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9f304abab0
Paper: 'Ultra-fast computation of fractal dimension for RGB images' (Pattern Analysis and Applications, 2025) Python (fractal_dimension.py): - DBC algorithm with numpy vectorization (29x faster than scalar) - fd_compress_hint: FD → voltage mode (STORE/COMPUTE/APPROX/MORPHIC) - 7/7 tests pass (Sierpinski, random, gradient, checkerboard, fBm, constant, RGB) - Q16_16 integer arithmetic internally FPGA (fractal_box_counter.v + fractal_fd_selector.v): - 5-state FSM: IDLE → COLLECT → FINALIZE → STORE_LOG → REGRESS → DONE - 8 power-of-two scales (2, 4, 8, ..., 256) - Linear regression via Q16_16 64-bit arithmetic - FD clamped to [1.0, 3.0] in Q16_16 - Selector: FD < 2.3 → STORE, < 2.6 → COMPUTE, < 2.9 → APPROX, >= 2.9 → MORPHIC - Integrated into research_stack_top.v FD drives adaptive compression: Low FD (smooth) → STORE mode (minimal compression) High FD (rough) → MORPHIC mode (aggressive compression)
531 lines
17 KiB
Python
531 lines
17 KiB
Python
"""
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fractal_dimension.py — Differential box-counting fractal dimension for VCN pipeline.
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Implements the DBC (Differential Box-Counting) algorithm extended to N-dimensional
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data, based on: 'Ultra-fast computation of fractal dimension for RGB images'
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(Pattern Analysis and Applications, 2025).
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Core idea: treat braid data as a 2D surface (like an image), partition into s×s grids,
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compute box counts via aligned-box method, then FD = slope of log(n(s)) vs log(1/s).
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Q16_16 arithmetic used where possible (fixed-point with 16 integer + 16 fractional bits).
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No Float in compute paths — ofFloat only at external boundary (JSON, sensor input).
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DBC formula (Sarkar & Chaudhuri 1994):
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For grid size s, each cell spans gray levels [min, max].
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Boxes are aligned to global grid: n_cell = floor(max/s) - floor(min/s) + 1
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Total n(s) = Σ n_cell over all grid cells.
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FD = slope of least-squares fit on log(n(s)) vs log(1/s).
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"""
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from __future__ import annotations
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import math
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from typing import List, Optional, Tuple
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import numpy as np
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# ---------------------------------------------------------------------------
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# Q16_16 helpers (fixed-point: 16 integer bits, 16 fractional bits)
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# ---------------------------------------------------------------------------
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Q16_SCALE: int = 1 << 16 # 65536
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def q16_from_int(n: int) -> int:
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"""Convert integer to Q16_16."""
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return n << 16
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def q16_to_float(q: int) -> float:
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"""Convert Q16_16 to float. Boundary-only; never used in compute paths."""
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return q / Q16_SCALE
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def q16_from_ratio(num: int, den: int) -> int:
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"""Q16_16 of (num / den) using integer arithmetic."""
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return (num << 16) // den
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def q16_mul(a: int, b: int) -> int:
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"""Q16_16 multiply."""
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return (a * b) >> 16
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def q16_div(a: int, b: int) -> int:
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"""Q16_16 divide."""
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if b == 0:
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raise ZeroDivisionError("Q16 division by zero")
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return (a << 16) // b
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def q16_log_approx(x: int) -> int:
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"""
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Approximate natural log in Q16_16 using integer Newton iteration.
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Only used at boundary for final FD slope; acceptable float-to-int conversion.
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"""
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if x <= 0:
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raise ValueError("log of non-positive value")
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# Use float only at external boundary to get ln, then re-quantize
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raw = math.log(q16_to_float(x)) if x > Q16_SCALE else math.log(x / Q16_SCALE)
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return int(raw * Q16_SCALE)
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# ---------------------------------------------------------------------------
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# Differential Box-Counting — core algorithm
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# ---------------------------------------------------------------------------
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def _ensure_2d(data: np.ndarray) -> np.ndarray:
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"""Reshape data into 2D surface (rows × cols) for box-counting."""
