Locality-sensitive hashing

Overview: The Computational Nightmare of All-White Puzzles Solving an all-white puzzle is the ultimate hardware and software stress test. Without visual cues or image features, you are left with nothing but geometry. A 4,000-piece puzzle requires approximately 30 million physical comparisons if approached manually—a task that would take a human years. This tutorial breaks down the computer vision and algorithmic techniques required to automate this process, moving from raw pixel data to a fully realized locality-sensitive hashing (LSH) system for high-speed geometric matching. Prerequisites To implement these concepts, you should be comfortable with: * **Python**: The primary language for prototyping computer vision logic. * **Image Processing**: Understanding thresholding and pixel-to-contour conversion. * **Geometry**: Familiarity with polygons and curve fitting. * **Data Structures**: Specifically hashes and multi-dimensional coordinate systems. Key Libraries & Tools * **OpenCV**: Used for initial image acquisition and extracting the contour (edge) of the puzzle pieces. * **NumPy**: Essential for handling the 128-dimensional arrays used in the hashing phase. * **Puget Systems Workstations**: High-performance hardware used to parallelize the pre-processing tasks. * **Telecentric Lens**: A critical piece of optics that eliminates perspective distortion, ensuring the puzzle piece dimensions remain accurate across the frame. Code Walkthrough: From Pixels to Hashing 1. Shape Extraction and Edge Normalization First, we isolate the puzzle piece from the background using thresholding. Once we have a binary blob, we identify the perimeter and convert it into a polygon. The real challenge is finding the corners to split the perimeter into four distinct edges. ```python Conceptual approach for finding clean corners for corner in approximate_corners: # Grab small segments of the polygon on either side of the corner curve_a = fit_curve(polygon_segment_left) curve_b = fit_curve(polygon_segment_right) # The intersection of these curves is the 'true' geometric corner true_corner = intersect(curve_a, curve_b) # Split the polygon at the point closest to this intersection split_point = find_closest_point(polygon, true_corner) ``` 2. Locality-Sensitive Hashing (LSH) To avoid comparing every edge to every other edge (an $O(n^2)$ nightmare), we use Locality-Sensitive Hashing. We project 128-dimensional edge features into a hashed sequence. Similar shapes fall into the same "bucket," allowing for $O(1)$ lookup. ```python 128-dimensional projection (Simplified logic) def get_lsh_hash(edge_points, hyperplanes): # edge_points is a vector approximating the curve shape # hyperplanes define the boundaries in high-dimensional space hash_sequence = "" for plane in hyperplanes: if dot_product(edge_points, plane) > 0: hash_sequence += "1" else: hash_sequence += "0" return hash_sequence ``` Syntax Notes: High-Dimensional Space While we visualize geometry in 2D or 3D, the algorithm uses a **128-dimensional space**. In this context, a "line" becomes a **hyperplane**. The syntax for determining which side of a hyperplane a point falls on is a simple dot product. This mathematical efficiency is what allows the robot to identify potential matches in milliseconds rather than hours. Practical Examples: Hierarchical Assembly Once potential matches are identified, the software doesn't just guess. It employs a **recursive block-building strategy**. It finds valid 2x2 blocks by ensuring pieces share mutual neighbors. These 2x2 blocks are then treated as "super-pieces" to form 4x4 blocks, continuing until the entire grid is solved. This dramatically reduces the search space by eliminating local geometric matches that don't fit the global grid. Tips & Gotchas * **Garbage In, Garbage Out**: If your puzzle pieces have fuzzy edges from a dull die-cutter, your algorithm will fail. Use a laser cutter or a vibratory tumbler to ensure sharp, clean edges. * **Pre-processing Latency**: Extracting 4,000 piece shapes can take days. Use Python's multiprocessing to split the job into smaller tasks across all available CPU cores. * **Cumulative Error**: In physical assembly, small misalignments add up. Your robot needs a secondary camera to "nudge" pieces into place as the grid grows.