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dlt.md

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+Direct Linear Transform (DLT) is fundamentally connected to triangulation through its ability to solve the intersection of projection rays. Here's how they relate:
+
+**Mathematical Foundation:**
+
+1. **Projection Model**
+- A 3D point $X = [X,Y,Z,1]^T$ projects to 2D point $x = [u,v,1]^T$ through:
+   $\lambda x = PX$
+   where $P$ is the 3×4 projection matrix
+   $\lambda$ is the projective depth
+
+2. **DLT Formulation**
+- For each image point, we can write:
+   $\lambda u = \frac{P_{1}^TX}{P_{3}^TX}$
+   $\lambda v = \frac{P_{2}^TX}{P_{3}^TX}$
+
+- This leads to linear equations:
+   $uP_{3}^TX - P_{1}^TX = 0$
+   $vP_{3}^TX - P_{2}^TX = 0$
+
+3. **Triangulation System**
+- For two views with points $x_1=(u_1,v_1)$ and $x_2=(u_2,v_2)$:
+
+$$\begin{bmatrix} 
+u_1P_{3,1}^T - P_{1,1}^T \\
+v_1P_{3,1}^T - P_{2,1}^T \\
+u_2P_{3,2}^T - P_{1,2}^T \\
+v_2P_{3,2}^T - P_{2,2}^T
+\end{bmatrix} X = 0$$
+
+**Why DLT Works for Triangulation:**
+
+1. **Geometric Interpretation**
+- DLT finds the 3D point that minimizes algebraic error
+- Each row in the equation system represents a constraint from one coordinate
+- The solution is the intersection of projection rays in 3D space
+
+2. **Solution Method**
+- The system $AX = 0$ is solved using SVD
+- The solution is the eigenvector corresponding to smallest singular value
+- This provides the optimal 3D point in a least-squares sense
+
+3. **Implementation Connection**
+In your code:
+
+```python
+A = [
+    point1[1]*P1[2,:] - P1[1,:],
+    P1[0,:] - point1[0]*P1[2,:],
+    point2[1]*P2[2,:] - P2[1,:],
+    P2[0,:] - point2[0]*P2[2,:],
+]
+```
+This directly implements the linear system described above.
+
+**Advantages:**
+- Linear solution
+- Simple to implement
+- Works with multiple views
+- Computationally efficient
+
+**Limitations:**
+- Sensitive to noise
+- Assumes perfect point correspondences
+- Minimizes algebraic (not geometric) error