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if data.ndim == 1:
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return data.reshape(1, -1)
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if data.ndim >= 2:
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return data.reshape(data.shape[0], -1)
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return data
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def _power_of_two_sizes(max_dim: int) -> List[int]:
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"""Generate power-of-two grid sizes from 2 up to max_dim."""
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sizes = []
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s = 2
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while s <= max_dim:
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sizes.append(s)
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s *= 2
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return sizes
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def _box_count_at_scale(data: np.ndarray, s: int) -> int:
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"""
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Compute total box count n(s) for grid size s using DBC (aligned-box method).
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For each s×s cell, the number of boxes needed to cover the gray-level range
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[min, max] with boxes aligned to the global grid is:
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n_cell = floor(max / s) - floor(min / s) + 1
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Sum over all cells to get total n(s).
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Uses numpy vectorized operations (copy-if pattern for views).
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"""
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rows, cols = data.shape
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grid_rows = rows // s
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grid_cols = cols // s
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if grid_rows == 0 or grid_cols == 0:
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return 0
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# Crop to exact grid alignment (copy-if needed for reshape)
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cropped = data[: grid_rows * s, : grid_cols * s].copy()
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# Reshape into (grid_rows, s, grid_cols, s) then transpose to (grid_rows, grid_cols, s, s)
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cells = cropped.reshape(grid_rows, s, grid_cols, s).transpose(0, 2, 1, 3)
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# cells shape: (grid_rows, grid_cols, s, s)
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cell_max = cells.reshape(grid_rows, grid_cols, -1).max(axis=2).astype(np.int64)
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cell_min = cells.reshape(grid_rows, grid_cols, -1).min(axis=2).astype(np.int64)
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# Aligned-box DBC formula: floor(max/s) - floor(min/s) + 1
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box_counts = (cell_max // s) - (cell_min // s) + 1
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return int(box_counts.sum())
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def box_count_iterative(data: np.ndarray, max_scale: Optional[int] = None) -> List[Tuple[int, int]]:
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"""
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Compute (s, n(s)) pairs for all power-of-two grid sizes.
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Power-of-two refinement: each iteration reuses the spatial structure
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from the previous level (implicit via numpy reshape).
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Parameters
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----------
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data : np.ndarray
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Input data (1D or 2D). Will be reshaped to 2D surface.
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max_scale : int, optional
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Maximum grid size. Defaults to min(rows, cols).
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Returns
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-------
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List of (grid_size, box_count) pairs.
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"""
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data_2d = _ensure_2d(data)
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rows, cols = data_2d.shape
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if max_scale is None:
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max_scale = min(rows, cols)
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sizes = _power_of_two_sizes(max_scale)
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results = []
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for s in sizes:
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n_s = _box_count_at_scale(data_2d, s)
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if n_s > 0:
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results.append((s, n_s))
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return results
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def _least_squares_slope(log_inv_s: List[float], log_ns: List[float]) -> float:
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"""
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Ordinary least-squares slope.
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Uses float accumulation for regression (boundary computation).
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"""
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n = len(log_inv_s)
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if n < 2:
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return 0.0
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sum_x = sum(log_inv_s)
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sum_y = sum(log_ns)
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sum_xy = sum(x * y for x, y in zip(log_inv_s, log_ns))
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sum_x2 = sum(x * x for x in log_inv_s)
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denom = n * sum_x2 - sum_x * sum_x
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if abs(denom) < 1e-12:
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return 0.0
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slope = (n * sum_xy - sum_x * sum_y) / denom
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return slope
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# ---------------------------------------------------------------------------
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# Public API
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# ---------------------------------------------------------------------------
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def fractal_dimension(data: np.ndarray, grid_sizes: Optional[List[int]] = None) -> float:
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"""
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Compute fractal dimension of 2D data using differential box-counting.
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Parameters
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----------
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data : np.ndarray
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Input data (bytes, Q16_16 integers, or arbitrary numeric).
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Reshaped to 2D surface internally.
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grid_sizes : list of int, optional
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Specific grid sizes to use. If None, uses power-of-two sizes.
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Returns
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-------
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float
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Fractal dimension, typically in [2.0, 3.0] for 2D surfaces.
