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-<!-- Please make sure you are opening a pull request against the `accepted` branch (not master!) of the STAGING repo (not 2023!) -->
 
-## OpenReview Submission Thread
-
-<!-- link to your OpenReview submission -->
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-## Checklist before requesting a review
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-<!-- To tick a box, put an 'x' inside it (e.g. [x]) -->
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2023-11-09-dof-visual-place-recognition-satellite/2023-11-09-dof-visual-place-recognition-satellite.md

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+---
+layout: distill
+title: 6-DOF estimation through visual place recognition
+description: A neural Visual Place Recognition solution is proposed which could help an agent with a downward-facing camera (such as a drone) to geolocate based on prior satellite imagery of terrain. The neural encoder infers extrinsic camera parameters from camera images, enabling estimation of 6 degrees of freedom (6-DOF), namely 3-space position and orientation. By encoding priors about satellite imagery in a neural network, the need for the agent to carry a satellite imagery dataset onboard is avoided.
+date: 2023-11-09
+htmlwidgets: true
+
+# Anonymize when submitting
+# authors:
+#   - name: Anonymous
+
+authors:
+  - name: Andrew Feldman
+    url: "https://andrew-feldman.com/"
+    affiliations:
+      name: MIT
+
+# must be the exact same name as your blogpost
+bibliography: 2023-11-09-dof-visual-place-recognition-satellite.bib  
+
+# Add a table of contents to your post.
+#   - make sure that TOC names match the actual section names
+#     for hyperlinks within the post to work correctly.
+toc:
+  - name: Introduction
+  - name: Background
+#  - name: Images and Figures
+#    subsections:
+#    - name: Interactive Figures
+  - name: Proposed solution
+    subsections:
+    - name: Image-to-extrinsics encoder architecture
+    - name: Data sources for offline training
+    - name: Training and evaluation
+      subsections:
+      - name: Data pipeline
+      - name: Training
+      - name: Hyperparameters
+      - name: Evaluation
+
+# Below is an example of injecting additional post-specific styles.
+# This is used in the 'Layouts' section of this post.
+# If you use this post as a template, delete this _styles block.
+_styles: >
+  .fake-img {
+    background: #bbb;
+    border: 1px solid rgba(0, 0, 0, 0.1);
+    box-shadow: 0 0px 4px rgba(0, 0, 0, 0.1);
+    margin-bottom: 12px;
+  }
+  .fake-img p {
+    font-family: monospace;
+    color: white;
+    text-align: left;
+    margin: 12px 0;
+    text-align: center;
+    font-size: 16px;
+  }
+---
+
+# Introduction
+
+The goal of this project is to demonstrate how a drone or other platform with a downward-facing camera could perform approximate geolocation through visual place recognition, using a neural scene representation of existing satellite imagery.
+
+Visual place recognition<d-cite key="Schubert_2023"></d-cite> refers to the ability of an agent to recognize a location which it has not previously seen, by exploiting a system for cross-referencing live camera footage against some ground-truth of prior image data.
+
+In this work, the goal is to compress the ground-truth image data into a neural model which maps live camera footage to geolocation coordinates.
+
+Twitter user Stephan Sturges demonstrates his solution<d-cite key="Sturges_2023"></d-cite> for allowing a drone with a downward-facing camera to geolocate through cross-referencing against a database of satellite images:
+
+<div class="row mt-3">
+    <div class="col-sm mt-3 mt-md-0">
+        {% include figure.html path="assets/img/2023-11-09-dof-visual-place-recognition-satellite/sturges_satellite_vpr.jpeg" class="img-fluid rounded z-depth-1" %}
+    </div>
+</div>
+<div class="caption">
+    Twitter user Stephan Sturges shows the results<d-cite key="Sturges_2023"></d-cite> of geolocation based on Visual Place Recognition.
+</div>
+
+The author of the above tweet employs a reference database of images. It would be interesting to eliminate the need for a raw dataset.
+
+Thus, this works seeks to develop a neural network which maps a terrain image from the agent's downward-facing camera, to a 6-DOF (position/rotation) representation of the agent in 3-space. Hopefully the neural network is more compact than the dataset itself - although aggressive DNN compression will not be a focus of this work.
+
+# Background
+
+The goal-statement - relating a camera image to a location and orientation in the world - has been deeply studied in computer vision and rendering<d-cite key="Anwar_2022"></d-cite>:
+
+<div class="row mt-3">
+    <div class="col-sm mt-3 mt-md-0">
+        {% include figure.html path="assets/img/2023-11-09-dof-visual-place-recognition-satellite/camera_intrinsic_extrinsic.png" class="img-fluid rounded z-depth-1" %}
+    </div>
+</div>
+<div class="caption">
+    Camera parameters, as described in<d-cite key="Anwar_2022"></d-cite>.
