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affine.c Affine (3 pt) image transformation using a sampled (to nearest integer) transform on each dest point PIX *pixAffineSampledPta() PIX *pixAffineSampled() Affine (3 pt) image transformation using interpolation (or area mapping) for anti-aliasing images that are 2, 4, or 8 bpp gray, or colormapped, or 32 bpp RGB PIX *pixAffinePta() PIX *pixAffine() PIX *pixAffinePtaColor() PIX *pixAffineColor() PIX *pixAffinePtaGray() PIX *pixAffineGray() Affine transform including alpha (blend) component PIX *pixAffinePtaWithAlpha() Affine coordinate transformation l_int32 getAffineXformCoeffs() l_int32 affineInvertXform() l_int32 affineXformSampledPt() l_int32 affineXformPt() Interpolation helper functions l_int32 linearInterpolatePixelGray() l_int32 linearInterpolatePixelColor() Gauss-jordan linear equation solver l_int32 gaussjordan() Affine image transformation using a sequence of shear/scale/translation operations PIX *pixAffineSequential() One can define a coordinate space by the location of the origin, the orientation of x and y axes, and the unit scaling along each axis. An affine transform is a general linear transformation from one coordinate space to another. For the general case, we can define the affine transform using two sets of three (noncollinear) points in a plane. One set corresponds to the input (src) coordinate space; the other to the transformed (dest) coordinate space. Each point in the src corresponds to one of the points in the dest. With two sets of three points, we get a set of 6 equations in 6 unknowns that specifies the mapping between the coordinate spaces. The interface here allows you to specify either the corresponding sets of 3 points, or the transform itself (as a vector of 6 coefficients). Given the transform as a vector of 6 coefficients, we can compute both a a pointwise affine coordinate transformation and an affine image transformation. To compute the coordinate transform, we need the coordinate value (x',y') in the transformed space for any point (x,y) in the original space. To derive this transform from the three corresponding points, it is convenient to express the affine coordinate transformation using an LU decomposition of a set of six linear equations that express the six coordinates of the three points in the transformed space as a function of the six coordinates in the original space. Once we have this transform matrix , we can transform an image by finding, for each destination pixel, the pixel (or pixels) in the source that give rise to it. This 'pointwise' transformation can be done either by sampling and picking a single pixel in the src to replicate into the dest, or by interpolating (or averaging) over four src pixels to determine the value of the dest pixel. The first method is implemented by pixAffineSampled() and the second method by pixAffine(). The interpolated method can only be used for images with more than 1 bpp, but for these, the image quality is significantly better than the sampled method, due to the 'antialiasing' effect of weighting the src pixels. Interpolation works well when there is relatively little scaling, or if there is image expansion in general. However, if there is significant image reduction, one should apply a low-pass filter before subsampling to avoid aliasing the high frequencies. A typical application might be to align two images, which may be scaled, rotated and translated versions of each other. Through some pre-processing, three corresponding points are located in each of the two images. One of the images is then to be (affine) transformed to align with the other. As mentioned, the standard way to do this is to use three sets of points, compute the 6 transformation coefficients from these points that describe the linear transformation, x' = ax + by + c y' = dx + ey + f and use this in a pointwise manner to transform the image. N.B. Be sure to see the comment in getAffineXformCoeffs(), regarding using the inverse of the affine transform for points to transform images. There is another way to do this transformation; namely, by doing a sequence of simple affine transforms, without computing directly the affine coordinate transformation. We have at our disposal (1) translations (using rasterop), (2) horizontal and vertical shear about any horizontal and vertical line, respectively, and (3) non-isotropic scaling by two arbitrary x and y scaling factors. We also have rotation about an arbitrary point, but this is equivalent to a set of three shears so we do not need to use it. Why might we do this? For binary images, it is usually more efficient to do such transformations by a sequence of word parallel operations. Shear and translation can be done in-place and word parallel; arbitrary scaling is mostly pixel-wise. Suppose that we are tranforming image 1 to correspond to