Lens Distortion Effects

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Imaging and Special Effects CE00376-2
Fac. of Comp., Eng. Tech. Staffordshire
University
Lens Distortion Effects
Dr. Claude C. Chibelushi
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Outline

• Introduction
• Transformation function
• Linear magnification
• Piecewise linear distortion
• Non-linear distortion
• Range problem
• Scaling of 2D image
• Orthogonal coordinate transformation
• Radial coordinate transformation
• Performance issues
• Summary

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Introduction

• Lens distortion effects image change by
  stretching and shrinking
• variable scaling coefficient applied to whole or
  part of image
• transformation of pixel position by varying
  amounts
• requires position transformation function
• type of function determines lens distortion
  effect wide variety of functions or function
  combinations
• scaling relative to chosen reference

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Introduction

• Geometric transformations (revisited)
• To solve problem of possible gaps in target image
  caused by magnification
• scan pixel position in target image
• compute corresponding position in source image
• apply inverse of desired transformation

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Transformation function
Linear magnification Image scaling
(revisited) for (row 0 row lt IM_HEIGHT
row) for (col 0 col lt IM_WIDTH col)
srcRow row / rowScale // inverse
scaling srcCol col / colScale if (
(srcRow gt 0 srcRow lt SRCIM_HEIGHT)
(srcCol gt 0 srcCol lt SRCIM_WIDTH)
) trgtImagerowcol srcImagesrcRowsrcCol
else trgtImagerowcol 255 //
arbitrary default
To be replaced by relevant inverse transformation
function for lens distortion
Assumption positive scaling factors
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Transformation function

• Linear magnification
• Position transformation function is straight line
  with slope greater than 1
• e.g. function for magnification by two
• t 2 s or s t / 2

position in source image
position in target image
slope
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Transformation function

• Piecewise linear distortion
• Position-dependent image scaling
• neighbouring image segments scaled using
  different linear functions
• typically combination of functions with slope
  greater than 1 (magnification) and slope less
  than 1 (demagnification)
• so that image size increase due to stretching
  balanced by size decrease due to shrinking

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Transformation function

• Example of non-linear distortion

http//www.pauck.de/marco/photo/stuff/peleng_fishe
ye/peleng_fisheye.html
http//www.omnitech.com/graphics/fisheye_macbeth.j
pg
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Transformation function

• Non-linear distortion
• Typical non-linear distortion fish-eye lens
  effect
• focus on region of interest
• magnification near centre of simulated lens (i.e.
  centre of scaling)
• demagnification (minification) near boundary of
  simulated lens

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Transformation function

• Non-linear distortion
• Position-dependent image scaling using non-linear
  function
• function whose slope is not constant
• e.g. sine transformation function
• t sin( s ) or s arcsin( t )

Range problem s might not be in legal range of
angles expected by sin
Range problem output range of sin not same as
expected range of t
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Transformation function

• Range problem
• Solution map value to fit into expected range
• Hence process for applying transformation from
  target to source position is
• convert target position to fit into input range
  of function
• apply inverse function
• convert function output to fit into range of
  source positions

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Transformation function

• Range problem
• Range fitting function similar to histogram
  stretch function (see lecture on Contrast
  Enhancement)
• for stretching values (old value v) from range
  minold, maxold to minnew, maxnew, about
  reference (ref)

1. Value shift
2. Value scaling
3. Value shift
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Transformation function

• Pseudo-code for position transformation
• // calculate source value for given target value
• scale(t)
• inVal fit2Range(t, tRange, inValRange, tRef,
  inValRef)
• outVal inverseFunction(inVal) // use suitable
  function
• s fit2Range(outVal, outValRange, sRange,
  outValRef, sRef)
• return s // return source value
• // convert value to fit into given range
• fit2Range(oldVal, oldRange, newRange, oldRef,
  newRef)
• newVal ((oldVal - oldRef) newRange /
  oldRange) newRef
• return newVal

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Scaling of 2D image

• Scaling of 2D image often based on 1D
  transformation function
• effect depends on how function is applied to x
  and y (i.e. column and row) coordinates of pixel
• coordinate transformation can be orthogonal,
  radial, ...
• for speed, function output values can be
  pre-calculated
• stored in look-up table

