.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/transform/plot_transform_types.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_transform_plot_transform_types.py: =================================== Types of homographies =================================== `Homographies `_ are transformations of a Euclidean space that preserve the alignment of points. Specific cases of homographies correspond to the conservation of more properties, such as parallelism (affine transformation), shape (similar transformation) or distances (Euclidean transformation). Homographies on a 2D Euclidean space (i.e., for 2D grayscale or multichannel images) are defined by a 3x3 matrix. All types of homographies can be defined by passing either the transformation matrix, or the parameters of the simpler transformations (rotation, scaling, ...) which compose the full transformation. The different types of homographies available in scikit-image are shown here, by increasing order of complexity (i.e. by reducing the number of constraints). While we focus here on the mathematical properties of transformations, tutorial :ref:`sphx_glr_auto_examples_transform_plot_geometric.py` explains how to use such transformations for various tasks such as image warping or parameter estimation. .. GENERATED FROM PYTHON SOURCE LINES 25-33 .. code-block:: Python import numpy as np import matplotlib.pyplot as plt from skimage import data from skimage import transform from skimage import img_as_float .. GENERATED FROM PYTHON SOURCE LINES 34-41 Euclidean (rigid) transformation ================================= A `Euclidean transformation `_, also called rigid transformation, preserves the Euclidean distance between pairs of points. It can be described as a rotation about the origin followed by a translation. .. GENERATED FROM PYTHON SOURCE LINES 41-45 .. code-block:: Python tform = transform.EuclideanTransform(rotation=np.pi / 12.0, translation=(100, -20)) print(tform.params) .. rst-class:: sphx-glr-script-out .. code-block:: none [[ 0.96592583 -0.25881905 100. ] [ 0.25881905 0.96592583 -20. ] [ 0. 0. 1. ]] .. GENERATED FROM PYTHON SOURCE LINES 46-53 Now let's apply this transformation to an image. Because we are trying to reconstruct the *image* after transformation, it is not useful to see where a *coordinate* from the input image ends up in the output, which is what the transform gives us. Instead, for every pixel (coordinate) in the output image, we want to figure out where in the input image it comes from. Therefore, we need to use the inverse of ``tform``, rather than ``tform`` directly. .. GENERATED FROM PYTHON SOURCE LINES 53-60 .. code-block:: Python img = img_as_float(data.chelsea()) tf_img = transform.warp(img, tform.inverse) fig, ax = plt.subplots() ax.imshow(tf_img) _ = ax.set_title('Euclidean transformation') .. image-sg:: /auto_examples/transform/images/sphx_glr_plot_transform_types_001.png :alt: Euclidean transformation :srcset: /auto_examples/transform/images/sphx_glr_plot_transform_types_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 61-64 For a rotation around the center of the image, one can compose a translation to change the origin, a rotation, and finally the inverse of the first translation. .. GENERATED FROM PYTHON SOURCE LINES 64-74 .. code-block:: Python rotation = transform.EuclideanTransform(rotation=np.pi / 3) shift = transform.EuclideanTransform(translation=-np.array(img.shape[:2]) / 2) # Compose transforms by multiplying their matrices matrix = np.linalg.inv(shift.params) @ rotation.params @ shift.params tform = transform.EuclideanTransform(matrix) tf_img = transform.warp(img, tform.inverse) fig, ax = plt.subplots() _ = ax.imshow(tf_img) .. image-sg:: /auto_examples/transform/images/sphx_glr_plot_transform_types_002.png :alt: plot transform types :srcset: /auto_examples/transform/images/sphx_glr_plot_transform_types_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 75-80 Similarity transformation ================================= A `similarity transformation `_ preserves the shape of objects. It combines scaling, translation and rotation. .. GENERATED FROM PYTHON SOURCE LINES 80-90 .. code-block:: Python tform = transform.SimilarityTransform( scale=0.5, rotation=np.pi / 12, translation=(100, 50) ) print(tform.params) tf_img = transform.warp(img, tform.inverse) fig, ax = plt.subplots() ax.imshow(tf_img) _ = ax.set_title('Similarity transformation') .. image-sg:: /auto_examples/transform/images/sphx_glr_plot_transform_types_003.png :alt: Similarity transformation :srcset: /auto_examples/transform/images/sphx_glr_plot_transform_types_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none [[ 0.48296291 -0.12940952 100. ] [ 0.12940952 0.48296291 50. ] [ 0. 0. 1. ]] .. GENERATED FROM PYTHON SOURCE LINES 91-98 Affine transformation ================================= An `affine transformation `_ preserves lines (hence the alignment of objects), as well as parallelism between lines. It can be decomposed into a similarity transform and a `shear transformation `_. .. GENERATED FROM PYTHON SOURCE LINES 98-109 .. code-block:: Python tform = transform.AffineTransform( shear=np.pi / 6, ) print(tform.params) tf_img = transform.warp(img, tform.inverse) fig, ax = plt.subplots() ax.imshow(tf_img) _ = ax.set_title('Affine transformation') .. image-sg:: /auto_examples/transform/images/sphx_glr_plot_transform_types_004.png :alt: Affine transformation :srcset: /auto_examples/transform/images/sphx_glr_plot_transform_types_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none [[ 1. -0.57735027 0. ] [ 0. 1. 0. ] [ 0. 0. 1. ]] .. GENERATED FROM PYTHON SOURCE LINES 110-116 Projective transformation (homographies) ======================================== A `homography `_, also called projective transformation, preserves lines but not necessarily parallelism. .. GENERATED FROM PYTHON SOURCE LINES 116-125 .. code-block:: Python matrix = np.array([[1, -0.5, 100], [0.1, 0.9, 50], [0.0015, 0.0015, 1]]) tform = transform.ProjectiveTransform(matrix=matrix) tf_img = transform.warp(img, tform.inverse) fig, ax = plt.subplots() ax.imshow(tf_img) ax.set_title('Projective transformation') plt.show() .. image-sg:: /auto_examples/transform/images/sphx_glr_plot_transform_types_005.png :alt: Projective transformation :srcset: /auto_examples/transform/images/sphx_glr_plot_transform_types_005.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 127-136 See also ======================================== * :ref:`sphx_glr_auto_examples_transform_plot_geometric.py` for composing transformations or estimating their parameters * :ref:`sphx_glr_auto_examples_transform_plot_rescale.py` for simple rescaling and resizing operations * :func:`skimage.transform.rotate` for rotating an image around its center .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.411 seconds) .. _sphx_glr_download_auto_examples_transform_plot_transform_types.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-image/scikit-image/v0.23.2?filepath=notebooks/auto_examples/transform/plot_transform_types.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_transform_types.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_transform_types.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_