for lens candidates.Source: §2.2 · pp. 3–4
  • The LRG selection yields 21,789 galaxies, about three times the count from the original Eisenstein et al. colour-magnitude cut.The sample averages 74 LRGs per tile; Figure 1 plots colour against photometric redshift and distinguishes the original and expanded selections.Source: Figure 1 · p. 3, §2.2 · p. 4

3 TRAINING THE CNN TO FIND LENSES

The CNN is trained on labeled lens and non-lens images built from real KiDS galaxies and simulated lensed sources. The training design uses a hybrid dataset, with mock lenses and image augmentation intended to help the network recognize lens morphology in survey data.

3.1 Input Samples

  • The available Masterlens sample of 657 systems is too small and heterogeneous for training, so the authors build mock lenses by combining simulated sources with real KiDS galaxies and their fields.The real fields retain seeing, noise, and lens-environment features that are difficult to simulate and whose omission could limit recognition in survey data.Source: §3.1 · p. 4

3 TRAINING THE CNN TO FIND LENSES

  • Six million labeled images train the CNN, which classifies an image as a lens when its output p exceeds 0.5.The set contains three million lens and three million non-lens images; the network learns image features from labeled examples during training.Source: §3.1 · p. 4, §3 · p. 4

3.1.1 Real Galaxy Sample

  • The real galaxy sample combines 6,326 LRGs, 218 visually identified contaminants, and 990 sources previously misclassified as lenses by CNNs.The 6,554 inspected LRGs were a third of the full sample; most identified contaminants were face-on spirals.Source: §3.1.1 · p. 5

3.1.2 Mock Lensed-Source Sample

  • The mock sample contains 10^6 simulated lensed images, with uniformly sampled model parameters rather than a distribution intended to match real lens populations.Sources use Sérsic profiles and lenses use Singular Isothermal Ellipsoid models; the simulations emphasize magnified rings, arcs, quads, folds, and cusps over harder-to-distinguish doubles. Einstein radii span 1.4–5 arcsec, with a minimum axis ratio of 0.3 and external shear below 0.05.Source: §3.1.2 · p. 5

3.2 Building the training examples

  • Positive examples combine a perturbed simulated source and real LRG, while negative examples use real galaxies, sampled with 60% probability as LRGs and 40% as contaminants or false positives.Positive images scale source peak brightness to 2–20% of the LRG peak before combining, clipping negative pixels, applying a square-root stretch, and normalizing; inputs are 60×60 pixels, about 12×12 arcsec.Source: §3.2 · pp. 5–6

3.3 Data augmentation

  • Augmentation applies random rotations, shifts, horizontal flips, and rescaling to both real-galaxy and mock-source images before extracting central 60×60-pixel regions.Shifts range from −4 to +4 pixels, horizontal flips have a 50% probability, and scale factors are sampled log-uniformly from 1/1.1 to 1.1.Source: §3.3 · p. 6

4 RESULTS

CNN screening followed by visual review reduces the 21,789-galaxy LRG sample to 56 lens candidates. Under the same color-magnitude and Einstein-radius cuts, LensPop forecasts about 50 detectable lenses in the 255-square-degree KiDS-DR3 area.

4.2 Final sample of candidates

  • 56 candidates remain after CNN and visual classification, down-selected from 21,789 galaxies.The final sample includes two of the three known lenses; the network missed the one with a 0.83-arcsec Einstein radius, below its training range.Source: §4.2 · pp. 6–7, Figure 7 · p. 9

4.1 Candidate selection

  • 761 of 21,789 LRGs scored p > 0.5 as lens candidates; visual reviewers used Sure, Maybe, and No categories and a score threshold of 17.The score distribution rises rapidly below the threshold and remains flat above it.Source: §4.1 · p. 6, Figure 5 · p. 7

4.2.1 Expected number of lenses

  • For the KiDS-DR3 effective area of 255 sq. deg, LensPop forecasts ∼50 lenses meeting the colour-magnitude cut and Einstein radius >1.4 arcsec, broadly comparable to the final candidate sample.Without those selection constraints, the forecast is ∼500 lenses for the same 255 sq. deg effective area.Source: §4.2.1 · p. 7

