{"id":6995,"date":"2021-08-02T14:47:34","date_gmt":"2021-08-02T13:47:34","guid":{"rendered":"https:\/\/www.blopig.com\/blog\/?p=6995"},"modified":"2021-08-03T16:12:52","modified_gmt":"2021-08-03T15:12:52","slug":"uniformly-sampled-3d-rotation-matrices","status":"publish","type":"post","link":"https:\/\/www.blopig.com\/blog\/2021\/08\/uniformly-sampled-3d-rotation-matrices\/","title":{"rendered":"Uniformly sampled 3D rotation matrices"},"content":{"rendered":"\n

It’s not as simple as you’d think.<\/strong><\/p>\n\n\n\n

If you want to skip the small talk, the code is at the bottom. <\/strong>Sampling 2D rotations uniformly is simple: rotate by an angle from the uniform distribution \"\theta. Extending this idea to 3D rotations, we could sample each of the three Euler angles from the same uniform distribution \"\phi,. This, however, gives more probability density to transformations which are clustered towards the poles:<\/p>\n\n\n\n

\"\"<\/a>
Sampling Euler angles uniformly does not give an even distribution across the sphere.<\/figcaption><\/figure><\/div>\n\n\n\n

In Fast Random Rotation Matrices (James Avro, 1992)<\/a>, a method for uniform random 3D rotation matrices is outlined, the main steps being:<\/p>\n\n\n\n\n\n\n\n

  1. A random rotation about the z<\/em> axis<\/li>
  2. Rotate the (unmoved) z<\/em> unit vector to a random position on the unit sphere<\/li><\/ol>\n\n\n\n

    The first of these is a simple 2D rotation matrix with a random angle, parameterised by \"x_1<\/p>\n\n\n\n

    \"\"<\/a>
    J. Avro, 1992<\/figcaption><\/figure><\/div>\n\n\n\n

    The second point is more tricky: it uses the Householder matrix with certain properties, given by:<\/p>\n\n\n\n

    \"\"<\/a>
    J. Avro, 1992<\/figcaption><\/figure><\/div>\n\n\n\n
    \"\"<\/a>
    J. Avro, 1992<\/figcaption><\/figure><\/div>\n\n\n\n

    where \"x_2\" and \"x_3\" are also drawn from \"U(0,. The final, completed random 3D rotation matrix is then (including a reflection through the origin to turn the Householder matrix transformation from a reflection into a rotation):<\/p>\n\n\n\n

    \"\"<\/a>
    J. Avro, 1992<\/figcaption><\/figure><\/div>\n\n\n\n

    I have included a python implementation which rotates any vector about the origin with the uniform 3D rotation matrix:<\/p>\n\n\n\n

    import numpy as np\n\ndef uniform_random_rotation(x):\n    \"\"\"Apply a random rotation in 3D, with a distribution uniform over the\n    sphere.\n\n    Arguments:\n        x: vector or set of vectors with dimension (n, 3), where n is the\n            number of vectors\n\n    Returns:\n        Array of shape (n, 3) containing the randomly rotated vectors of x,\n        about the mean coordinate of x.\n\n    Algorithm taken from \"Fast Random Rotation Matrices\" (James Avro, 1992):\n    https:\/\/doi.org\/10.1016\/B978-0-08-050755-2.50034-8\n    \"\"\"\n\n    def generate_random_z_axis_rotation():\n        \"\"\"Generate random rotation matrix about the z axis.\"\"\"\n        R = np.eye(3)\n        x1 = np.random.rand()\n        R[0, 0] = R[1, 1] = np.cos(2 * np.pi * x1)\n        R[0, 1] = -np.sin(2 * np.pi * x1)\n        R[1, 0] = np.sin(2 * np.pi * x1)\n        return R\n\n    # There are two random variables in [0, 1) here (naming is same as paper)\n    x2 = 2 * np.pi * np.random.rand()\n    x3 = np.random.rand()\n\n    # Rotation of all points around x axis using matrix\n    R = generate_random_z_axis_rotation()\n    v = np.array([\n        np.cos(x2) * np.sqrt(x3),\n        np.sin(x2) * np.sqrt(x3),\n        np.sqrt(1 - x3)\n    ])\n    H = np.eye(3) - (2 * np.outer(v, v))\n    M = -(H @ R)\n    x = x.reshape((-1, 3))\n    mean_coord = np.mean(x, axis=0)\n    return ((x - mean_coord) @ M) + mean_coord @ M<\/pre>\n","protected":false},"excerpt":{"rendered":"
    It’s not as simple as you’d think. If you want to skip the small talk, the code is at the bottom. Sampling 2D rotations uniformly is simple: rotate by an angle from the uniform distribution . Extending this idea to 3D rotations, we could sample each of the three Euler angles from the same uniform […]<\/p>\n","protected":false},"author":61,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","wikipediapreview_detectlinks":true,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"ngg_post_thumbnail":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[29,14,221,227,15],"tags":[416,418,420,421,422,415],"ppma_author":[541],"class_list":["post-6995","post","type-post","status-publish","format-standard","hentry","category-code","category-howto","category-python","category-python-code","category-technical","tag-matrices","tag-matrix","tag-rotation","tag-sample","tag-sphere","tag-vision"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"authors":[{"term_id":541,"user_id":61,"is_guest":0,"slug":"jack","display_name":"Jack Scantlebury","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/2d30962dbe9d08db0ac110abbf9ffd5bd52f4eb7da79636d286fb584280feb2c?s=96&d=mm&r=g","0":null,"1":"","2":"","3":"","4":"","5":"","6":"","7":"","8":""}],"_links":{"self":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/6995","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/users\/61"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/comments?post=6995"}],"version-history":[{"count":5,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/6995\/revisions"}],"predecessor-version":[{"id":7265,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/6995\/revisions\/7265"}],"wp:attachment":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/media?parent=6995"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/categories?post=6995"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/tags?post=6995"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/ppma_author?post=6995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}