\n
![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F25747348%2F61b7f20998538db19346d6d34c961920%2F2025-04-14%20083034.jpg?generation=1744608643370046&alt=media\")
Each dataset consists of multiple scenes. A scene is a group of related images that belong together (e.g., different viewpoints of the same object). Some images in a dataset may be outliers, meaning they donβt belong to any scene (e.g., random photos taken in the area).
\nIn the test data, all images are mixed in one folder. Your task is:
\noutlier \"Ground truth\" is the true known pose of the camera when the image was taken. In training data, this is given to you. In test data, you try to predict it.
\nCamera center C in world coordinates is calculated as:
\nC = βR Β· T
\nThis transforms the camera pose into a 3D point in space.
Camera centres of one scene:
\n
Your predicted camera centers may be rotated differently, shifted (translated), and scaled.
\nThis is normal in SfM - reconstructions can be geometrically correct but misaligned.
So, to compare your predicted camera centers C to the ground truth centers Cg, a similarity transformation T (i.e. scale, rotation and translation altogether) is applied. This aligns your 3D camera predictions to the ground truth up to scale and orientation.
\n(Some of your predicted camera poses may be wrong - far from ground truth - and we want to ignore bad data and only use good matches to compute the best alignment.)
To compute a similarity transform (rotation, translation, and scale) between two sets of 3D points, you need at least 3 point correspondences that are not collinear.
\nA triplet is a group of three matching camera centers. Each one has:
\nSo a triplet is:
\n{(C1, Cg1), (C2, Cg2), (C3, Cg3)}
RANSAC = Random Sample Consensus - Algorithm to randomly test many triplets, ignore outliers, and find the best transformation.
\nThe thresholds column lists multiple numeric values per scene. These represent distance or error thresholds that are used to evaluate how accurate predicted 3D camera poses are (compared to ground truth). Each threshold is a maximum allowable error (in meters) to consider a pose prediction as correct.
So use Hornβs method (algorithm used to compute the best similarity transformation between two sets of 3D points) to compute a similarity transformation.
\nFind scale s, rotation R, translation t so that:
= s β
R β
Ci + tThis is our first estimate Tβ².
\nThen apply Tβ² to all predicted camera centers Ci and for each one count how many of your predictions become registered cameras (i.e., within error threshold of ground truth):
= (β T(Ci) β Cgi β < threshold)Where β Β· β denotes Euclidean (L2) distance between two 3D points.
Try many random triplets and get many candidate transformations Tβ².
\nFor each one, count how many registered cameras it produces and save the best one.
\nTake the best transformation Tβ², and re-estimate it using all the inliers (not just the original triplet).
It measures how accurately your predicted camera centers match the ground truth, after alignment with a similarity transformation.
\nAs discussed before, you use triplets and Hornβs method inside RANSAC.
\nYou find the best transformation:
= s β
R β
Ci + tThis aligns your predicted camera centers to the same coordinate frame as the ground-truth.
\nThen apply this transformation - for each predicted camera center Ci, compute:
\nCi_aligned = T(Ci)
Now for each threshold t_j, check if the difference between Ci_aligned and ground-truth center Cgi is less than t_j:
\n = (β T(Ci) β Cgi β < threshold)Then compute registration ratio per threshold for scene:
\nFor each N' = N β 3 images of the scene and each threshold t_j, compute the percentage of registered cameras:
rj = ( / ') * (registered)That gives you the Average Accuracy (AA) for that scene.
\nThen find average across all thresholds:
( / k) * sum(rj)Where:
\nOnce you compute mAA per scene, you calculate final mAA for all scenes by averaging across scenes.
\nEach scene from the ground-truth data is denoted as S<sub>ki</sub>, and your predicted clusters are C<sub>kj</sub>.
The evaluation will:
\nEach ground-truth scene S<sub>ki</sub> is matched to the user-submitted cluster C<sub>kj</sub> that gives the highest mAA after alignment.
\nAll images labeled as outliers are excluded from this matching step.
