{"id":117,"date":"2026-03-24T11:15:08","date_gmt":"2026-03-24T09:15:08","guid":{"rendered":"https:\/\/blogs.uef.fi\/photogrammetry\/?page_id=117"},"modified":"2026-03-24T11:15:10","modified_gmt":"2026-03-24T09:15:10","slug":"understanding-affine-and-projective-transformations-in-photogrammetry","status":"publish","type":"page","link":"https:\/\/blogs.uef.fi\/photogrammetry\/understanding-affine-and-projective-transformations-in-photogrammetry\/","title":{"rendered":"Understanding Affine and Projective Transformations in Photogrammetry"},"content":{"rendered":"\n

In photogrammetry, transformations are essential tools for mapping points from one coordinate system to another. Two of the most widely used transformations are Affine<\/strong> and Projective Transformations<\/strong>. Both play a crucial role in tasks like image rectification, 3D modeling, and georeferencing. Here\u2019s a breakdown of these transformations:<\/p>\n\n\n\n

1. Affine Transformation<\/h3>\n\n\n\n

An Affine Transformation<\/strong> is a linear mapping method that preserves points, straight lines, and planes. It allows for translation, rotation, scaling, and shearing, making it versatile for many photogrammetric applications.<\/p>\n\n\n\n

Mathematically, a 2D affine transformation can be expressed as:[<\/mo>x<\/mi>\u2032<\/mo><\/msup><\/mstyle><\/mtd><\/mtr>y<\/mi>\u2032<\/mo><\/msup><\/mstyle><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>a<\/mi>1<\/mn><\/msub><\/mstyle><\/mtd>a<\/mi>2<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>4<\/mn><\/msub><\/mstyle><\/mtd>a<\/mi>5<\/mn><\/msub><\/mstyle><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>[<\/mo>x<\/mi><\/mstyle><\/mtd><\/mtr>y<\/mi><\/mstyle><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>+<\/mo>[<\/mo>a<\/mi>3<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>6<\/mn><\/msub><\/mstyle><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mrow>\\begin{bmatrix} x’ \\\\ y’ \\end{bmatrix} = \\begin{bmatrix} a_1 & a_2 \\\\ a_4 & a_5 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} + \\begin{bmatrix} a_3 \\\\ a_6 \\end{bmatrix}<\/annotation><\/semantics><\/math>[x\u2032y\u2032\u200b]=[a1\u200ba4\u200b\u200ba2\u200ba5\u200b\u200b][xy\u200b]+[a3\u200ba6\u200b\u200b]<\/p>\n\n\n\n

Here:<\/p>\n\n\n\n

    \n
  • x<\/mi>,<\/mo>y<\/mi><\/mrow>x, y<\/annotation><\/semantics><\/math>x,y are the original coordinates<\/li>\n\n\n\n
  • x<\/mi>\u2032<\/mo><\/msup>,<\/mo>y<\/mi>\u2032<\/mo><\/msup><\/mrow>x’, y’<\/annotation><\/semantics><\/math>x\u2032,y\u2032 are the transformed coordinates<\/li>\n\n\n\n
  • a<\/mi>1<\/mn><\/msub><\/mrow>a_1<\/annotation><\/semantics><\/math>a1\u200b to a<\/mi>6<\/mn><\/msub><\/mrow>a_6<\/annotation><\/semantics><\/math>a6\u200b are transformation parameters representing rotation, scaling, shearing, and translation.<\/li>\n<\/ul>\n\n\n\n

    Affine transformations require at least 3 non-collinear points<\/strong> to solve for the parameters using least-squares estimation.<\/p>\n\n\n\n

    Key Features:<\/strong><\/p>\n\n\n\n

      \n
    • Preserves parallelism of lines<\/li>\n\n\n\n
    • Can handle non-uniform scaling and skewing<\/li>\n\n\n\n
    • Suitable for most 2D-to-2D georeferencing tasks<\/li>\n<\/ul>\n\n\n\n

      2. Projective Transformation<\/h3>\n\n\n\n

      A Projective Transformation<\/strong>, also known as a homography, goes a step further by accounting for perspective distortions. This makes it ideal for mapping between images taken from different viewpoints, such as in aerial or satellite imagery.<\/p>\n\n\n\n

      The 2D projective transformation is given by:x<\/mi>\u2032<\/mo><\/msup>=<\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>2<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>3<\/mn><\/msub><\/mrow>a<\/mi>7<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>8<\/mn><\/msub>y<\/mi>+<\/mo>1<\/mn><\/mrow><\/mfrac>,<\/mo><\/mspace>y<\/mi>\u2032<\/mo><\/msup>=<\/mo>a<\/mi>4<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>5<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>6<\/mn><\/msub><\/mrow>a<\/mi>7<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>8<\/mn><\/msub>y<\/mi>+<\/mo>1<\/mn><\/mrow><\/mfrac><\/mrow>x’ = \\frac{a_1 x + a_2 y + a_3}{a_7 x + a_8 y + 1}, \\quad y’ = \\frac{a_4 x + a_5 y + a_6}{a_7 x + a_8 y + 1}<\/annotation><\/semantics><\/math>x\u2032=a7\u200bx+a8\u200by+1a1\u200bx+a2\u200by+a3\u200b\u200b,y\u2032=a7\u200bx+a8\u200by+1a4\u200bx+a5\u200by+a6\u200b\u200b<\/p>\n\n\n\n

      Here:<\/p>\n\n\n\n

        \n
      • The denominator introduces perspective effects<\/li>\n\n\n\n
      • At least 4 points<\/strong> are needed to compute the 8 parameters<\/li>\n<\/ul>\n\n\n\n

        Applications:<\/strong><\/p>\n\n\n\n

          \n
        • Correcting image distortions due to camera perspective<\/li>\n\n\n\n
        • Transforming images to a common plane for mosaicking<\/li>\n\n\n\n
        • Key step in 3D reconstruction and space resection techniques<\/li>\n<\/ul>\n\n\n\n

          Choosing Between Affine and Projective<\/h3>\n\n\n\n
            \n
          • Use Affine<\/strong> when images or coordinate sets differ only by rotation, scaling, and translation. It\u2019s simpler and computationally faster.<\/li>\n\n\n\n
          • Use Projective<\/strong> when perspective distortion is significant, such as aerial images captured at different angles or heights.<\/li>\n<\/ul>\n\n\n\n

            In essence, affine transformations handle linear adjustments, while projective transformations account for the complexities of perspective. Understanding both allows photogrammetrists to align images accurately and build reliable 3D models.<\/p>\n","protected":false},"excerpt":{"rendered":"

            In photogrammetry, transformations are essential tools for mapping points from one coordinate system to another. Two of the most widely used transformations are Affine and Projective Transformations. Both play a crucial role in tasks like image rectification, 3D modeling, and georeferencing. Here\u2019s a breakdown of these transformations: 1. Affine Transformation An Affine Transformation is a […]<\/p>\n","protected":false},"author":746,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-117","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nUnderstanding Affine and Projective Transformations in Photogrammetry - Learn Photogrammetry<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.uef.fi\/photogrammetry\/understanding-affine-and-projective-transformations-in-photogrammetry\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Understanding Affine and Projective Transformations in Photogrammetry - Learn Photogrammetry\" \/>\n<meta property=\"og:description\" content=\"In photogrammetry, transformations are essential tools for mapping points from one coordinate system to another. Two of the most widely used transformations are Affine and Projective Transformations. Both play a crucial role in tasks like image rectification, 3D modeling, and georeferencing. Here\u2019s a breakdown of these transformations: 1. 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