巴特沃斯滤波器

巴特沃斯滤波器在频域中实现，旨在没有通带或阻带纹波。 它可以用于低通或高通变体。 cutoff_frequency_ratio 参数用于将截止频率设置为采样频率的一部分。 鉴于奈奎斯特频率是采样频率的一半，这意味着此参数应该是一个大于 0 的浮点数，小于 0.5。 滤波器的 order 可以调整以控制过渡宽度，较高的值会导致通带和阻带之间的过渡更急剧。

巴特沃斯滤波示例

这里我们定义了一个 get_filtered 辅助函数，用于在指定的截止频率序列上重复低通和高通滤波。

import matplotlib.pyplot as plt

from skimage import data, filters

image = data.camera()

# cutoff frequencies as a fraction of the maximum frequency
cutoffs = [0.02, 0.08, 0.16]


def get_filtered(image, cutoffs, squared_butterworth=True, order=3.0, npad=0):
    """Lowpass and highpass butterworth filtering at all specified cutoffs.

    Parameters
    ----------
    image : ndarray
        The image to be filtered.
    cutoffs : sequence of int
        Both lowpass and highpass filtering will be performed for each cutoff
        frequency in `cutoffs`.
    squared_butterworth : bool, optional
        Whether the traditional Butterworth filter or its square is used.
    order : float, optional
        The order of the Butterworth filter

    Returns
    -------
    lowpass_filtered : list of ndarray
        List of images lowpass filtered at the frequencies in `cutoffs`.
    highpass_filtered : list of ndarray
        List of images highpass filtered at the frequencies in `cutoffs`.
    """

    lowpass_filtered = []
    highpass_filtered = []
    for cutoff in cutoffs:
        lowpass_filtered.append(
            filters.butterworth(
                image,
                cutoff_frequency_ratio=cutoff,
                order=order,
                high_pass=False,
                squared_butterworth=squared_butterworth,
                npad=npad,
            )
        )
        highpass_filtered.append(
            filters.butterworth(
                image,
                cutoff_frequency_ratio=cutoff,
                order=order,
                high_pass=True,
                squared_butterworth=squared_butterworth,
                npad=npad,
            )
        )
    return lowpass_filtered, highpass_filtered


def plot_filtered(lowpass_filtered, highpass_filtered, cutoffs):
    """Generate plots for paired lists of lowpass and highpass images."""
    fig, axes = plt.subplots(2, 1 + len(cutoffs), figsize=(12, 8))
    fontdict = dict(fontsize=14, fontweight='bold')

    axes[0, 0].imshow(image, cmap='gray')
    axes[0, 0].set_title('original', fontdict=fontdict)
    axes[1, 0].set_axis_off()

    for i, c in enumerate(cutoffs):
        axes[0, i + 1].imshow(lowpass_filtered[i], cmap='gray')
        axes[0, i + 1].set_title(f'lowpass, c={c}', fontdict=fontdict)
        axes[1, i + 1].imshow(highpass_filtered[i], cmap='gray')
        axes[1, i + 1].set_title(f'highpass, c={c}', fontdict=fontdict)

    for ax in axes.ravel():
        ax.set_xticks([])
        ax.set_yticks([])
    plt.tight_layout()
    return fig, axes


# Perform filtering with the (squared) Butterworth filter at a range of
# cutoffs.
lowpasses, highpasses = get_filtered(image, cutoffs, squared_butterworth=True)

fig, axes = plot_filtered(lowpasses, highpasses, cutoffs)
titledict = dict(fontsize=18, fontweight='bold')
fig.text(
    0.5,
    0.95,
    '(squared) Butterworth filtering (order=3.0, npad=0)',
    fontdict=titledict,
    horizontalalignment='center',
)

Text(0.5, 0.95, '(squared) Butterworth filtering (order=3.0, npad=0)')

避免边界伪影

从上面的图像中可以看出，图像边缘附近存在伪影（特别是对于较小的截止值）。 这是由于 DFT 的周期性，可以通过在滤波之前对边缘进行一些填充来减少，这样图像的周期性扩展就不会有尖锐的边缘。 这可以通过 npad 参数完成，传入 butterworth。

请注意，通过填充，图像边缘处的阴影被大大减少。

lowpasses, highpasses = get_filtered(image, cutoffs, squared_butterworth=True, npad=32)

fig, axes = plot_filtered(lowpasses, highpasses, cutoffs)
fig.text(
    0.5,
    0.95,
    '(squared) Butterworth filtering (order=3.0, npad=32)',
    fontdict=titledict,
    horizontalalignment='center',
)

Text(0.5, 0.95, '(squared) Butterworth filtering (order=3.0, npad=32)')

真正的巴特沃斯滤波器

要使用巴特沃斯滤波器的传统信号处理定义，请设置 squared_butterworth=False。 此变体的频域振幅曲线是默认情况下的平方根。 这会导致在给定 order 时，从通带到阻带的过渡更加平缓。 这可以在下面的图像中看到，与上面的平方巴特沃斯对应物相比，低通情况下的图像略微更清晰。

lowpasses, highpasses = get_filtered(image, cutoffs, squared_butterworth=False, npad=32)

fig, axes = plot_filtered(lowpasses, highpasses, cutoffs)
fig.text(
    0.5,
    0.95,
    'Butterworth filtering (order=3.0, npad=32)',
    fontdict=titledict,
    horizontalalignment='center',
)

plt.show()