After the best cluster C<sub>kji</sub> is selected for each scene S<sub>ki</sub>, compute the clustering score as:
clustering_score = ||||Where:
\n|S β© C| is the number of images that correctly overlap between your predicted cluster and the real scene |C| is the total number of images in your predicted clusterThis measures how pure your cluster is - i.e., out of everything you said belongs to this scene, how many actually do?
\nAs explained earlier, for each image:
\nThis mAA represents recall - how many real scene images you found and aligned correctly.
\nNow both mAA (recall-like) and clustering score (precision-like) are combined using the harmonic mean, similar to the F1-score in classification:
\n = * (mAA * clustering_score) / (mAA + clustering_score)This ensures that:
\nYou will repeat this evaluation for each dataset D<sub>k</sub>.
\nThe final score for your submission is:
i.e., the average harmonic mean across all datasets.
\nExtract global embeddings using models like CLIP, DINOv2, etc., then cluster them using algorithms like DBSCAN, KMeans, Agglomerative Clusteringβ¦
π Image features are compact vector representations of the visual content of the image.
\nGlobal descriptors are single vectors that represent the entire image - its objects, textures, and layout.
\nβ Use cases:
\nDINOv2 is part of the DINO (Distillation with No Labels) family of self-supervised vision transformers:
\nβ In our pipeline:
\n0.3).\nThis approach is faster and more scalable than DBSCAN for large datasets, and does not require knowing the number of clusters beforehand.\n
\nπ Note:
\nI do not use ALIKED or LightGlue at this stage.
\nThey are used later in the pipeline for local feature extraction and matching - when reconstructing 3D camera poses.
While my solution clusters images using global descriptors, another common approach in image matching is to use local features:
\nKeypoints are distinctive image regions (corners, blobs, edges) that:
\nπ§ Common keypoint detectors:
\nLocal Descriptors:
\nSmall feature vectors extracted around keypoints describing their local visual neighborhood.
\nUsed for matching keypoints between different images.
In our case:
\n\nStructure-from-Motion (SfM) is a computer vision pipeline that reconstructs:\n
\n
Triangulation - Estimating 3D coordinates of a point from two or more images of it (2D β 3D)
\nPnP (Perspective-n-Point) - Given 2D-3D correspondences, estimate the camera pose (R, T)
\nIncremental SfM - Build the scene step by step: triangulate β PnP β repeat
h5_to_db utilities.RTNaN to its pose fields.\nUsing COLMAP, pycolmap, or another SfM engine, we perform the following:\n
R (3x3)T (3x1)Each dataset consists of multiple scenes. A scene is a group of related images that belong together (e.g., different viewpoints of the same object). Some images in a dataset may be outliers, meaning they donβt belong to any scene (e.g., random photos taken in the area).
\nIn the test data, all images are mixed in one folder. Your task is:
\noutlier \"Ground truth\" is the true known pose of the camera when the image was taken. In training data, this is given to you. In test data, you try to predict it.
\nCamera center C in world coordinates is calculated as:
\nC = βR Β· T
\nThis transforms the camera pose into a 3D point in space.
Camera centres of one scene:
\n
Your predicted camera centers may be rotated differently, shifted (translated), and scaled.
\nThis is normal in SfM - reconstructions can be geometrically correct but misaligned.
So, to compare your predicted camera centers C to the ground truth centers Cg, a similarity transformation T (i.e. scale, rotation and translation altogether) is applied. This aligns your 3D camera predictions to the ground truth up to scale and orientation.
\n(Some of your predicted camera poses may be wrong - far from ground truth - and we want to ignore bad data and only use good matches to compute the best alignment.)
To compute a similarity transform (rotation, translation, and scale) between two sets of 3D points, you need at least 3 point correspondences that are not collinear.
\nA triplet is a group of three matching camera centers. Each one has:
\nSo a triplet is:
\n{(C1, Cg1), (C2, Cg2), (C3, Cg3)}
RANSAC = Random Sample Consensus - Algorithm to randomly test many triplets, ignore outliers, and find the best transformation.
\nThe thresholds column lists multiple numeric values per scene. These represent distance or error thresholds that are used to evaluate how accurate predicted 3D camera poses are (compared to ground truth). Each threshold is a maximum allowable error (in meters) to consider a pose prediction as correct.