4.3 Sample Characterization and Comparison

  • The candidates have median stellar mass log M⋆/M⊙∼11.2 dex, consistent within scatter and uncertainties with SLACS and SL2S.Their median redshift is 0.28+0.12−0.08, and the z < 0.4 color cut helps explain why it is below the SL2S median.Source: §4.3 · p. 9

4.3.1 A sanity check of the candidates

  • About half of candidates with measured velocity dispersions overlap the predicted and visually estimated Einstein-radius ranges; KSL627 and KSL376 are more likely ring galaxies.KSL627 and KSL376 have blue rings of ∼4.3 and ∼5.7 arcsec radii, respectively.Source: §4.3.1 · p. 10
  • The dynamical checks indicate candidate nature but do not validate it; group membership can make stellar velocity dispersion fail to trace group dynamics.The authors call for spectroscopic validation and follow-up, including arc redshift estimates.Source: §4.3.1 · p. 10

5 CONCLUSIONS

The CNN-based search retrieved 761 candidates from 21,789 LRGs and visual inspection retained 56. The results support a lower-limit forecast of about 100 LRG lenses in full KiDS, with a much larger optimistic yield if the search is widened.

5 CONCLUSIONS

  • Einstein-radius checks compare dynamical estimates across source redshift with visual estimates; for candidates without measured velocity dispersion, it is inferred from stellar mass.The Figure 9 comparison includes a Jeans-dynamical estimate and an estimate fixing σSIS to the stellar velocity dispersion; Figure 10 covers candidates without measured dispersion.Source: Figure 9 · p. 11, Figure 10 · p. 12
  • 761 candidates (3.6% of 21,789 LRGs) were retrieved, and joint visual inspection retained 56.Seven human classifiers performed the visual inspection.Source: §5 · p. 11
  • The CNN correctly classified two of three known lenses; the missed lens's 0.83-arcsec Einstein radius was below its 1.4–5.0-arcsec training range.The missed lens also had less prominent lensed images than the other two.Source: §5 · p. 11
  • A sanity check suggests ~22 solid candidates, roughly consistent with the ~50 expected for the KiDS-DR3 survey area.The check used candidates with velocity-dispersion measurements or stellar-mass estimates.Source: §5 · p. 11
  • The full KiDS survey is expected to yield at least ~100 LRG lens candidates; widening the restricted selection could raise the optimistic estimate to ~2,400.The optimistic scenario assumes nearly all lenses have signal-to-noise above 20 and at least three PSF resolution elements for arc-like images.Source: §5 · p. 11
  • False positives currently outnumber true positives by ~40-fold; reducing their rate below 0.1% without substantially lowering true positives remains difficult.Human visual inspection suggests at least 0.25% can be reached, possibly with additional RGB colour information.Source: §5 · pp. 11–12
  • A realistic validation set should reproduce survey conditions, where negatives far outnumber positives.Source: §5 · p. 12
  • Potential improvements include training on lens-like false positives, model averaging, galaxy-subtracted images, or multiband images.The authors also plan spectroscopic follow-up of their best candidates and application to the full KiDS survey.Source: §5 · pp. 12–13

APPENDIX A: NEURAL NETWORKS

The appendix introduces the theory of Convolutional Neural Networks.

APPENDIX A: NEURAL NETWORKS

  • The appendix provides a short introduction to Convolutional Neural Networks (CNNs).Source: Appendix A · p. 17

A1 Feed-forward neural network

A feed-forward neural network transforms a training example through hidden units toward a desired output. It learns weights and biases by minimizing a loss function.

A1 Feed-forward neural network

A2 Convolutional Neural Network

A CNN processes image-like inputs as matrices, using convolutional filters to build feature maps. Stacked layers learn increasingly complex representations that the output layer classifies.

A2 Convolutional Neural Network

APPENDIX B: CNN IMPLEMENTATION

The appendix describes the CNN implementation and training setup, including its software, hardware, and processing times.