+</div>
+
+Formally<d-cite key="Anwar_2022"></d-cite>,
+* The image-formation problem is modeled as a camera forming an image of the world using a planar sensor.
+* **World coordinates** refer to 3-space coordinates in the Earth or world reference frame.
+* **Image coordinates** refer to 2-space planar coordinates in the camera image plane.
+* **Pixel coordinates** refer to 2-space coordinates in the final image output from the image sensor, taking into account any translation or skew of pixel coordinates with respect to the image coordinates.
+
+The mapping from world coordinates to pixel coordinates is framed as two composed transformations, described as sets of parameters<d-cite key="Anwar_2022"></d-cite>:
+* **Extrinsic camera parameters** - the transformation from world coordinates to image coordinates (affected by factors "extrinsic" to the camera internals, i.e. position and orientation.)
+* **Intrinsic camera parameters** - the transformation from image coordinates to pixel coordinates (affected by factors "intrinsic" to the camera's design.)
+
+And so broadly speaking, this work strives to design a neural network that can map from an image (taken by the agent's downward-facing camera) to camera parameters of the agent's camera. With camera parameters in hand, geolocation parameters automatically drop out from extracting extrinsic translation parameters.
+
+To simplify the task, assume that camera intrinsic characteristics are consistent from image to image, and thus could easily be calibrated out in any application use-case. Therefore, this work focuses on inferring **extrinsic camera parameters** from an image. We assume that pixels map directly into image space.
+
+The structure of extrinsic camera parameters is as follows<d-cite key="Anwar_2022"></d-cite>:
+
+$$
+\mathbf{E}_{4 \times 4} = \begin{bmatrix} \mathbf{R}_{3 \times 3} & \mathbf{t}_{3 \times 1} \\ \mathbf{0}_{1 \times 3} & 1 \end{bmatrix}
+$$
+
+where $$\mathbf{R}_{3 \times 3} \in \mathbb{R^{3 \times 3}}$$ is rotation matrix representing the rotation from the world reference frame to the camera reference frame, and $$\mathbf{t}_{3 \times 1} \in \mathbb{R^{3 \times 1}}$$ represents a translation vector from the world origin to the image/camera origin.
+
+Then the image coordinates (a.k.a. camera coordinates) $$P_c$$ of a world point $$P_w$$ can be computed as<d-cite key="Anwar_2022"></d-cite>:
+
+$$
+\mathbf{P_c} = \mathbf{E}_{4 \times 4} \cdot \mathbf{P_w}
+$$
+
+# Proposed solution
+
+## Image-to-extrinsics encoder architecture
+
+The goal of this work, is to train a neural network which maps an image drawn from $$R^{3 \times S \times S}$$ (where $$S$$ is pixel side-length of an image matrix) to a pair of camera extrinsic parameters $$R_{3 \times 3}$$ and $$t_{3 \times 1}$$:
+
+$$
+\mathbb{R^{3 \times S \times S}} \rightarrow \mathbb{R^{3 \times 3}} \times \mathbb{R^3}
+$$
+
+The proposed solution is a CNN-based encoder which maps the image into a length-12 vector (the flattened extrinsic parameters); a hypothetical architecture sketch is shown below:
+
+<div class="row mt-3">
+    <div class="col-sm mt-3 mt-md-0">
+        {% include figure.html path="assets/img/2023-11-09-dof-visual-place-recognition-satellite/nn.svg" class="img-fluid rounded z-depth-1" %}
+    </div>
+</div>
+<div class="caption">
+    Image encoder architecture.
+</div>
+
+## Data sources for offline training
+
+Online sources<d-cite key="Geller_2022"></d-cite> provide downloadable satellite terrain images.
+
+## Training and evaluation
+
+The scope of the model's evaluation is, that it will be trained to recognize aerial views of some constrained area i.e. Atlantic City New Jersey; this constrained area will be referred to as the "area of interest."
+
+### Data pipeline
+
+The input to the data pipeline is a single aerial image of the area of interest. The output of the pipeline is a data loader which generates augmented images.
+
+The image of the area of interest is $$\mathbb{R^{3 \times T \times T}}$$ where $$T$$ is the image side-length in pixels.
+
+Camera images will be of the form $$\mathbb{R^{3 \times S \times S}}$$ where $$S$$ is the image side-length in pixels, which may differ from $$T$$.