So use Hornβs method (algorithm used to compute the best similarity transformation between two sets of 3D points) to compute a similarity transformation.
\nFind scale s, rotation R, translation t so that:
= s β
R β
Ci + tThis is our first estimate Tβ².
\nThen apply Tβ² to all predicted camera centers Ci and for each one count how many of your predictions become registered cameras (i.e., within error threshold of ground truth):
= (β T(Ci) β Cgi β < threshold)Where β Β· β denotes Euclidean (L2) distance between two 3D points.
Try many random triplets and get many candidate transformations Tβ².
\nFor each one, count how many registered cameras it produces and save the best one.
\nTake the best transformation Tβ², and re-estimate it using all the inliers (not just the original triplet).
It measures how accurately your predicted camera centers match the ground truth, after alignment with a similarity transformation.
\nAs discussed before, you use triplets and Hornβs method inside RANSAC.
\nYou find the best transformation:
= s β
R β
Ci + tThis aligns your predicted camera centers to the same coordinate frame as the ground-truth.
\nThen apply this transformation - for each predicted camera center Ci, compute:
\nCi_aligned = T(Ci)
Now for each threshold t_j, check if the difference between Ci_aligned and ground-truth center Cgi is less than t_j:
\n = (β T(Ci) β Cgi β < threshold)Then compute registration ratio per threshold for scene:
\nFor each N' = N β 3 images of the scene and each threshold t_j, compute the percentage of registered cameras:
rj = ( / ') * (registered)That gives you the Average Accuracy (AA) for that scene.
\nThen find average across all thresholds:
( / k) * sum(rj)Where:
\nOnce you compute mAA per scene, you calculate final mAA for all scenes by averaging across scenes.
\nEach scene from the ground-truth data is denoted as S<sub>ki</sub>, and your predicted clusters are C<sub>kj</sub>.
The evaluation will:
\nEach ground-truth scene S<sub>ki</sub> is matched to the user-submitted cluster C<sub>kj</sub> that gives the highest mAA after alignment.
\nAll images labeled as outliers are excluded from this matching step.
After the best cluster C<sub>kji</sub> is selected for each scene S<sub>ki</sub>, compute the clustering score as:
clustering_score = ||||Where:
\n|S β© C| is the number of images that correctly overlap between your predicted cluster and the real scene |C| is the total number of images in your predicted clusterThis measures how pure your cluster is - i.e., out of everything you said belongs to this scene, how many actually do?
\nAs explained earlier, for each image:
\nThis mAA represents recall - how many real scene images you found and aligned correctly.
\nNow both mAA (recall-like) and clustering score (precision-like) are combined using the harmonic mean, similar to the F1-score in classification:
\n = * (mAA * clustering_score) / (mAA + clustering_score)This ensures that:
\nYou will repeat this evaluation for each dataset D<sub>k</sub>.
\nThe final score for your submission is:
i.e., the average harmonic mean across all datasets.
\nExtract global embeddings using models like CLIP, DINOv2, etc., then cluster them using algorithms like DBSCAN, KMeans, Agglomerative Clusteringβ¦
π Image features are compact vector representations of the visual content of the image.
\nGlobal descriptors are single vectors that represent the entire image - its objects, textures, and layout.
\nβ Use cases:
\nDINOv2 is part of the DINO (Distillation with No Labels) family of self-supervised vision transformers:
\nβ In our pipeline:
\n0.3).\nThis approach is faster and more scalable than DBSCAN for large datasets, and does not require knowing the number of clusters beforehand.\n
\nπ Note:
\nI do not use ALIKED or LightGlue at this stage.
\nThey are used later in the pipeline for local feature extraction and matching - when reconstructing 3D camera poses.
While my solution clusters images using global descriptors, another common approach in image matching is to use local features:
\nKeypoints are distinctive image regions (corners, blobs, edges) that:
\nπ§ Common keypoint detectors:
\nLocal Descriptors:
\nSmall feature vectors extracted around keypoints describing their local visual neighborhood.
