From d8739754bec1b6d63fc9bfba3608f88f0d7d7f4f Mon Sep 17 00:00:00 2001
From: crosstyan <crosstyan@outlook.com>
Date: Thu, 5 Dec 2024 17:06:54 +0800
Subject: [PATCH] x

---
 README.md |  8 +++++++
 dlt.md    | 64 +++++++++++++++++++++++++++++++++++++++++++++++++++++++
 gen.sh    | 47 ++++++++++++++++++++++++++++++++++++++++
 3 files changed, 119 insertions(+)
 create mode 100644 README.md
 create mode 100644 dlt.md
 create mode 100644 gen.sh

diff --git a/README.md b/README.md
new file mode 100644
index 0000000..978d868
--- /dev/null
+++ b/README.md
@@ -0,0 +1,8 @@
+# ChArUco experiments
+
+go to [`opencv_contrib/modules/aruco/misc/pattern_generator`](https://github.com/opencv/opencv_contrib/tree/4.x/modules/aruco/misc/pattern_generator) to generate custom patterns.
+
+See also [gen.sh](gen.sh) (put it in the same directory as the pattern generator).
+
+- A0 size, 10x7, square 115mm, marker 90mm
+- 400mm x 400mm ArUco board (4x4, 0-5 id)
diff --git a/dlt.md b/dlt.md
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index 0000000..baf58ef
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+++ b/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
diff --git a/gen.sh b/gen.sh
new file mode 100644
index 0000000..abcb36c
--- /dev/null
+++ b/gen.sh
@@ -0,0 +1,47 @@
+SQUARE_MM=115
+MARKER_MM=90
+MARKER_SEP_MM=$(echo "scale=4; $SQUARE_MM - $MARKER_MM" | bc)
+
+if [ $MARKER_SEP_MM -lt 0 ]; then
+    echo "Marker size must be smaller than square size"
+    exit 1
+fi
+
+SQUARE_SIZE=$(echo "scale=4; $SQUARE_MM / 1000" | bc)
+MARKER_SIZE=$(echo "scale=4; $MARKER_MM / 1000" | bc)
+MARKER_SEP=$(echo "scale=4; $MARKER_SEP_MM / 1000" | bc)
+
+PAGE_PADDING_MM=10
+PAGE_PADDING=$(echo "scale=4; $PAGE_PADDING_MM / 1000" | bc)
+
+# A0
+PAGE_WIDTH_MM=1189
+PAGE_HEIGHT_MM=841
+
+BOARD_WIDTH_IN_SQUARE=$(echo "scale=0; ($PAGE_WIDTH_MM - $PAGE_PADDING_MM) / $SQUARE_MM" | bc)
+BOARD_HEIGHT_IN_SQUARE=$(echo "scale=0; ($PAGE_HEIGHT_MM - $PAGE_PADDING_MM) / $SQUARE_MM" | bc)
+START_MARKER_ID=10
+
+OUTPUT_FILENAME="charuco_${PAGE_WIDTH_MM}x${PAGE_HEIGHT_MM}_${BOARD_WIDTH_IN_SQUARE}x${BOARD_HEIGHT_IN_SQUARE}_s${SQUARE_MM}_m${MARKER_MM}.pdf"
+
+# DICT_4X4_1000
+# DICT_5X5_1000
+# DICT_6X6_1000
+# DICT_7X7_1000
+# DICT_ARUCO_ORIGINAL
+# DICT_APRILTAG_16h5
+# DICT_APRILTAG_25h9
+# DICT_APRILTAG_36h10
+# DICT_APRILTAG_36h11
+
+python MarkerPrinter.py --charuco \
+    --file $OUTPUT_FILENAME \
+    --dictionary DICT_4X4_1000 \
+    --page_border_x $PAGE_PADDING \
+    --page_border_y $PAGE_PADDING \
+    --square_length $SQUARE_SIZE \
+    --marker_length $MARKER_SIZE \
+    --marker_separation $MARKER_SEP \
+    --charuco_size_x $BOARD_WIDTH_IN_SQUARE \
+    --charuco_size_y $BOARD_HEIGHT_IN_SQUARE \
+    --first_marker $START_MARKER_ID