{"id":122,"date":"2026-03-24T11:20:44","date_gmt":"2026-03-24T09:20:44","guid":{"rendered":"https:\/\/blogs.uef.fi\/photogrammetry\/?page_id=122"},"modified":"2026-03-24T11:20:46","modified_gmt":"2026-03-24T09:20:46","slug":"projective-transformation-in-photogrammetry","status":"publish","type":"page","link":"https:\/\/blogs.uef.fi\/photogrammetry\/projective-transformation-in-photogrammetry\/","title":{"rendered":"Projective Transformation in Photogrammetry"},"content":{"rendered":"\n


Projective transformation is a fundamental mapping in photogrammetry that models the perspective geometry of imaging systems.
2D Projective Transformation
A point (<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>(x, y)<\/annotation><\/semantics><\/math>(x,y) is mapped to (<\/mo>x<\/mi>\u2032<\/mo><\/msup>,<\/mo>y<\/mi>\u2032<\/mo><\/msup>)<\/mo><\/mrow>(x’, y’)<\/annotation><\/semantics><\/math>(x\u2032,y\u2032) as:
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><\/mrow>x’ = \\frac{a_1 x + a_2 y + a_3}{a_7 x + a_8 y + 1}<\/annotation><\/semantics><\/math>x\u2032=a7\u200bx+a8\u200by+1a1\u200bx+a2\u200by+a3\u200b\u200b 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>y’ = \\frac{a_4 x + a_5 y + a_6}{a_7 x + a_8 y + 1}<\/annotation><\/semantics><\/math>y\u2032=a7\u200bx+a8\u200by+1a4\u200bx+a5\u200by+a6\u200b\u200b This rational form allows modeling of perspective distortion.

Linearized Form for Estimation
To estimate the parameters, we rearrange the equations:
x<\/mi>\u2032<\/mo><\/msup>(<\/mo>a<\/mi>7<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>8<\/mn><\/msub>y<\/mi>+<\/mo>1<\/mn>)<\/mo>\u2212<\/mo>(<\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>2<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>3<\/mn><\/msub>)<\/mo>=<\/mo>0<\/mn><\/mrow>x'(a_7 x + a_8 y + 1) – (a_1 x + a_2 y + a_3) = 0<\/annotation><\/semantics><\/math>x\u2032(a7\u200bx+a8\u200by+1)\u2212(a1\u200bx+a2\u200by+a3\u200b)=0 y<\/mi>\u2032<\/mo><\/msup>(<\/mo>a<\/mi>7<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>8<\/mn><\/msub>y<\/mi>+<\/mo>1<\/mn>)<\/mo>\u2212<\/mo>(<\/mo>a<\/mi>4<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>5<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>6<\/mn><\/msub>)<\/mo>=<\/mo>0<\/mn><\/mrow>y'(a_7 x + a_8 y + 1) – (a_4 x + a_5 y + a_6) = 0<\/annotation><\/semantics><\/math>y\u2032(a7\u200bx+a8\u200by+1)\u2212(a4\u200bx+a5\u200by+a6\u200b)=0 Expanding:
x<\/mi>\u2032<\/mo><\/msup>a<\/mi>7<\/mn><\/msub>x<\/mi>+<\/mo>x<\/mi>\u2032<\/mo><\/msup>a<\/mi>8<\/mn><\/msub>y<\/mi>+<\/mo>x<\/mi>\u2032<\/mo><\/msup>\u2212<\/mo>a<\/mi>1<\/mn><\/msub>x<\/mi>\u2212<\/mo>a<\/mi>2<\/mn><\/msub>y<\/mi>\u2212<\/mo>a<\/mi>3<\/mn><\/msub>=<\/mo>0<\/mn><\/mrow>x’ a_7 x + x’ a_8 y + x’ – a_1 x – a_2 y – a_3 = 0<\/annotation><\/semantics><\/math>x\u2032a7\u200bx+x\u2032a8\u200by+x\u2032\u2212a1\u200bx\u2212a2\u200by\u2212a3\u200b=0 y<\/mi>\u2032<\/mo><\/msup>a<\/mi>7<\/mn><\/msub>x<\/mi>+<\/mo>y<\/mi>\u2032<\/mo><\/msup>a<\/mi>8<\/mn><\/msub>y<\/mi>+<\/mo>y<\/mi>\u2032<\/mo><\/msup>\u2212<\/mo>a<\/mi>4<\/mn><\/msub>x<\/mi>\u2212<\/mo>a<\/mi>5<\/mn><\/msub>y<\/mi>\u2212<\/mo>a<\/mi>6<\/mn><\/msub>=<\/mo>0<\/mn><\/mrow>y’ a_7 x + y’ a_8 y + y’ – a_4 x – a_5 y – a_6 = 0<\/annotation><\/semantics><\/math>y\u2032a7\u200bx+y\u2032a8\u200by+y\u2032\u2212a4\u200bx\u2212a5\u200by\u2212a6\u200b=0 These equations are linear with respect to the unknown parameters<\/strong>.