\nUsed for matching keypoints between different images.
In our case:
\n\nStructure-from-Motion (SfM) is a computer vision pipeline that reconstructs:\n
\n
Triangulation - Estimating 3D coordinates of a point from two or more images of it (2D β 3D)
\nPnP (Perspective-n-Point) - Given 2D-3D correspondences, estimate the camera pose (R, T)
\nIncremental SfM - Build the scene step by step: triangulate β PnP β repeat
h5_to_db utilities.RTNaN to its pose fields.\nUsing COLMAP, pycolmap, or another SfM engine, we perform the following:\n
R (3x3)T (3x1)outlier \n3. Reconstruct the camera pose (R, T) for each image (except outliers)\n\n---\n\n{(C1, Cg1), (C2, Cg2), (C3, Cg3)}\n\nRANSAC = Random Sample Consensus - Algorithm to randomly test many triplets, ignore outliers, and find the best transformation.\n\nThe thresholds column lists multiple numeric values per scene. These represent distance or error thresholds that are used to evaluate how accurate predicted 3D camera poses are (compared to ground truth). Each threshold is a maximum allowable error (in meters) to consider a pose prediction as correct.\n\n---\n\nSo use Hornβs method (algorithm used to compute the best similarity transformation between two sets of 3D points) to compute a similarity transformation. \nFind scale s, rotation R, translation t so that:\n\nT(Ci) = s β
R β
Ci + tis_registered = (β T(Ci) β Cgi β < threshold)β Β· β denotes Euclidean (L2) distance between two 3D points.\n\nTry many random triplets and get many candidate transformations Tβ². \nFor each one, count how many registered cameras it produces and save the best one. \nTake the best transformation Tβ², and re-estimate it using all the inliers (not just the original triplet).\n\n---\n\nT(Ci) = s β
R β
Ci + tCi_aligned = T(Ci)\n\nNow for each threshold t_j, check if the difference between Ci_aligned and ground-truth center Cgi is less than t_j:\n\nis_registered = (β T(Ci) β Cgi β < threshold)rj = (1 / N') * sum(registered)mAA = (1 / k) * sum(rj)Ski, and your predicted clusters are Ckj.\n\nThe evaluation will:\n- Compare your clusters with the true scenes \n- Check how well your clustering matches the original grouping \n- Measure how accurate your pose predictions are (as explained earlier with mAA)\n\nEach ground-truth scene Ski is matched to the user-submitted cluster Ckj that gives the highest mAA after alignment. \nAll images labeled as outliers are excluded from this matching step.\n\n---\n\nCkji is selected for each scene Ski, compute the clustering score as:\n\nclustering_score = |S β© C| / |C||S β© C| is the number of images that correctly overlap between your predicted cluster and the real scene \n- |C| is the total number of images in your predicted cluster\n\nThis measures how pure your cluster is - i.e., out of everything you said belongs to this scene, how many actually do?\n\n---\n\nfinal_score_per_dataset = 2 * (mAA * clustering_score) / (mAA + clustering_score)Dk. \nThe final score for your submission is:\n\nmean(final_score_per_dataset)CLIP, DINOv2, etc., then cluster them using algorithms like DBSCAN, KMeans, Agglomerative Clustering...\n\nπ **Image features** are compact vector representations of the visual content of the image.\n\n---\n\n##### 0.3).\n- **Connected components** of this graph are treated as **predicted scenes**.\n- Images with no edges (isolated nodes) are labeled as **outliers**.\n\n\nThis approach is faster and more scalable than DBSCAN for large datasets, and does not require knowing the number of clusters beforehand.\n
\n\nπ **Note**: \nI do not use ALIKED or LightGlue at this stage. \nThey are used later in the pipeline for **local feature extraction and matching** - when reconstructing 3D camera poses.\n\n---\n\n#####\nStructure-from-Motion (SfM) is a computer vision pipeline that reconstructs:\n
\n
h5_to_db utilities.RTNaN to its pose fields.\nUsing COLMAP, pycolmap, or another SfM engine, we perform the following:\n
R (3x3)T (3x1)Great stuff! And all in very details, just what I needed to get started! Thanks @dmytrobuhai !