+
+* **Generate an image from the agent camera's vantage-point**
+    * Convert the area-of-interest image tensor ($$\mathbb{R^{3 \times T \times T}}$$) to a matrix of homogenous world coordinates ($$\mathbb{R^{pixels \times 4}}$$) and an associated matrix of RGB values for each point ($$\mathbb{R^{pixels \times 3}}$$)
+        * For simplicity, assume that all features in the image have an altitutde of zero
+        * Thus, all of the pixel world coordinates will lie in a plane
+    * Generate random extrinsic camera parameters $$R_{3 \times 3}$$ and $$t_{3 \times 1}$$
+    * Transform the world coordinates into image coordinates ($$\mathbb{R^{pixels \times 3}}$$) (note, this does not affect the RGB matrix)
+    * Note - this implicitly accomplishes the commonly-used image augmentations such as shrink/expand, crop, rotate, skew
+* **Additional data augmentation** - to prevent overfitting
+    * Added noise
+    * Color/brightness adjustment
+    * TBD
+* **Convert the image coordinates and the RGB matrix into a camera image tensor ($$\mathbb{R^{3 \times S \times S}}$$)**
+
+Each element of a batch from this dataloader, will be a tuple of (extrinsic parameters,camera image).
+
+## Training
+
+* For each epoch, and each mini-batch...
+* unpack batch elements into camera images and ground-truth extrinsic parameters
+* Apply the encoder to the camera images
+* Loss: MSE between encoder estimates of extrinsic parameters, and the ground-truth values
+
+### Hyperparameters
+* Architecture
+    * Encoder architecture - CNN vs MLP vs ViT(?) vs ..., number of layers, ...
+    * Output normalizations
+    * Nonlinearities - ReLU, tanh, ...
+* Learning-rate
+* Optimizer - ADAM, etc.
+* Regularizations - dropout, L1, L2, ...
+
+## Evaluation
+
+For a single epoch, measure the total MSE loss of the model's extrinsic parameter estimates relative to the ground-truth. 
+
+## Feasibility
+
+Note that I am concurrently taking 6.s980 "Machine learning for inverse graphics" so I already have background in working with camera parameters, which should help me to complete this project on time.

2023-11-09-dof-visual-place-recognition-satellite/assets/bibliography/2023-11-09-dof-visual-place-recognition-satellite.bib

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+@article{Schubert_2023,
+	doi = {10.1109/mra.2023.3310859},
+
+	url = {https://doi.org/10.1109%2Fmra.2023.3310859},
+
+	year = 2023,
+	publisher = {Institute of Electrical and Electronics Engineers ({IEEE})},
+
+	pages = {2--16},
+
+	author = {Stefan Schubert and Peer Neubert and Sourav Garg and Michael Milford and Tobias Fischer},
+
+	title = {Visual Place Recognition: A Tutorial},
+
+	journal = {{IEEE} Robotics {\&}amp$\mathsemicolon$ Automation Magazine}
+}
+
+@misc{Sturges_2023, title={Demo: One Minute of UAV visual navigation from satellite imagery where GPS coordinates are derived by comparing an image from a downward facing camera to a prior satellite image 😄 this is a solution for high-altitude navigation with no GPS. pic.twitter.com/y9i0qg4zng}, url={https://twitter.com/StephanSturges/status/1722236377990082632?t=1Yf69YRUIkydn8lu0g8Epw}, journal={Twitter}, publisher={Twitter}, author={Sturges, Stephan}, year={2023}, month={Nov}}
+
+@misc{Anwar_2022, title={What are Intrinsic and Extrinsic Camera Parameters in Computer Vision?}, url={https://towardsdatascience.com/what-are-intrinsic-and-extrinsic-camera-parameters-in-computer-vision-7071b72fb8ec}, journal={Medium}, author={Anwar, Aqeel}, year={2022}, month={Feb}} 
+
+@misc{Geller_2022, title={Downloading satellite images made “Easy” hero image}, url={https://sites.northwestern.edu/researchcomputing/2021/11/19/downloading-satellite-images-made-easy/}, journal={Research Computing and Data Services Updates}, publisher={© 2023 Northwestern University}, author={Geller, Aaron}, year={2022}, month={Nov}} 
+
+@misc{Taylor_2020, title={New Jersey: Images of the Garden State}, url={https://www.theatlantic.com/photo/2020/08/new-jersey-photos/614872/}, journal={The Atlantic}, publisher={Atlantic Media Company}, author={Taylor, Alan}, year={2020}, month={Aug}} 
+
+@misc{sitzmann2020implicit,
+      title={Implicit Neural Representations with Periodic Activation Functions}, 
+      author={Vincent Sitzmann and Julien N. P. Martel and Alexander W. Bergman and David B. Lindell and Gordon Wetzstein},
+      year={2020},
+      eprint={2006.09661},
+      archivePrefix={arXiv},
+      primaryClass={cs.CV}
+}
+
+@misc{xiang2018posecnn,
+      title={PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes}, 
+      author={Yu Xiang and Tanner Schmidt and Venkatraman Narayanan and Dieter Fox},
+      year={2018},
+      eprint={1711.00199},
+      archivePrefix={arXiv},
+      primaryClass={cs.CV}
+}