最大树

最大树是图像的层次表示，它是大量形态滤波器的基础。

如果我们将阈值操作应用于图像，我们将获得一个包含一个或多个连通分量的二进制图像。如果我们应用一个较低的阈值，则来自较高阈值的所有连通分量都包含在来自较低阈值的连通分量中。这自然地定义了一个嵌套分量的层次结构，可以用树来表示。每当由阈值 t1 获得的连通分量 A 包含在由阈值 t1 < t2 获得的分量 B 中时，我们就说 B 是 A 的父节点。生成的树结构称为分量树。最大树是对这种分量树的紧凑表示。 [1], [2], [3], [4]

在这个例子中，我们直观地了解了最大树是什么。

参考文献

[1]

Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive Connected Operators for Image and Sequence Processing. IEEE Transactions on Image Processing, 7(4), 555-570. DOI:10.1109/83.663500

[2]

Berger, C., Geraud, T., Levillain, R., Widynski, N., Baillard, A., Bertin, E. (2007). Effective Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging. In International Conference on Image Processing (ICIP) (pp. 41-44). DOI:10.1109/ICIP.2007.4379949

[3]

Najman, L., & Couprie, M. (2006). Building the component tree in quasi-linear time. IEEE Transactions on Image Processing, 15(11), 3531-3539. DOI:10.1109/TIP.2006.877518

[4]

Carlinet, E., & Geraud, T. (2014). A Comparative Review of Component Tree Computation Algorithms. IEEE Transactions on Image Processing, 23(9), 3885-3895. DOI:10.1109/TIP.2014.2336551

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.lines import Line2D
from skimage.morphology import max_tree
import networkx as nx

在我们开始之前：一些辅助函数

def plot_img(ax, image, title, plot_text, image_values):
    """Plot an image, overlaying image values or indices."""
    ax.imshow(image, cmap='gray', aspect='equal', vmin=0, vmax=np.max(image))
    ax.set_title(title)
    ax.set_yticks([])
    ax.set_xticks([])

    for x in np.arange(-0.5, image.shape[0], 1.0):
        ax.add_artist(
            Line2D((x, x), (-0.5, image.shape[0] - 0.5), color='blue', linewidth=2)
        )

    for y in np.arange(-0.5, image.shape[1], 1.0):
        ax.add_artist(Line2D((-0.5, image.shape[1]), (y, y), color='blue', linewidth=2))

    if plot_text:
        for i, j in np.ndindex(*image_values.shape):
            ax.text(
                j,
                i,
                image_values[i, j],
                fontsize=8,
                horizontalalignment='center',
                verticalalignment='center',
                color='red',
            )
    return


def prune(G, node, res):
    """Transform a canonical max tree to a max tree."""
    value = G.nodes[node]['value']
    res[node] = str(node)
    preds = [p for p in G.predecessors(node)]
    for p in preds:
        if G.nodes[p]['value'] == value:
            res[node] += f", {p}"
            G.remove_node(p)
        else:
            prune(G, p, res)
    G.nodes[node]['label'] = res[node]
    return


def accumulate(G, node, res):
    """Transform a max tree to a component tree."""
    total = G.nodes[node]['label']
    parents = G.predecessors(node)
    for p in parents:
        total += ', ' + accumulate(G, p, res)
    res[node] = total
    return total


def position_nodes_for_max_tree(G, image_rav, root_x=4, delta_x=1.2):
    """Set the position of nodes of a max-tree.

    This function helps to visually distinguish between nodes at the same
    level of the hierarchy and nodes at different levels.
    """
    pos = {}
    for node in reversed(list(nx.topological_sort(canonical_max_tree))):
        value = G.nodes[node]['value']
        if canonical_max_tree.out_degree(node) == 0:
            # root
            pos[node] = (root_x, value)

        in_nodes = [y for y in canonical_max_tree.predecessors(node)]

        # place the nodes at the same level
        level_nodes = [y for y in filter(lambda x: image_rav[x] == value, in_nodes)]
        nb_level_nodes = len(level_nodes) + 1

        c = nb_level_nodes // 2
        i = -c
        if len(level_nodes) < 3:
            hy = 0
            m = 0
        else:
            hy = 0.25
            m = hy / (c - 1)

        for level_node in level_nodes:
            if i == 0:
                i += 1
            if len(level_nodes) < 3:
                pos[level_node] = (pos[node][0] + i * 0.6 * delta_x, value)
            else:
                pos[level_node] = (
                    pos[node][0] + i * 0.6 * delta_x,
                    value + m * (2 * np.abs(i) - c - 1),
                )
            i += 1