", "rawMarkdown": "Great stuff! And all in very details, just what I needed to get started! Thanks @dmytrobuhai !", "votes": 1 }, { "id": 3203991, "postDate": "2025-05-17T16:35:55.340Z", "content": "Thank you a lot for the extremely useful and concise explanations! It is my first day during which I am deeply diving into the competition details and this topic definitely helped me to quickly grasp main theoretical concepts. π₯
", "rawMarkdown": "Thank you a lot for the extremely useful and concise explanations! It is my first day during which I am deeply diving into the competition details and this topic definitely helped me to quickly grasp main theoretical concepts. π₯", "votes": 1 }, { "id": 3186803, "postDate": "2025-04-25T07:32:38.420Z", "content": "I have some confusion about the calculation formula for the Camera center C in world coordinates. Could you please explain how it is derived? I would be very grateful if you could reply.
", "rawMarkdown": "I have some confusion about the calculation formula for the Camera center C in world coordinates. Could you please explain how it is derived? I would be very grateful if you could reply.", "votes": 1, "replies": [ { "id": 3187230, "postDate": "2025-04-25T17:32:22.147Z", "content": "hello, this formula:
\nX_cam = R β X_world + T\n\n
transforms a point from the world coordinate system (global 3D space) into the camera coordinate system.
\n\n\nThink of it this way: the world is fixed, you have a camera somewhere in that world, looking in a specific direction. You want to understand how a 3D point in the world appears from the perspective of the camera.
\n
That means take the point in world space, rotate it into the camera's view direction, and then translate it so that the camera becomes the new origin
\nIf the camera is not aligned with the world axes, then we need to rotate the world to the cameraβs orientation. Each row (or column, depending on convention) of R is a unit vector representing one axis of the rotated frame.
\nFirstly, let's clarify that vectors are orthonormal if they satisfy two conditions:
\nA rotation matrix R is an orthonormal linear transformation that:
a. Preserves lengths:
\n
||R x|| = ||x||\n
b. Preserves angles between vectors:
\n
β¨R x, R yβ© = β¨x, yβ©\n
c. Is orthogonal:
\n
R^T = Rβ»ΒΉ\n
For any orthogonal matrix:
\n
R^T β R = I\n
\n So multiplying by the inverse matrix on the right gives:
\n
R^T = Rβ»ΒΉ\n
The transpose undoes the transformation - it rotates back.
\nd. Has determinant:
\n
det(R) = +1\n
\n (i.e., it's a proper rotation, not a reflection)
Let X_world be a 3D point in the world frame:
X_world = [x, y, z]^T\n
And let:
\nR = [r1 r2 r3]\n
\nwhere r1, r2, r3 are the unit vectors of the new basis (camera axes).
Then the rotated point becomes:
\nX_cam = R β X_world = x β r1 + y β r2 + z β r3\n\n
This expresses the vector X_world in the new frame defined by R.
\n![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F25747348%2F9adb1e1db134ce9caec75bda1361cd27%2F1_rkaT6sY7OHDworLvSiWv4w.jpg?generation=1745602417430547&alt=media\")
![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F25747348%2F4c52edbc3d78990c5f8e1d2481f073d2%2FRotation-and-translation-parameters-of-the-camera.png?generation=1745602430125479&alt=media\"){kind=spacer}
After rotation, we shift the entire coordinate frame to the camera's origin.
\nT is the position of the world origin in the camera's coordinate system.
So:
\n+T translates the rotated point to the cameraβs local coordinate frame\n\n
This moves the origin so that the camera becomes the new reference point.
