Persistent Images of Gravitionally Lensed Galaxies

• this is an image of a galaxy cluster
• collection of \(10^2\)-\(10^3\) galaxies
• believe it or not, 2 parts to pay attention to:
  • foreground galaxy
  • background galaxy
• but what exactly are they and why do we care about them?

What?

• Predicted by GR
• Light from distant galaxies gets warped by foreground galaxies + dark matter

Why?

• Natural way to see futher
• Map
  • the universe
  • dark matter
  • dark energy

[explain dark matter and dark energy]

• For this presentation, they are
  • predicted by GR
  • result of when light from distant…
• Why do we care about them?
  • allow us to see more distant galaxies
  • map further into universe
  • more datapoints of places with dark matter –> better understand dark matter
  • further we see, the better we can estimate expansion, better we can constrain/characterize dark energy
  • …[pause]
  • Now we know what and why…

Euclid

• here’s the how
• Euclid:
  • telescope that was launched in 2023
  • goals to study dark matter like never before
• …[pause]
• to give you an idea of what I mean by “like never before”, I included the following video

Video

• In March of 2025, Euclid released its first data products [gesture to the screen]
• …[pause]
• What you you watching is 1% of the area Euclid plans on covering in its mission
• …[pause]
• What are Euclid’s goals?

Euclid Goals

• 3D map of \(\frac{1}{3}\) of the sky
• dark matter and dark energy
• 10 billion light years

(Collaboration et al. 2025)

• 3D map of \(\frac{1}{3}\) of the sky (why 1/3? turns out lots of dust in galaxy; we want to avoid)
• dark matter and dark energy (get better understanding)
• 10 billion light years (very big)

Overall Goals of Euclid

• What is the structure and history of the cosmic web?

• What is the nature of dark matter?

• How has the expansion of the Universe changed over time?

• What is the nature of dark energy?

• Is our understanding of gravity complete?

• That brings us to my project

Research Goal

Can persistence images help classify features in natural images better than just natual image classification on its own?

[read off slide]

• In Astro, TDA is applied to large point cloud data (cosmic web) (my original goal)
• Not really applied to individual images/small scale Sousbie, Pichon, and Kawahara (2011)
• Let’s try it
• Could be useful in other classifications (ie everything is based on how we see things)
• …[pause]
• So how we do this? More specifically, how did I do this?

Data

• Coordinates from Walmsley et al. (2025)
• Only select top candidates
• Get image cutouts of each target
• Preprocess the images (resize, clip, scale, and invert)

• The Euclid Collaboration published a list of lensed galaxies they identified through ML algorithm
• Take coordinates from that
• Filter by only grade “A” candidates ~250
• Get image cutouts of each target (242 now)
• Preprocess the images (resize, clip, scale, and invert (h0))
• …[pause]
• those images look like this [next slide]

• on left, we have the “natural” images of the cluster
• Note that most of the image is a value of 0
• can pose a problem for sublevel persistence algorithms
  • start from the lowest pixel value and end at the highest
  • like the Cubical persistence algorithm in the python library giotto, which I used for this project
  • Note cubical persistence can be thought of as the cubical analog to simplicial persistence
• Note that the sublevel persistence issue is more of a problem when identifying connected components
  • If everything is zero, then everything will be connected (we can see that in the diagram too)
• We can get around this by inverting the image for our h0 calculations
  • We can see the rest of the process is straightforward: compute persistence diagram, then convert the persistence diagram into a persistence image using the Persim python library
• Unlike the h0 superlevel persistence, the h1 calculations can be done with sublevel persistence
• Once we have our images, we can vectorize and concatenate h0 and h1
• Convert vector to dataframe
• Perform Hierarchical Clustering

• Here are the results for the “natural” images
• These were just the raw images from the previous slide
• If you haven’t seen a dendrogram before, essentially: [explain dendrogram]
• The key thing to take note of for this is the hight at which we start seeing connections
• That is: when do clusters start being similar?
• Keep that hight in mind as we switch to the results from the persistence image clustering

• what do you notice?
• Did anyone catch the scale? (0-600) now (0-100)
• The persistence image appears to have found similar clusters more easily than the natural images

Discussion

Issues

• Dimensionality: natural vs persistence
• Natural: Curse of Dimensionality
• PCA: highly correlated variables in natural images

Future Work

• Finish PCA
• Alternative clustering algorithms
• Reducing size of natural images
• Exploring parameter space of persistence images

• Overall, I’m not ready to definitively say yes: persistence images perform better than natural images for clustering
• Why?
• The natural images themselves are 119x119 grids collapsed into 242 ~14000 length vectors (very wide = bad)
• Persistence images are only 242 200 length vectors (more long = good)
• So we have a dimensionality issue
• Speaking a little more on that, hierarchical clustering suffers from the curse of dimensionality, where more variables causes things to naturally become further apart
• Additionally, my first look into PCA showed a lot of highly correlated variables
  • Natural clustering process could be likely simplified

Questions?

References

Chen, Yen-Chi, Shirley Ho, Peter E. Freeman, Christopher R. Genovese, and Larry Wasserman. 2015. “Cosmic Web Reconstruction Through Density Ridges: Method and Algorithm.” Monthly Notices of the Royal Astronomical Society 454 (1): 1140–56. https://doi.org/10.1093/mnras/stv1996.

Collaboration, Euclid, M. Walmsley, P. Holloway, N. E. P. Lines, K. Rojas, T. E. Collett, A. Verma, et al. 2025. “Euclid Quick Data Release (Q1): The Strong Lensing Discovery Engine a – System Overview and Lens Catalogue.” https://arxiv.org/abs/2503.15324.

Sousbie, T., C. Pichon, and H. Kawahara. 2011. “The Persistent Cosmic Web and Its Filamentary Structure – II. Illustrations.” Monthly Notices of the Royal Astronomical Society 414 (1): 384–403. https://doi.org/10.1111/j.1365-2966.2011.18395.x.

Walmsley, Mike, Philip Holloway, Natalie Lines, Karina Rojas, Thomas Collett, Aprajita Verma, Tian Li, James Nightingale, Giulia Despali, and Stefan Schuldt. 2025. “Euclid Quick Data Release (Q1): The Strong Lensing Discovery Engine.” Zenodo. https://doi.org/10.5281/zenodo.15025832.