forked from HQU-gxy/camera-extrinsic-play
163 lines
6.1 KiB
Markdown
163 lines
6.1 KiB
Markdown
I'll write down the transformation sequence using LaTeX notation. Let me break down the complete operation:
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For a point $P$ in the original world space, the final transformation can be written as:
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$$
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P_{final} = (C_{GL} \cdot E \cdot W_{Y})^{-1} \cdot P
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$$
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Where:
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- $E$ is your original extrinsic matrix (world-to-camera transform)
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- $W_{Y}$ is the Z-up to Y-up world conversion matrix
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- $C_{GL}$ is the OpenCV to OpenGL camera conversion matrix
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- $^{-1}$ denotes matrix inversion
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Breaking down each matrix:
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$$
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W_{Y} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & 0 & 1 & 0 \\
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0 & -1 & 0 & 0 \\
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0 & 0 & 0 & 1
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\end{bmatrix}
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$$
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$$
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C_{GL} = \begin{bmatrix}
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1 & 0 & 0 & 0 \\
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0 & -1 & 0 & 0 \\
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0 & 0 & -1 & 0 \\
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0 & 0 & 0 & 1
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\end{bmatrix}
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$$
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The order of operations (from right to left) is:
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1. Convert world from Z-up to Y-up ($W_{Y}$)
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2. Apply camera transform ($E$)
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3. Convert camera space to OpenGL convention ($C_{GL}$)
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4. Invert the whole transformation ($^{-1}$)
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This matches our code where we write:
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```typescript
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finalTransform
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.multiply(cameraConversion) // C_GL
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.multiply(worldToCamera) // E
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.multiply(worldZupToYup) // W_Y
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.invert() // ^{-1}
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```
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Let me explain the reasoning behind this order:
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1. First, $W_Y$ (Z-up to Y-up):
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- We need to convert the world coordinates BEFORE applying the camera transform
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- Because your original extrinsic matrix $E$ was defined in Z-up world coordinates
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- If we didn't do this first, the camera transform would be applied in the wrong coordinate system
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2. Then, $E$ (your extrinsic matrix):
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- This transforms points from the (now Y-up) world space into camera space
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- The camera space at this point is still in OpenCV convention
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3. Finally, $C_{GL}$ (OpenCV to OpenGL):
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- This converts the camera space from OpenCV convention to OpenGL convention
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- It must be applied AFTER the camera transform because it's converting the camera's view space
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- If we applied it before, we'd be incorrectly modifying the world coordinates
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Think about it like this:
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1. We first need to fix the world coordinate system to match Three.js ($W_Y$)
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2. Then we can apply your camera's view of this world ($E$)
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3. Finally, we adapt how the camera "sees" to match OpenGL conventions ($C_{GL}$)
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If we did it in a different order:
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- Applying $C_{GL}$ before $E$ would incorrectly transform the world coordinates instead of the camera space
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- Applying $W_Y$ after $E$ would transform the camera space instead of the world coordinates
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- The transformations would effectively be applied to the wrong spaces
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The inversion at the end ($^{-1}$) is needed because Three.js expects camera-to-world transforms rather than world-to-camera transforms.
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Ah yes, you're absolutely right! I made a mistake in my explanation. In matrix multiplication, the operations are indeed applied from right to left.
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When we write:
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$$
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P_{final} = (C_{GL} \cdot E \cdot W_{Y})^{-1} \cdot P
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$$
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The actual order of operations on point $P$ (before inversion) would be:
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1. $W_Y \cdot P$: Convert from Z-up to Y-up world coordinates
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2. $E \cdot (W_Y \cdot P)$: Apply camera transform
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3. $C_{GL} \cdot (E \cdot (W_Y \cdot P))$: Convert to OpenGL camera convention
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In our code:
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```typescript
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finalTransform
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.multiply(cameraConversion) // C_GL
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.multiply(worldToCamera) // E
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.multiply(worldZupToYup) // W_Y
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```
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Three.js's `multiply()` method actually applies the matrices in the opposite order of how we write them (it pre-multiplies), which is why our code worked. But my explanation of the order was incorrect.
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The actual sequence is:
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1. $C_{GL}$: Convert camera space to OpenGL convention
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2. $E$: Transform from world to camera
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3. $W_Y$: Convert world from Z-up to Y-up
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Thank you for catching this! It's crucial to understand the correct order of operations, especially when dealing with coordinate system transformations.
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Let me break this down more clearly:
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## Why do we need these conversions?
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1. **World Coordinate Conversion (Z-up to Y-up)**:
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- Your original data uses a Z-up coordinate system (Z is "up" in the real world)
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- Three.js uses a Y-up coordinate system (Y is "up" in Three.js)
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- We need to convert between these two world coordinate systems
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2. **Camera Coordinate Conversion (OpenCV to OpenGL)**:
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- Camera coordinate systems define how the camera "sees" the world
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- OpenCV uses a right-handed system where Z points forward from the camera
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- OpenGL/Three.js uses a system where Z points backward from the camera
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- We need to convert between these camera conventions
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## What's the difference?
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- **World conversion** (Z-up to Y-up) affects how we represent points in the world
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- **Camera conversion** (OpenCV to OpenGL) affects how the camera interprets what it sees
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These are completely separate transforms that operate on different coordinate spaces:
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- One operates on the world before the camera sees it
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- The other operates on the camera's view of the world
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## Why this specific order?
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The order matters because we're dealing with a transformation pipeline:
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1. First, we want to convert the world from Z-up to Y-up (`worldCvt`)
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- This makes the world match Three.js's expectations
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2. Then, apply the camera's extrinsic matrix (`Rt`)
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- This transforms world points into the camera's coordinate system
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- At this point, we're in OpenCV camera space
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3. Finally, convert from OpenCV to OpenGL camera conventions (`cameraCvt`)
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- This converts the camera space to match Three.js's expectations
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If we changed the order:
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- Applying camera conversion before the camera transformation would incorrectly transform world points
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- Applying world conversion after the camera transformation would transform points that are already in camera space
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Think of it as a pipeline:
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1. Fix the world coordinates (world conversion)
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2. View the world through the camera (camera extrinsic)
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3. Adjust how the camera interprets what it sees (camera conversion)
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The fact that matrix multiplication is associative means we can compute this entire pipeline as a single matrix operation, but the conceptual order still matters for getting the correct result.
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---
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[Matrix4.multiply](https://threejs.org/docs/#api/en/math/Matrix4.multiply)
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