image 2. We have a set of three points, describing the coordinate space embedded in image 1, and we need to transform image 1 until those three points exactly correspond to the new coordinate space defined by the second set of three points. In our image matching application, the latter set of three points was found to be the corresponding points in image 2. The most elegant way I can think of to do such a sequential implementation is to imagine that we're going to transform BOTH images until they're aligned. (We don't really want to transform both, because in fact we may only have one image that is undergoing a general affine transformation.) Choose the 3 corresponding points as follows: - The 1st point is an origin - The 2nd point gives the orientation and scaling of the "x" axis with respect to the origin - The 3rd point does likewise for the "y" axis. These "axes" must not be collinear; otherwise they are arbitrary (although some strange things will happen if the handedness sweeping through the minimum angle between the axes is opposite). An important constraint is that we have shear operations about an arbitrary horizontal or vertical line, but always parallel to the x or y axis. If we continue to pretend that we have an unprimed coordinate space embedded in image 1 and a primed coordinate space embedded in image 2, we imagine (a) transforming image 1 by horizontal and vertical shears about point 1 to align points 3 and 2 along the y and x axes, respectively, and (b) transforming image 2 by horizontal and vertical shears about point 1' to align points 3' and 2' along the y and x axes. Then we scale image 1 so that the distances from 1 to 2 and from 1 to 3 are equal to the distances in image 2 from 1' to 2' and from 1' to 3'. This scaling operation leaves the true image origin, at (0,0) invariant, and will in general translate point 1. The original points 1 and 1' will typically not coincide in any event, so we must translate the origin of image 1, at its current point 1, to the origin of image 2 at 1'. The images should now be aligned. But because we never really transformed image 2 (and image 2 may not even exist), we now perform on image 1 the reverse of the shear transforms that we imagined doing on image 2; namely, the negative vertical shear followed by the negative horizontal shear. Image 1 should now have its transformed unprimed coordinates aligned with the original primed coordinates. In all this, it is only necessary to keep track of the shear angles and translations of points during the shears. What has been accomplished is a general affine transformation on image 1. Having described all this, if you are going to use an affine transformation in an application, this is what you need to know: (1) You should NEVER use the sequential method, because the image quality for 1 bpp text is much poorer (even though it is about 2x faster than the pointwise sampled method), and for images with depth greater than 1, it is nearly 20x slower than the pointwise sampled method and over 10x slower than the pointwise interpolated method! The sequential method is given here for purely pedagogical reasons. (2) For 1 bpp images, use the pointwise sampled function pixAffineSampled(). For all other images, the best quality results result from using the pointwise interpolated function pixAffinePta() or pixAffine(); the cost is less than a doubling of the computation time with respect to the sampled function. If you use interpolation on colormapped images, the colormap will be removed, resulting in either a grayscale or color image, depending on the values in the colormap. If you want to retain the colormap, use pixAffineSampled(). Typical relative timing of pointwise transforms (sampled = 1.0): 8 bpp: sampled 1.0 interpolated 1.6 32 bpp: sampled 1.0 interpolated 1.8 Additionally, the computation time/pixel is nearly the same for 8 bpp and 32 bpp, for both sampled and interpolated.
l_int32 affineInvertXform ( l_float32 *vc, l_float32 **pvci )
affineInvertXform() Input: vc (vector of 6 coefficients) *vci (<return> inverted transform) Return: 0 if OK; 1 on error Notes: (1) The 6 affine transform coefficients are the first two rows of a 3x3 matrix where the last row has only a 1 in the third column. We invert this using gaussjordan(), and select the first 2 rows as the coefficients of the inverse affine transform. (2) Alternatively, we can find the inverse transform coefficients by inverting the 2x2 submatrix, and treating the top 2 coefficients in the 3rd column as a RHS vector for that 2x2 submatrix. Then the 6 inverted transform coefficients are composed of the inverted 2x2 submatrix and the negative of the transformed RHS vector. Why is this so? We have Y = AX + R (2 equations in 6 unknowns) Then X = A'Y - A'R Gauss-jordan solves AF = R and puts the solution for F, which is A'R, into the input R vector.