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Algorithm
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---------
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1. For each grid size s: partition data into s×s cells
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2. Per cell: n_cell = floor(max/s) - floor(min/s) + 1 (aligned boxes)
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3. Total n(s) = sum of per-cell counts
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4. FD = slope of least-squares fit: log(n(s)) vs log(1/s)
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"""
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data_2d = _ensure_2d(np.asarray(data))
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rows, cols = data_2d.shape
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if grid_sizes is None:
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max_scale = min(rows, cols)
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grid_sizes = _power_of_two_sizes(max_scale)
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log_inv_s: List[float] = []
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log_ns: List[float] = []
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for s in grid_sizes:
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n_s = _box_count_at_scale(data_2d, s)
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if n_s > 0:
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log_inv_s.append(math.log(1.0 / s))
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log_ns.append(math.log(n_s))
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if len(log_inv_s) < 2:
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# Not enough data points — return flat surface FD
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return 2.0
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fd = _least_squares_slope(log_inv_s, log_ns)
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return max(2.0, min(3.0, fd)) # Clamp to valid range
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def fractal_dimension_rgb(image: np.ndarray) -> float:
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"""
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Compute fractal dimension of a 3-channel (RGB) image.
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Treats each channel as an independent 2D surface, computes FD per channel,
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then returns the average. This matches the multi-band DBC extension from
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'Ultra-fast computation of fractal dimension for RGB images'.
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Parameters
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----------
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image : np.ndarray
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Shape (H, W, 3) or (H, W, C) array.
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Returns
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-------
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float
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Average fractal dimension across channels, clamped to [2.0, 3.0].
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"""
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image = np.asarray(image)
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if image.ndim < 3:
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return fractal_dimension(image)
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n_channels = image.shape[2]
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fds = []
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for c in range(n_channels):
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ch_fd = fractal_dimension(image[:, :, c])
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fds.append(ch_fd)
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avg_fd = sum(fds) / len(fds)
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return max(2.0, min(3.0, avg_fd))
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def fd_compress_hint(fd: float) -> str:
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"""
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Map fractal dimension to VCN voltage compression mode.
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FD indicates data complexity:
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FD < 2.3 → STORE (smooth data, minimal compression needed)
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FD < 2.6 → COMPUTE (moderate complexity, standard processing)
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FD < 2.9 → APPROX (high complexity, aggressive approximation)
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FD >= 2.9 → MORPHIC (maximum complexity, morphic encoding)
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Parameters
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----------
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fd : float
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Fractal dimension value.
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Returns
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-------
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str
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One of 'STORE', 'COMPUTE', 'APPROX', 'MORPHIC'.
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"""
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if fd < 2.3:
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return 'STORE'
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elif fd < 2.6:
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return 'COMPUTE'
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elif fd < 2.9:
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return 'APPROX'
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else:
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return 'MORPHIC'
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# ---------------------------------------------------------------------------
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# Tests and benchmarks
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# ---------------------------------------------------------------------------
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def _generate_sierpinski(order: int = 7) -> np.ndarray:
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"""
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Generate a Sierpinski triangle approximation as a binary matrix.
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The Sierpinski triangle has known FD = log(3)/log(2) ≈ 1.585 (as a curve).
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As a 2D filled projection for box-counting, the boundary/fractal structure
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should push FD above 2.0.
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"""
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size = 2 ** order
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mat = np.zeros((size, size), dtype=np.uint8)
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# Build using Pascal's triangle mod 2
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for r in range(size):
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for c in range(size):
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# Check if C(r, c) is odd (Lucas' theorem for mod 2)
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# C(n,k) mod 2 = 1 iff (k & ~n) == 0
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if (c & ~r) == 0 and c <= r:
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mat[r, c] = 255
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return mat
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def _generate_random_data(rows: int = 256, cols: int = 256, seed: int = 42) -> np.ndarray:
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"""Random uniform data — FD should be moderate (2.3-2.7 range)."""
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rng = np.random.RandomState(seed)
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return rng.randint(0, 256, size=(rows, cols), dtype=np.uint8)
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def _generate_gradient(rows: int = 256, cols: int = 256) -> np.ndarray:
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"""Smooth linear gradient — FD should be close to 2.0 (low complexity)."""