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Scaling of 2D image

• Orthogonal coordinate transformation
• x and y coordinates transformed independently
• Preserves horizontal and vertical lines

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Scaling of 2D image

• Pseudo-code for orthogonal coordinate
  transformation
• for (row 0 row lt IM_HEIGHT row)
• for (col 0 col lt IM_WIDTH col)
• // inverse transformation of row and col
• srcRow scale(row)
• srcCol scale(col)
• if ( (srcRow gt 0 srcRow lt SRCIM_HEIGHT)
• (srcCol gt 0 srcCol lt SRCIM_WIDTH) )
• trgtImagerowcol srcImagesrcRowsrcCol
  
• else
• trgtImagerowcol 255 // arbitrary
  default

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Scaling of 2D image

• Radial coordinate transformation
• x and y coordinates transformed in step
• so that target and source position lie along
  radial line from lens centre
• Preserves lines that radiate from centre of
  scaling

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Scaling of 2D image

• Radial coordinate transformation
• Process
• polar coordinate (i.e. radius relative to centre
  of lens) calculated from x and y coordinates
• radius transformed using transformation function
• x and y coordinates scaled to match new radius
• pixel copied to calculated position

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Scaling of 2D image

• Radial coordinate transformation
• Radial distance r between position (x, y) and
  lens centre (xc, yc)

Pythagoras theorem
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Scaling of 2D image

• Radial coordinate transformation
• Scale x and y coordinates to match new radius

Similar triangles formula
, hence
and
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Scaling of 2D image

• Radial coordinate transformation
• Calculation of x and y coordinates
• replace h and w by (y - yc) and (x - xc),
  respectively
• rearrange formulas to get unknown variables
• these are xs and ys, for inverse transformation

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Scaling of 2D image

• Pseudo-code for radial coordinate transformation
• // scale coordinate so that transformed point
  lies along
• // radius from centre of scaling to untransformed
  position
• radialScaleCoord(radius, oldRadius, oldCoord,
  centreCoord)
• // inverse transformation of coordinate
• coord ((oldCoord - centreCoord) radius /
  oldRadius)
• centreCoord
• return coord

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Scaling of 2D image

• Pseudo-code for radial coordinate transformation
  (ctd.)
• for (row 0 row lt IM_HEIGHT row)
• for (col 0 col lt IM_WIDTH col)
• radius sqrt( square(row - magCentreRow)
• square(col - magCentreCol) )
• // inverse transformation of radius
• srcRadius scale(radius)
• // scale row and col radially
• srcRow radialScaleCoord(srcRadius, radius,
  row, magCentreRow)
• srcCol radialScaleCoord(srcRadius, radius,
  col, magCentreCol)
• if ( (srcRow gt 0 srcRow lt SRCIM_HEIGHT)
• (srcCol gt 0 srcCol lt SRCIM_WIDTH) )
• trgtImagerowcol srcImagesrcRowsrcCol
  
• else
• trgtImagerowcol 255 // arbitrary
  default

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Scaling of 2D image

• Performance issues
• Aliasing artefacts arise when high resolution
  pattern (e.g. image, sound) changed into lower
  resolution pattern
• typical artefacts
• in still images jagged (jaggies) blocky, or
  Moiré patterns
• in moving images wagon-wheel effect, pixel
  popping (or crawlies)

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Scaling of 2D image

• Performance issues
• To minimise aliasing artefacts due to
  magnification
• source image should have higher resolution than
  target image
• source resolution higher than target resolution
  times maximum slope of transformation function
• apply anti-aliasing technique / filter

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Scaling of 2D image

• Performance issues
• Pseudo-code given earlier not optimised for
  performance, e.g.
• repeated function calls during raster scanning
  overhead due to context switching associated with
  function call
• solution minimise number of functions called
  repeatedly
• repeated recalculation of fixed values in
  functions e.g. ratio of value ranges,
  coordinates relative to lens centre
• solution pre-calculate and store fixed values
• repeated costly operations e.g. square root,
  sine
• solution pre-calculate and store in look-up table

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Summary

• Lens distortion effects
• image change by stretching and shrinking whole or
  part of image
• Image scaling based on position transformation
  function
• function may be linear, piecewise linear,
  non-linear
• scaling of 2D image using 1D function
• coordinate transformation can be orthogonal,
  radial, ...