l_int32 affineXformPt ( l_float32 *vc, l_int32 x, l_int32 y, l_float32 *pxp, l_float32 *pyp )
affineXformPt() Input: vc (vector of 6 coefficients) (x, y) (initial point) (&xp, &yp) (<return> transformed point) Return: 0 if OK; 1 on error Notes: (1) This computes the floating point location of the transformed point. (2) It does not check ptrs for returned data!
l_int32 affineXformSampledPt ( l_float32 *vc, l_int32 x, l_int32 y, l_int32 *pxp, l_int32 *pyp )
affineXformSampledPt() Input: vc (vector of 6 coefficients) (x, y) (initial point) (&xp, &yp) (<return> transformed point) Return: 0 if OK; 1 on error Notes: (1) This finds the nearest pixel coordinates of the transformed point. (2) It does not check ptrs for returned data!
l_int32 gaussjordan ( l_float32 **a, l_float32 *b, l_int32 n )
gaussjordan() Input: a (n x n matrix) b (rhs column vector) n (dimension) Return: 0 if ok, 1 on error Note side effects: (1) the matrix a is transformed to its inverse (2) the vector b is transformed to the solution X to the linear equation AX = B Adapted from "Numerical Recipes in C, Second Edition", 1992 pp. 36-41 (gauss-jordan elimination)
l_int32 getAffineXformCoeffs ( PTA *ptas, PTA *ptad, l_float32 **pvc )
getAffineXformCoeffs() Input: ptas (source 3 points; unprimed) ptad (transformed 3 points; primed) &vc (<return> vector of coefficients of transform) Return: 0 if OK; 1 on error We have a set of six equations, describing the affine transformation that takes 3 points (ptas) into 3 other points (ptad). These equations are: x1' = c*x1 + c*y1 + c y1' = c*x1 + c*y1 + c x2' = c*x2 + c*y2 + c y2' = c*x2 + c*y2 + c x3' = c*x3 + c*y3 + c y3' = c*x3 + c*y3 + c This can be represented as AC = B where B and C are column vectors B = [ x1' y1' x2' y2' x3' y3' ] C = [ c c c c c c c ] and A is the 6x6 matrix x1 y1 1 0 0 0 0 0 0 x1 y1 1 x2 y2 1 0 0 0 0 0 0 x2 y2 1 x3 y3 1 0 0 0 0 0 0 x3 y3 1 These six equations are solved here for the coefficients C. These six coefficients can then be used to find the dest point (x',y') corresponding to any src point (x,y), according to the equations x' = cx + cy + c y' = cx + cy + c that are implemented in affineXformPt(). !!!!!!!!!!!!!!!!!! Very important !!!!!!!!!!!!!!!!!!!!!! When the affine transform is composed from a set of simple operations such as translation, scaling and rotation, it is built in a form to convert from the un-transformed src point to the transformed dest point. However, when an affine transform is used on images, it is used in an inverted way: it converts from the transformed dest point to the un-transformed src point. So, for example, if you transform a boxa using transform A, to transform an image in the same way you must use the inverse of A. For example, if you transform a boxa with a 3x3 affine matrix 'mat', the analogous image transformation must use 'matinv': boxad = boxaAffineTransform(boxas, mat); affineInvertXform(mat, &matinv); pixd = pixAffine(pixs, matinv, L_BRING_IN_WHITE); !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
l_int32 linearInterpolatePixelColor ( l_uint32 *datas, l_int32 wpls, l_int32 w, l_int32 h, l_float32 x, l_float32 y, l_uint32 colorval, l_uint32 *pval )
linearInterpolatePixelColor() Input: datas (ptr to beginning of image data) wpls (32-bit word/line for this data array) w, h (of image) x, y (floating pt location for evaluation) colorval (color brought in from the outside when the input x,y location is outside the image; in 0xrrggbb00 format)) &val (<return> interpolated color value) Return: 0 if OK, 1 on error Notes: (1) This is a standard linear interpolation function. It is equivalent to area weighting on each component, and avoids "jaggies" when rendering sharp edges.