APPENDIX B: CNN IMPLEMENTATION

  • Training used Python 2.7 with Lasagne and Theano, while CPU data augmentation used scikit-image.Source: Appendix B · p. 18
  • A GeForce GTX 760 trained the network in parallel with CPU-based data augmentation.Source: Appendix B · p. 18
  • Training took about 2 hours, and classifying the LRG sample took about 20 minutes.Source: Appendix B · p. 18

B1 Network architecture

The CNN uses convolutional and fully connected layers to process 60-by-60 inputs and output a strong-lens probability.

B1 Network architecture

  • A 60-by-60 input feeds four convolutional layers, followed by two fully connected layers of 1024 units each.The convolutional layers use 32, 64, 128, and 128 filters, respectively.Source: B1 Network architecture · p. 18
  • Max-pooling follows the first two convolutional layers and the second fully connected layer, reducing dimensionality and teaching translational invariance.Source: B1 Network architecture · p. 18
  • A sigmoid output gives a value from 0 to 1 representing the probability that an input is a strong gravitational lens.Batch normalization is applied before each layer's non-linearity; the architecture has about 30 million trainable parameters.Source: B1 Network architecture · p. 18
  • Figure A2 schematically depicts the CNN architecture, and Table A1 summarizes the characteristics of its layers.Source: Figure A2 · p. 19, Table A1 · p. 19

B2 Training

The network is trained with binary crossentropy and mini-batch stochastic gradient descent using ADAM updates, alongside regularization.

B2 Training

  • Training minimizes binary crossentropy between lens/non-lens targets and predictions from the network's sigmoid output.Source: B2 Training · p. 19
  • Training used 10,000 ADAM updates on batches of 600 images, each containing 300 lens and 300 non-lens examples.The learning rate decreased from 0.004 to 0.0004 after 3 million examples and to 0.00004 after 5.5 million.Source: B2 Training · p. 19
  • Dropout in fully connected layers reduces computed parameters and the possibility of over-fitting, while L2 regularization uses λ = 10^-4.The L2 penalty favors small diffuse weights over peaky ones.Source: B2 Training · p. 19
  • Figure B1 shows network-output distributions for lens and non-lens validation examples.Source: Figure B1 · p. 19
  • Figure B2 compares output distributions for the full LRG sample and the 56 visually selected candidates.Source: Figure B2 · p. 20

B3 Analysis

The network performs well on a balanced validation set, but its output varies with simulated lens properties and the real LRG sample differs from validation.

B3 Analysis

  • 96% accuracy was reached for both lens and non-lens examples on a 5,000-image validation set split evenly between the classes.Source: B3 Analysis · p. 20
  • Lens-example output distributions skew higher for larger Einstein radii, higher magnification, and more luminous sources relative to the KiDS galaxy.The paper reports more accurate classification for training examples with these higher parameter values.Source: B3 Analysis · p. 20
  • The 56 visually selected candidates are a subset of 761 candidates with p > 0.5, but the full LRG sample lacks a significant high-output peak like validation.Source: B3 Analysis · p. 20
  • The authors attribute the real-versus-simulated distribution difference partly to fewer lenses in the LRG sample and recommend validation sets matching real-lens counts and parameter distributions.Source: B3 Analysis · p. 20

APPENDIX C: R-BAND IMAGES OF THE CANDIDATES

Appendix C presents r-band KiDS images of the 56 visually selected CNN candidates, the images used as the network’s input. Figure B3 separately shows CNN-output distributions across bins of mock-source parameters.

APPENDIX C: R-BAND IMAGES OF THE CANDIDATES

  • The r-band KiDS images shown in Appendix C were the actual inputs to the CNN.Source: Appendix C · p. 20
  • Figure B3 plots CNN outputs across mock-source parameter bins, with colors marking brightness ratio, Einstein radius, and magnification.Source: Figure B3 · p. 21
  • Figure C1 shows square-root-stretched r-band KiDS images of the 56 candidates selected by visual inspection.Each image is 20 by 20 arcsec and is labelled with an internal ID and visual classification score, out of 70.Source: Figure C1 · p. 22
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