\nTo go back (camera β world), you need to undo the transformation:
\nStart with:
\nX_cam = R β X_world + T\n\n
Subtract T and multiply by Rβ»ΒΉ on the left:
Rβ»ΒΉ β (X_cam - T)= Rβ»ΒΉ β R β X_word \n\n
We discuss early that R^T = Rβ»ΒΉ , so
\nR^T β (X_cam - T)= I β X_word (where I is unit matrix)\n\n
Than:
\nX_world = R^T β (X_cam β T)\n\n
Now apply this inverse transformation to the camera's own center in its local frame:
\nC_cam = [0, 0, 0]^T (camera origin in its own frame)\nC_world = R^T β (C_cam β T) = R^T β (0 β T) = β R^T β T\n\n
\n\nNote:
\nTin superscript is the transpose, not exponentiation.
This formula gives the camera center C in world coordinates - exactly what we use during evaluation.
\nX_cam = R β X_world + T\n\n\ntransforms a point from the world coordinate system (global 3D space) into the camera coordinate system.\n\n> Think of it this way: the world is fixed, you have a camera somewhere in that world, looking in a specific direction. You want to understand how a 3D point in the world appears from the perspective of the camera.\n\nThat means take the point in world space, rotate it into the camera's view direction, and then translate it so that the camera becomes the new origin\n\n---\n\nIf the camera is not aligned with the world axes, then we need to rotate the world to the cameraβs orientation. Each row (or column, depending on convention) of **R** is a unit vector representing one axis of the rotated frame.\n\nFirstly, let's clarify that **vectors are orthonormal** if they satisfy two conditions:\n\n1. Vectors are **perpendicular** to each other \n2. Each vector has **unit length**\n\nA **rotation matrix `R`** is an **orthonormal linear transformation** that:\n\na. **Preserves lengths**: \n
||R x|| = ||x||\n\nb. **Preserves angles between vectors**: \n
β¨R x, R yβ© = β¨x, yβ©\n\nc. **Is orthogonal**: \n
R^T = Rβ»ΒΉ\n\n For any orthogonal matrix: \n
R^T β R = I\n So multiplying by the inverse matrix on the right gives: \n
R^T = Rβ»ΒΉ\n The transpose undoes the transformation - it rotates back.\n\nd. **Has determinant**: \n
det(R) = +1\n (i.e., it's a **proper rotation**, not a reflection)\n\n---\n\nLet `X_world` be a 3D point in the world frame: \n
X_world = [x, y, z]^T\n\nAnd let: \n
R = [r1 r2 r3]\nwhere `r1`, `r2`, `r3` are the **unit vectors** of the new basis (camera axes).\n\nThen the rotated point becomes:\n\n
\nX_cam = R β X_world = x β r1 + y β r2 + z β r3\n\n\nThis expresses the vector `X_world` in the new frame defined by `R`.\n\n\n\n---\n\nAfter rotation, we **shift the entire coordinate frame to the camera's origin**.\n\n`T` is the **position of the world origin in the camera's coordinate system**. \n\nSo:\n\n
\n+T translates the rotated point to the cameraβs local coordinate frame\n\n\nThis **moves the origin** so that the camera becomes the new reference point.\n\n---\n\n### Inverse Transformation\n\nTo go back (camera β world), you need to **undo the transformation**:\n\nStart with:\n\n
\nX_cam = R β X_world + T\n\n\nSubtract `T` and multiply by `Rβ»ΒΉ` on the left:\n\n
\nRβ»ΒΉ β (X_cam - T)= Rβ»ΒΉ β R β X_word \n\nWe discuss early that R^T = Rβ»ΒΉ , so\n
\nR^T β (X_cam - T)= I β X_word (where I is unit matrix)\n\nThan:\n
\nX_world = R^T β (X_cam β T)\n\n\nNow apply this **inverse transformation** to the **camera's own center** in its local frame:\n\n
\nC_cam = [0, 0, 0]^T (camera origin in its own frame)\nC_world = R^T β (C_cam β T) = R^T β (0 β T) = β R^T β T\n\n\n> Note: `T` in superscript is the **transpose**, not exponentiation.\n\nThis formula gives the **camera center `C` in world coordinates** - exactly what we use during evaluation.\n\n", "votes": 10, "replies": [ { "id": 3187537, "postDate": "2025-04-26T07:01:57.993Z", "content": "
Thank you so much for your detailed answer! It really helped me understand everything. I truly appreciate your kindness and patience!