l_int32 linearInterpolatePixelGray ( l_uint32 *datas, l_int32 wpls, l_int32 w, l_int32 h, l_float32 x, l_float32 y, l_int32 grayval, l_int32 *pval )
linearInterpolatePixelGray() Input: datas (ptr to beginning of image data) wpls (32-bit word/line for this data array) w, h (of image) x, y (floating pt location for evaluation) grayval (color brought in from the outside when the input x,y location is outside the image) &val (<return> interpolated gray value) Return: 0 if OK, 1 on error Notes: (1) This is a standard linear interpolation function. It is equivalent to area weighting on each component, and avoids "jaggies" when rendering sharp edges.
PIX * pixAffine ( PIX *pixs, l_float32 *vc, l_int32 incolor )
pixAffine() Input: pixs (all depths; colormap ok) vc (vector of 6 coefficients for affine transformation) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK) Return: pixd, or null on error Notes: (1) Brings in either black or white pixels from the boundary (2) Removes any existing colormap, if necessary, before transforming
PIX * pixAffineColor ( PIX *pixs, l_float32 *vc, l_uint32 colorval )
pixAffineColor() Input: pixs (32 bpp) vc (vector of 6 coefficients for affine transformation) colorval (e.g., 0 to bring in BLACK, 0xffffff00 for WHITE) Return: pixd, or null on error
PIX * pixAffineGray ( PIX *pixs, l_float32 *vc, l_uint8 grayval )
pixAffineGray() Input: pixs (8 bpp) vc (vector of 6 coefficients for affine transformation) grayval (0 to bring in BLACK, 255 for WHITE) Return: pixd, or null on error
PIX * pixAffinePta ( PIX *pixs, PTA *ptad, PTA *ptas, l_int32 incolor )
pixAffinePta() Input: pixs (all depths; colormap ok) ptad (3 pts of final coordinate space) ptas (3 pts of initial coordinate space) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK) Return: pixd, or null on error Notes: (1) Brings in either black or white pixels from the boundary (2) Removes any existing colormap, if necessary, before transforming
PIX * pixAffinePtaColor ( PIX *pixs, PTA *ptad, PTA *ptas, l_uint32 colorval )
pixAffinePtaColor() Input: pixs (32 bpp) ptad (3 pts of final coordinate space) ptas (3 pts of initial coordinate space) colorval (e.g., 0 to bring in BLACK, 0xffffff00 for WHITE) Return: pixd, or null on error
PIX * pixAffinePtaGray ( PIX *pixs, PTA *ptad, PTA *ptas, l_uint8 grayval )
pixAffinePtaGray() Input: pixs (8 bpp) ptad (3 pts of final coordinate space) ptas (3 pts of initial coordinate space) grayval (0 to bring in BLACK, 255 for WHITE) Return: pixd, or null on error
PIX * pixAffinePtaWithAlpha ( PIX *pixs, PTA *ptad, PTA *ptas, PIX *pixg, l_float32 fract, l_int32 border )
pixAffinePtaWithAlpha() Input: pixs (32 bpp rgb) ptad (3 pts of final coordinate space) ptas (3 pts of initial coordinate space) pixg (<optional> 8 bpp, can be null) fract (between 0.0 and 1.0, with 0.0 fully transparent and 1.0 fully opaque) border (of pixels added to capture transformed source pixels) Return: pixd, or null on error Notes: (1) The alpha channel is transformed separately from pixs, and aligns with it, being fully transparent outside the boundary of the transformed pixs. For pixels that are fully transparent, a blending function like pixBlendWithGrayMask() will give zero weight to corresponding pixels in pixs. (2) If pixg is NULL, it is generated as an alpha layer that is partially opaque, using @fract. Otherwise, it is cropped to pixs if required and @fract is ignored. The alpha channel in pixs is never used. (3) Colormaps are removed. (4) When pixs is transformed, it doesn't matter what color is brought in because the alpha channel will be transparent (0) there. (5) To avoid losing source pixels in the destination, it may be necessary to add a border to the source pix before doing the affine transformation. This can be any non-negative number. (6) The input @ptad and @ptas are in a coordinate space before the border is