一、说明

        到目前为止，在我们的讨论中，我已经交替使用了“傅里叶变换”和“快速傅里叶变换（FFT）”。在这一点上，值得注意的是区别！FFT 是“离散”傅里叶变换 （DFT） 的有效算法实现。“离散”表示我们可以将变换应用于一系列点，而不是完整的连续信号。在数据科学应用中，我们通常有一组样本，而不是一个连续的输入函数，所以我们通常对DFT感兴趣！

# see full context at https://github.com/peterbbryan/fourier-sketcher@staticmethod
def _from_jpeg(path_str: str) -> Image:""" Load image at path. """return Image.open(path_str)@property
def binary_mask(self):""" Binary mask from image. """im = np.array(self._image)  # pylint: disable=invalid-nameedges = cv.Canny(im, 100, 150)  # pylint:disable=no-memberreturn np.array(edges).astype(np.bool)

# see full context at https://github.com/peterbbryan/fourier-sketcher@property
def complex_sequence(self) -> np.ndarray:""" Binary pixel coordinates as complex sequence. """# row/col to x/y will impart a directional flip on coordinateslocs = np.argwhere(np.flipud(self.binary_mask)).astype(np.complex128)n_rows, n_cols = self.binary_mask.shape# center about the originrows = locs[:, 0] - (n_rows // 2)columns = locs[:, 1] - (n_cols // 2)# represent y values with irows *= 1j# restructure to x, y orderreturn rows + columns

# see full context at https://github.com/peterbbryan/fourier-sketcher@staticmethod
def _sort_points(complex_sequence: np.ndarray, stopping_threshold: float = 10
) -> np.ndarray:"""Arrange points so that neighbors are near each other in sequence.Args:complex_sequence: Points as complex array.Returns:Sorted points to minimize sequential differences."""reordered: List[np.complex128] = []current_point = complex_sequence[0]complex_sequence[0] = np.infwhile len(reordered) < complex_sequence.shape[0]:if np.min(np.abs(complex_sequence - current_point)) > stopping_threshold:breakmin_dist_ind = np.argmin(np.abs(complex_sequence - current_point))reordered.append(current_point)current_point = complex_sequence[min_dist_ind]complex_sequence[min_dist_ind] = np.infreturn np.array(reordered)

# see full context at https://github.com/peterbbryan/fourier-sketcherdef _get_frequency_components(self) -> Tuple[np.ndarray, np.ndarray]:"""Get frequency components from outline.Returns:Tuple of frequency values and bin centers."""complex_sequence = self._sort_points(self._edger.complex_sequence)fft_values = np.fft.fft(complex_sequence)fft_bins = np.fft.fftfreq(len(fft_values), d=1 / (len(fft_values)))# make this less stupidfft_values = fft_values[np.abs(fft_bins) <= (self._n_oscillators / 2)]fft_bins = fft_bins[np.abs(fft_bins) <= (self._n_oscillators / 2)]fft_bins = fft_bins[np.argsort(-np.abs(fft_values))]fft_values = fft_values[np.argsort(-np.abs(fft_values))]return fft_values, fft_bins

# see full context at https://github.com/peterbbryan/fourier-sketcherdef _get_arr_from_fig(self, dpi: int = 90) -> np.ndarray:"""Get np.ndarray from a figure.Args:fig: Matplotlib Figure.dpi: Resolution.Returns:Numpy array of image contents."""buf = io.BytesIO()self._fig.savefig(buf, format="png", dpi=dpi)buf.seek(0)img_arr = np.frombuffer(buf.getvalue(), dtype=np.uint8)buf.close()img = cv2.imdecode(img_arr, 1)  # pylint: disable=no-memberimg = cv2.cvtColor(img, cv2.COLOR_BGR2RGB)  # pylint: disable=no-memberreturn img@staticmethod
def _make_plot(points: List[Tuple[float, float]]) -> None:"""Plot outline so far.Args:points: x, y points to plot."""plt.scatter(*list(zip(*points)), linewidth=1)plt.draw()plt.axis("square")plt.axis("off")def make_gif(self, path_str: str) -> None:"""Make gif of outline.Args:path_str: Path to write gif to."""plt.ion()fft_values, fft_bins = self._get_frequency_components()plots, points = [], []for step in tqdm.tqdm(np.linspace(0, 2 * np.pi, self._n_steps)):edge = (0.0, 0.0)for i in range(self._n_oscillators):mag = np.abs(fft_values[i])ang = np.angle(fft_values[i])oscillator = self._get_oscillator(frequency=fft_bins[i], phase=ang)circle = self._get_circle(center=edge, radius=mag, angle=oscillator.angle(step))circle.draw()edge = circle.edge_pointif i == self._n_oscillators - 1:points.append(edge)self._make_plot(points)plot_img_np = self._get_arr_from_fig()plots.append(Image.fromarray(plot_img_np))plt.clf()self._write_gif(plots, path_str)

# see full context at https://github.com/peterbbryan/fourier-sketcher@staticmethod
def _write_gif(plots: List[Image.Image],path_str: str,last_frame_repeats: int = 100,duration: int = 60,
) -> None:"""Write animated gif.Args:plots: Output plots to write.path_str: Path to output gif.last_frame_repeats: Number of repeats of last still.duration: Duration in seconds."""for _ in range(last_frame_repeats):plots.append(plots[-1])plots[0].save(path_str, save_all=True, append_images=plots[1:], duration=duration, loop=0,)