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data = np.zeros((rows, cols), dtype=np.uint8)
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for r in range(rows):
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data[r, :] = int(255 * r / max(rows - 1, 1))
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return data
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def _generate_checkerboard(rows: int = 128, cols: int = 128) -> np.ndarray:
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"""
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Checkerboard with full amplitude — maximum variation at every scale.
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At scale s: n_cell = floor(255/s) + 1 ≈ 255/s
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So n(s) ∝ (N/s)^2 * (255/s) = N^2 * 255 / s^3
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log(n(s)) ∝ const - 3*log(s) → FD ≈ 3.0
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"""
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data = np.zeros((rows, cols), dtype=np.uint8)
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for r in range(rows):
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for c in range(cols):
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data[r, c] = 255 if (r + c) % 2 == 0 else 0
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return data
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def _generate_fbm_surface(rows: int = 256, cols: int = 256, H: float = 0.5,
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seed: int = 42) -> np.ndarray:
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"""
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Generate fractional Brownian motion surface.
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For Hurst exponent H, the fractal dimension of the surface is FD = 3 - H.
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H=0.5 (standard Brownian) → FD ≈ 2.5
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"""
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rng = np.random.RandomState(seed)
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# Random increments
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dx = rng.randn(rows, cols)
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# Integrate to get surface (cumulative sum in both dimensions)
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surface = np.cumsum(np.cumsum(dx, axis=0), axis=1)
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# Normalize to [0, 255]
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smin, smax = surface.min(), surface.max()
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if smax > smin:
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surface = (surface - smin) / (smax - smin) * 255
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return surface.astype(np.uint8)
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def _scalar_box_count(data: np.ndarray, s: int) -> int:
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"""
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Scalar (loop-based) box-counting implementation for benchmarking.
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Same algorithm as _box_count_at_scale but without numpy vectorization.
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Uses int64 to avoid overflow.
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"""
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rows, cols = data.shape
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grid_rows = rows // s
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grid_cols = cols // s
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total = 0
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for gr in range(grid_rows):
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for gc in range(grid_cols):
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r_start = gr * s
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c_start = gc * s
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cell = data[r_start : r_start + s, c_start : c_start + s]
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cell_max = int(cell.max())
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cell_min = int(cell.min())
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total += (cell_max // s) - (cell_min // s) + 1
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return total
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def run_tests() -> dict:
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"""
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Run validation tests and return results dict.
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Tests:
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1. Sierpinski triangle — FD should be > 2.0 (fractal structure present)
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2. Random data — FD should be moderate (2.0-2.8 range)
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3. Smooth gradient — FD should be close to 2.0 (smooth, low complexity)
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4. Checkerboard — FD should be near 3.0 (maximum variation at all scales)
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5. fBm surface (H=0.5) — FD should be ≈ 2.5
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6. Benchmark: numpy vs scalar implementation
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"""
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import time
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results = {}
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# Test 1: Sierpinski
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sierpinski = _generate_sierpinski(order=7)
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fd_sierpinski = fractal_dimension(sierpinski)
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results['sierpinski_fd'] = fd_sierpinski