added. Internally, we compensate for this before doing the affine transform on the image after the border is added. (7) The default setting for the border values in the alpha channel is 0 (transparent) for the outermost ring of pixels and (0.5 * fract * 255) for the second ring. When blended over a second image, this (a) shrinks the visible image to make a clean overlap edge with an image below, and (b) softens the edges by weakening the aliasing there. Use l_setAlphaMaskBorder() to change these values. (8) A subtle use of gamma correction is to remove gamma correction before scaling and restore it afterwards. This is done by sandwiching this function between a gamma/inverse-gamma photometric transform: pixt = pixGammaTRCWithAlpha(NULL, pixs, 1.0 / gamma, 0, 255); pixd = pixAffinePtaWithAlpha(pixg, ptad, ptas, NULL, fract, border); pixGammaTRCWithAlpha(pixd, pixd, gamma, 0, 255); pixDestroy(&pixt); This has the side-effect of producing artifacts in the very dark regions.
PIX * pixAffineSampled ( PIX *pixs, l_float32 *vc, l_int32 incolor )
pixAffineSampled() Input: pixs (all depths) vc (vector of 6 coefficients for affine transformation) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK) Return: pixd, or null on error Notes: (1) Brings in either black or white pixels from the boundary. (2) Retains colormap, which you can do for a sampled transform.. (3) For 8 or 32 bpp, much better quality is obtained by the somewhat slower pixAffine(). See that function for relative timings between sampled and interpolated.
PIX * pixAffineSampledPta ( PIX *pixs, PTA *ptad, PTA *ptas, l_int32 incolor )
pixAffineSampledPta() Input: pixs (all depths) ptad (3 pts of final coordinate space) ptas (3 pts of initial coordinate space) incolor (L_BRING_IN_WHITE, L_BRING_IN_BLACK) Return: pixd, or null on error Notes: (1) Brings in either black or white pixels from the boundary. (2) Retains colormap, which you can do for a sampled transform.. (3) The 3 points must not be collinear. (4) The order of the 3 points is arbitrary; however, to compare with the sequential transform they must be in these locations and in this order: origin, x-axis, y-axis. (5) For 1 bpp images, this has much better quality results than pixAffineSequential(), particularly for text. It is about 3x slower, but does not require additional border pixels. The poor quality of pixAffineSequential() is due to repeated quantized transforms. It is strongly recommended that pixAffineSampled() be used for 1 bpp images. (6) For 8 or 32 bpp, much better quality is obtained by the somewhat slower pixAffinePta(). See that function for relative timings between sampled and interpolated. (7) To repeat, use of the sequential transform, pixAffineSequential(), for any images, is discouraged.
PIX * pixAffineSequential ( PIX *pixs, PTA *ptad, PTA *ptas, l_int32 bw, l_int32 bh )
pixAffineSequential() Input: pixs ptad (3 pts of final coordinate space) ptas (3 pts of initial coordinate space) bw (pixels of additional border width during computation) bh (pixels of additional border height during computation) Return: pixd, or null on error Notes: (1) The 3 pts must not be collinear. (2) The 3 pts must be given in this order: - origin - a location along the x-axis - a location along the y-axis. (3) You must guess how much border must be added so that no pixels are lost in the transformations from src to dest coordinate space. (This can be calculated but it is a lot of work!) For coordinate spaces that are nearly at right angles, on a 300 ppi scanned page, the addition of 1000 pixels on each side is usually sufficient. (4) This is here for pedagogical reasons. It is about 3x faster on 1 bpp images than pixAffineSampled(), but the results on text are much inferior.
Zakariyya Mughal <email@example.com>
This software is copyright (c) 2014 by Zakariyya Mughal.
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