", "rawMarkdown": "Thank you so much for your detailed answer! It really helped me understand everything. I truly appreciate your kindness and patience!", "votes": 2 } ] } ] }, { "id": 3178395, "postDate": "2025-04-14T05:50:12.840Z", "rawMarkdown": "", "isDeleted": true } ], "comments": [ { "id": 3204068, "author_name": "Diganta", "author_url": "", "post_date": "2025-05-17T18:17:30.153000", "content": "Great stuff! And all in very details, just what I needed to get started! Thanks @dmytrobuhai !
", "votes": 1, "replies": [] }, { "id": 3203991, "author_name": "Vyacheslav Efimov", "author_url": "", "post_date": "2025-05-17T16:35:55.340000", "content": "Thank you a lot for the extremely useful and concise explanations! It is my first day during which I am deeply diving into the competition details and this topic definitely helped me to quickly grasp main theoretical concepts. π₯
", "votes": 1, "replies": [] }, { "id": 3186803, "author_name": "stardustlove", "author_url": "", "post_date": "2025-04-25T07:32:38.420000", "content": "I have some confusion about the calculation formula for the Camera center C in world coordinates. Could you please explain how it is derived? I would be very grateful if you could reply.
", "votes": 1, "replies": [ { "id": 3187230, "author_name": "Dmytro Buhai", "author_url": "", "post_date": "2025-04-25T17:32:22.147000", "content": "hello, this formula:
\nX_cam = R β X_world + T\n\n
transforms a point from the world coordinate system (global 3D space) into the camera coordinate system.
\n\n\nThink of it this way: the world is fixed, you have a camera somewhere in that world, looking in a specific direction. You want to understand how a 3D point in the world appears from the perspective of the camera.
\n
That means take the point in world space, rotate it into the camera's view direction, and then translate it so that the camera becomes the new origin
\nIf the camera is not aligned with the world axes, then we need to rotate the world to the cameraβs orientation. Each row (or column, depending on convention) of R is a unit vector representing one axis of the rotated frame.
\nFirstly, let's clarify that vectors are orthonormal if they satisfy two conditions:
\nA rotation matrix R is an orthonormal linear transformation that:
a. Preserves lengths:
\n
||R x|| = ||x||\n
b. Preserves angles between vectors:
\n
β¨R x, R yβ© = β¨x, yβ©\n
c. Is orthogonal:
\n
R^T = Rβ»ΒΉ\n
For any orthogonal matrix:
\n
R^T β R = I\n
\n So multiplying by the inverse matrix on the right gives:
\n
R^T = Rβ»ΒΉ\n
The transpose undoes the transformation - it rotates back.
\nd. Has determinant:
\n
det(R) = +1\n
\n (i.e., it's a proper rotation, not a reflection)
Let X_world be a 3D point in the world frame:
X_world = [x, y, z]^T\n
And let:
\nR = [r1 r2 r3]\n
\nwhere r1, r2, r3 are the unit vectors of the new basis (camera axes).
Then the rotated point becomes:
\nX_cam = R β X_world = x β r1 + y β r2 + z β r3\n\n
This expresses the vector X_world in the new frame defined by R.
\n
After rotation, we shift the entire coordinate frame to the camera's origin.
\nT is the position of the world origin in the camera's coordinate system.
So:
\n+T translates the rotated point to the cameraβs local coordinate frame\n\n
This moves the origin so that the camera becomes the new reference point.
\nTo go back (camera β world), you need to undo the transformation:
\nStart with:
\nX_cam = R β X_world + T\n\n
Subtract T and multiply by Rβ»ΒΉ on the left:
Rβ»ΒΉ β (X_cam - T)= Rβ»ΒΉ β R β X_word \n\n
We discuss early that R^T = Rβ»ΒΉ , so
\nR^T β (X_cam - T)= I β X_word (where I is unit matrix)\n\n
Than:
\nX_world = R^T β (X_cam β T)\n\n
Now apply this inverse transformation to the camera's own center in its local frame:
\nC_cam = [0, 0, 0]^T (camera origin in its own frame)\nC_world = R^T β (C_cam β T) = R^T β (0 β T) = β R^T β T\n\n
\n\nNote:
\nTin superscript is the transpose, not exponentiation.