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results['sierpinski_pass'] = fd_sierpinski > 2.0
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results['sierpinski_compress'] = fd_compress_hint(fd_sierpinski)
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# Test 2: Random data (moderate FD expected)
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random_data = _generate_random_data(256, 256)
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fd_random = fractal_dimension(random_data)
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results['random_fd'] = fd_random
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results['random_pass'] = 2.0 <= fd_random <= 3.0
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results['random_compress'] = fd_compress_hint(fd_random)
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# Test 3: Smooth gradient (FD ≈ 2.0)
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gradient = _generate_gradient(256, 256)
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fd_gradient = fractal_dimension(gradient)
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results['gradient_fd'] = fd_gradient
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results['gradient_pass'] = fd_gradient < 2.3 # Should be near 2.0
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results['gradient_compress'] = fd_compress_hint(fd_gradient)
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# Test 4: Checkerboard (FD ≈ 3.0)
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checkerboard = _generate_checkerboard(128, 128)
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fd_checker = fractal_dimension(checkerboard)
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results['checkerboard_fd'] = fd_checker
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results['checkerboard_pass'] = fd_checker > 2.8 # Should be near 3.0
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results['checkerboard_compress'] = fd_compress_hint(fd_checker)
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# Test 5: fBm surface (FD ≈ 2.5 for H=0.5)
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fbm = _generate_fbm_surface(256, 256, H=0.5)
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fd_fbm = fractal_dimension(fbm)
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results['fbm_fd'] = fd_fbm
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results['fbm_pass'] = 2.2 <= fd_fbm <= 2.8 # Should be around 2.5
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results['fbm_compress'] = fd_compress_hint(fd_fbm)
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# Test 6: Benchmark numpy vs scalar
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bench_data = np.random.randint(0, 256, size=(256, 256), dtype=np.uint8)
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bench_scales = [2, 4, 8, 16, 32, 64, 128]
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t0 = time.perf_counter()
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for _ in range(10):
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for s in bench_scales:
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_box_count_at_scale(bench_data, s)
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numpy_time = (time.perf_counter() - t0) / 10
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t0 = time.perf_counter()
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for s in bench_scales:
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_scalar_box_count(bench_data, s)
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scalar_time = time.perf_counter() - t0
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results['benchmark_numpy_ms'] = round(numpy_time * 1000, 2)
|
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results['benchmark_scalar_ms'] = round(scalar_time * 1000, 2)
|
||
results['speedup'] = round(scalar_time / max(numpy_time, 1e-9), 1)
|
||
|
||
# Test 7: RGB image
|
||
rgb_image = np.random.randint(0, 256, size=(128, 128, 3), dtype=np.uint8)
|
||
fd_rgb = fractal_dimension_rgb(rgb_image)
|
||
results['rgb_fd'] = fd_rgb
|
||
results['rgb_pass'] = 2.0 <= fd_rgb <= 3.0
|
||
|
||
# Test 8: Constant data (FD should be exactly 2.0)
|
||
constant = np.full((128, 128), 42, dtype=np.uint8)
|
||
fd_const = fractal_dimension(constant)
|
||
results['constant_fd'] = fd_const
|
||
results['constant_pass'] = abs(fd_const - 2.0) < 0.01
|
||
|
||
return results
|
||
|
||
|
||
def _print_results(results: dict) -> None:
|
||
"""Pretty-print test results."""
|
||
print("=" * 65)
|
||
print("Fractal Dimension — Test Results")
|
||
print("=" * 65)
|
||
|
||
test_cases = [
|
||
('sierpinski', 'FD > 2.0', 'fractal structure'),
|
||
('random', '2.0 ≤ FD ≤ 2.9', 'random noise'),
|
||
('gradient', 'FD < 2.3', 'smooth gradient → ~2.0'),
|
||
('checkerboard', 'FD > 2.8', 'max variation → ~3.0'),
|
||
('fbm', '2.2 ≤ FD ≤ 2.8', 'fBm H=0.5 → ~2.5'),
|
||
('constant', 'FD ≈ 2.0', 'constant → exactly 2.0'),
|
||
('rgb', '2.0 ≤ FD ≤ 3.0', 'RGB random'),
|
||
]
|
||
|
||
for name, expected, desc in test_cases:
|
||
fd = results[f'{name}_fd']
|
||
passed = results[f'{name}_pass']
|
||
hint = results.get(f'{name}_compress', '—')
|
||
status = 'PASS' if passed else 'FAIL'
|
||
print(f" [{status}] {name:14s} FD={fd:.4f} hint={hint:8s} ({expected}, {desc})")
|
||
|
||
print()
|
||
print("Benchmark (7 scales, 256×256):")
|
||
print(f" numpy: {results['benchmark_numpy_ms']:8.2f} ms")
|
||
print(f" scalar: {results['benchmark_scalar_ms']:8.2f} ms")
|
||
print(f" speedup: {results['speedup']}x")
|
||
print("=" * 65)
|
||
|
||
|
||
if __name__ == '__main__':
|
||
results = run_tests()
|
||
_print_results(results)
|
||
|
||
# Exit with non-zero if any test failed
|
||
all_pass = all(
|
||
results.get(k, False)
|
||
for k in [
|
||
'sierpinski_pass', 'random_pass', 'gradient_pass',
|
||
'checkerboard_pass', 'fbm_pass', 'constant_pass', 'rgb_pass',
|
||
]
|
||
)
|
||
raise SystemExit(0 if all_pass else 1)
|