        # place the nodes at different levels
        other_level_nodes = [
            y for y in filter(lambda x: image_rav[x] > value, in_nodes)
        ]
        if len(other_level_nodes) == 1:
            i = 0
        else:
            i = -len(other_level_nodes) // 2
        for other_level_node in other_level_nodes:
            if (len(other_level_nodes) % 2 == 0) and (i == 0):
                i += 1
            pos[other_level_node] = (
                pos[node][0] + i * delta_x,
                image_rav[other_level_node],
            )
            i += 1

    return pos


def plot_tree(graph, positions, ax, *, title='', labels=None, font_size=8, text_size=8):
    """Plot max and component trees."""
    nx.draw_networkx(
        graph,
        pos=positions,
        ax=ax,
        node_size=40,
        node_shape='s',
        node_color='white',
        font_size=font_size,
        labels=labels,
    )
    for v in range(image_rav.min(), image_rav.max() + 1):
        ax.hlines(v - 0.5, -3, 10, linestyles='dotted')
        ax.text(-3, v - 0.15, f"val: {v}", fontsize=text_size)
    ax.hlines(v + 0.5, -3, 10, linestyles='dotted')
    ax.set_xlim(-3, 10)
    ax.set_title(title)
    ax.set_axis_off()

图像定义

我们定义了一个小的测试图像。为了清晰起见，我们选择一个示例图像，其中图像值不会与索引混淆（不同的范围）。

image = np.array(
    [
        [40, 40, 39, 39, 38],
        [40, 41, 39, 39, 39],
        [30, 30, 30, 32, 32],
        [33, 33, 30, 32, 35],
        [30, 30, 30, 33, 36],
    ],
    dtype=np.uint8,
)

最大树

接下来，我们计算这个图像的最大树。图像的最大树

P, S = max_tree(image)

P_rav = P.ravel()

图像图

然后，我们可视化图像及其展开的索引。具体来说，我们使用以下叠加来绘制图像：- 图像值- 展开的索引（用作像素标识符）- max_tree 函数的输出

# raveled image
image_rav = image.ravel()

# raveled indices of the example image (for display purpose)
raveled_indices = np.arange(image.size).reshape(image.shape)

fig, (ax1, ax2, ax3) = plt.subplots(1, 3, sharey=True, figsize=(9, 3))

plot_img(ax1, image - image.min(), 'Image Values', plot_text=True, image_values=image)
plot_img(
    ax2,
    image - image.min(),
    'Raveled Indices',
    plot_text=True,
    image_values=raveled_indices,
)
plot_img(ax3, image - image.min(), 'Max-tree indices', plot_text=True, image_values=P)

可视化阈值操作

现在，我们研究一系列阈值操作的结果。分量树（和最大树）提供了不同级别上连通分量之间包含关系的表示。

fig, axes = plt.subplots(3, 3, sharey=True, sharex=True, figsize=(6, 6))
thresholds = np.unique(image)
for k, threshold in enumerate(thresholds):
    bin_img = image >= threshold
    plot_img(
        axes[(k // 3), (k % 3)],
        bin_img,
        f"Threshold : {threshold}",
        plot_text=True,
        image_values=raveled_indices,
    )

最大树图

现在，我们绘制分量树和最大树。分量树将所有可能的阈值操作产生的不同像素集相互关联。如果一个级别上的分量包含在较低级别上的分量中，则图中将有一个箭头。最大树只是像素集的不同编码。

1. 分量树：显式地写出像素集。例如，我们看到 {6}（在 41 处应用阈值的结果）是 {0, 1, 5, 6}（在 40 处应用阈值的结果）的父节点。

2. 最大树：只写出在此级别上进入集合的像素。因此，我们将写 {6} -> {0,1,5} 而不是 {6} -> {0, 1, 5, 6}

3. 规范最大树：这是我们的实现提供的表示。这里，每个像素都是一个节点。多个像素的连通分量由其中一个像素表示。因此，我们用 {6} -> {5}、{1} -> {5}、{0} -> {5} 来代替 {6} -> {0,1,5} 这使我们能够通过图像（顶行，第三列）来表示图。

# the canonical max-tree graph
canonical_max_tree = nx.DiGraph()
canonical_max_tree.add_nodes_from(S)
for node in canonical_max_tree.nodes():
    canonical_max_tree.nodes[node]['value'] = image_rav[node]
canonical_max_tree.add_edges_from([(n, P_rav[n]) for n in S[1:]])

# max-tree from the canonical max-tree
nx_max_tree = nx.DiGraph(canonical_max_tree)
labels = {}
prune(nx_max_tree, S[0], labels)

# component tree from the max-tree
labels_ct = {}
total = accumulate(nx_max_tree, S[0], labels_ct)

# positions of nodes : canonical max-tree (CMT)
pos_cmt = position_nodes_for_max_tree(canonical_max_tree, image_rav)

# positions of nodes : max-tree (MT)
pos_mt = dict(zip(nx_max_tree.nodes, [pos_cmt[node] for node in nx_max_tree.nodes]))

# plot the trees with networkx and matplotlib
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, sharey=True, figsize=(20, 8))

plot_tree(
    nx_max_tree,
    pos_mt,
    ax1,
    title='Component tree',
    labels=labels_ct,
    font_size=6,
    text_size=8,
)

plot_tree(nx_max_tree, pos_mt, ax2, title='Max tree', labels=labels)

plot_tree(canonical_max_tree, pos_cmt, ax3, title='Canonical max tree')

fig.tight_layout()

plt.show()