This formula gives the camera center C in world coordinates - exactly what we use during evaluation.
Thank you so much for your detailed answer! It really helped me understand everything. I truly appreciate your kindness and patience!
", "votes": 2, "replies": [] } ] } ] }, { "id": 3178395, "author_name": "", "author_url": "", "post_date": "2025-04-14T05:50:12.840000", "content": "", "votes": 0, "replies": [] } ], "raw_markdown_by_id": { "3178394": "outlier \n3. Reconstruct the camera pose (R, T) for each image (except outliers)\n\n---\n\n{(C1, Cg1), (C2, Cg2), (C3, Cg3)}\n\nRANSAC = Random Sample Consensus - Algorithm to randomly test many triplets, ignore outliers, and find the best transformation.\n\nThe thresholds column lists multiple numeric values per scene. These represent distance or error thresholds that are used to evaluate how accurate predicted 3D camera poses are (compared to ground truth). Each threshold is a maximum allowable error (in meters) to consider a pose prediction as correct.\n\n---\n\nSo use Hornβs method (algorithm used to compute the best similarity transformation between two sets of 3D points) to compute a similarity transformation. \nFind scale s, rotation R, translation t so that:\n\nT(Ci) = s β
R β
Ci + tis_registered = (β T(Ci) β Cgi β < threshold)β Β· β denotes Euclidean (L2) distance between two 3D points.\n\nTry many random triplets and get many candidate transformations Tβ². \nFor each one, count how many registered cameras it produces and save the best one. \nTake the best transformation Tβ², and re-estimate it using all the inliers (not just the original triplet).\n\n---\n\nT(Ci) = s β
R β
Ci + tCi_aligned = T(Ci)\n\nNow for each threshold t_j, check if the difference between Ci_aligned and ground-truth center Cgi is less than t_j:\n\nis_registered = (β T(Ci) β Cgi β < threshold)rj = (1 / N') * sum(registered)mAA = (1 / k) * sum(rj)Ski, and your predicted clusters are Ckj.\n\nThe evaluation will:\n- Compare your clusters with the true scenes \n- Check how well your clustering matches the original grouping \n- Measure how accurate your pose predictions are (as explained earlier with mAA)\n\nEach ground-truth scene Ski is matched to the user-submitted cluster Ckj that gives the highest mAA after alignment. \nAll images labeled as outliers are excluded from this matching step.\n\n---\n\nCkji is selected for each scene Ski, compute the clustering score as:\n\nclustering_score = |S β© C| / |C||S β© C| is the number of images that correctly overlap between your predicted cluster and the real scene \n- |C| is the total number of images in your predicted cluster\n\nThis measures how pure your cluster is - i.e., out of everything you said belongs to this scene, how many actually do?\n\n---\n\nfinal_score_per_dataset = 2 * (mAA * clustering_score) / (mAA + clustering_score)Dk. \nThe final score for your submission is:\n\nmean(final_score_per_dataset)CLIP, DINOv2, etc., then cluster them using algorithms like DBSCAN, KMeans, Agglomerative Clustering...\n\nπ **Image features** are compact vector representations of the visual content of the image.\n\n---\n\n##### 0.3).\n- **Connected components** of this graph are treated as **predicted scenes**.\n- Images with no edges (isolated nodes) are labeled as **outliers**.\n\n\nThis approach is faster and more scalable than DBSCAN for large datasets, and does not require knowing the number of clusters beforehand.\n
\n\nπ **Note**: \nI do not use ALIKED or LightGlue at this stage. \nThey are used later in the pipeline for **local feature extraction and matching** - when reconstructing 3D camera poses.\n\n---\n\n#####\nStructure-from-Motion (SfM) is a computer vision pipeline that reconstructs:\n
\n
h5_to_db utilities.RTNaN to its pose fields.\nUsing COLMAP, pycolmap, or another SfM engine, we perform the following:\n
R (3x3)T (3x1)