Matrix Form
For multiple points, the system is written as:
L<\/mi>=<\/mo>A<\/mi>X<\/mi><\/mrow>\\mathbf{L} = \\mathbf{A}\\mathbf{X}<\/annotation><\/semantics><\/math>L=AX where:
X<\/mi>=<\/mo>[<\/mo>a<\/mi>1<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>2<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>3<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>4<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>5<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>6<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>7<\/mn><\/msub><\/mstyle><\/mtd><\/mtr>a<\/mi>8<\/mn><\/msub><\/mstyle><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mrow>\\mathbf{X} = \\begin{bmatrix} a_1 \\\\ a_2 \\\\ a_3 \\\\ a_4 \\\\ a_5 \\\\ a_6 \\\\ a_7 \\\\ a_8 \\end{bmatrix}<\/annotation><\/semantics><\/math>X=\u200ba1\u200ba2\u200ba3\u200ba4\u200ba5\u200ba6\u200ba7\u200ba8\u200b\u200b\u200b The least squares solution is:
X<\/mi>^<\/mo><\/mover>=<\/mo>(<\/mo>A<\/mi>T<\/mi><\/msup>A<\/mi>)<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>T<\/mi><\/msup>L<\/mi><\/mrow>\\hat{\\mathbf{X}} = (\\mathbf{A}^T \\mathbf{A})^{-1} \\mathbf{A}^T \\mathbf{L}<\/annotation><\/semantics><\/math>X^=(ATA)\u22121ATL
3D to 2D Projective Transformation
For mapping (<\/mo>x<\/mi>,<\/mo>y<\/mi>,<\/mo>z<\/mi>)<\/mo><\/mrow>(x, y, z)<\/annotation><\/semantics><\/math>(x,y,z) to image coordinates:
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>z<\/mi>+<\/mo>a<\/mi>4<\/mn><\/msub><\/mrow>a<\/mi>9<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>10<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>11<\/mn><\/msub>z<\/mi>+<\/mo>1<\/mn><\/mrow><\/mfrac><\/mrow>x’ = \\frac{a_1 x + a_2 y + a_3 z + a_4}{a_9 x + a_{10} y + a_{11} z + 1}<\/annotation><\/semantics><\/math>x\u2032=a9\u200bx+a10\u200by+a11\u200bz+1a1\u200bx+a2\u200by+a3\u200bz+a4\u200b\u200b y<\/mi>\u2032<\/mo><\/msup>=<\/mo>a<\/mi>5<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>6<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>7<\/mn><\/msub>z<\/mi>+<\/mo>a<\/mi>8<\/mn><\/msub><\/mrow>a<\/mi>9<\/mn><\/msub>x<\/mi>+<\/mo>a<\/mi>10<\/mn><\/msub>y<\/mi>+<\/mo>a<\/mi>11<\/mn><\/msub>z<\/mi>+<\/mo>1<\/mn><\/mrow><\/mfrac><\/mrow>y’ = \\frac{a_5 x + a_6 y + a_7 z + a_8}{a_9 x + a_{10} y + a_{11} z + 1}<\/annotation><\/semantics><\/math>y\u2032=a9\u200bx+a10\u200by+a11\u200bz+1a5\u200bx+a6\u200by+a7\u200bz+a8\u200b\u200b This model contains 11 parameters<\/strong> and is widely used in photogrammetry (e.g., DLT).

Key Notes<\/strong><\/p>\n\n\n\n


Minimum of 4 points required for 2D case
Parameters solved using least squares
Captures perspective effects
More general than affine transformation<\/p>\n","protected":false},"excerpt":{"rendered":"

Projective transformation is a fundamental mapping in photogrammetry that models the perspective geometry of imaging systems.2D Projective TransformationA point (x,y)(x, y)(x,y) is mapped to (x\u2032,y\u2032)(x’, y’)(x\u2032,y\u2032) as:x\u2032=a1x+a2y+a3a7x+a8y+1x’ = \\frac{a_1 x + a_2 y + a_3}{a_7 x + a_8 y + 1}x\u2032=a7\u200bx+a8\u200by+1a1\u200bx+a2\u200by+a3\u200b\u200b y\u2032=a4x+a5y+a6a7x+a8y+1y’ = \\frac{a_4 x + a_5 y + a_6}{a_7 x + a_8 y + […]<\/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-122","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nProjective Transformation 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\/projective-transformation-in-photogrammetry\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Projective Transformation in Photogrammetry - Learn Photogrammetry\" \/>\n<meta property=\"og:description\" content=\"Projective transformation is a fundamental mapping in photogrammetry that models the perspective geometry of imaging systems.2D Projective TransformationA point (x,y)(x, y)(x,y) is mapped to (x\u2032,y\u2032)(x’, y’)(x\u2032,y\u2032) as:x\u2032=a1x+a2y+a3a7x+a8y+1x’ = frac{a_1 x + a_2 y + a_3}{a_7 x + a_8 y + 1}x\u2032=a7\u200bx+a8\u200by+1a1\u200bx+a2\u200by+a3\u200b\u200b y\u2032=a4x+a5y+a6a7x+a8y+1y’ = frac{a_4 x + a_5 y + a_6}{a_7 x + a_8 y + […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blogs.uef.fi\/photogrammetry\/projective-transformation-in-photogrammetry\/\" \/>\n<meta property=\"og:site_name\" content=\"Learn Photogrammetry\" \/>\n<meta property=\"article:modified_time\" content=\"2026-03-24T09:20:46+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/blogs.uef.fi\\\/photogrammetry\\\/projective-transformation-in-photogrammetry\\\/\",\"url\":\"https:\\\/\\\/blogs.uef.fi\\\/photogrammetry\\\/projective-transformation-in-photogrammetry\\\/\",\"name\":\"Projective Transformation in Photogrammetry - 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