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Topic: Bicubic interpolation

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	Bicubic interpolation - Wikipedia, the free encyclopedia
		In numerical analysis, a branch of mathematics, bicubic interpolation is one of the most common interpolation methods in two dimensions.
		Bicubic interpolation results in an interpolating function which is continuous, has continuous first partial derivatives, and has continuous cross derivatives everywhere.
		If we are interpolating a function with known derivatives, the common approach is to use the height at the four vertices, together with three derivatives at each vertex.
	en.wikipedia.org /wiki/Bicubic_interpolation (266 words)

	Interpolation - Wikipedia, the free encyclopedia
		Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points.
		Furthermore, the interpolant is a polynomial and thus infinitely differentiable.
		For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials.
	en.wikipedia.org /wiki/Interpolation (1046 words)

	Interpolation
		Bicubic: Bicubic interpolation uses the information from an original pixel and sixteen of the surrounding pixels to determine the color of the new pixels that are created from the original pixel.
		Bicubic interpolation is a big improvement over the previous two interpolation methods for two reasons: (1) Bicubic interpolation uses data from a larger number of pixels and (2) Bicubic interpolation uses a Bicubic calculation that is more sophisticated than the calculations of the previous interpolation methods.
		Bicubic interpolation is capable of producing photo quality results and is probably the method most commonly used.
	ronbigelow.com /articles/interpolation/interpolation.htm (2878 words)

	Understanding Digital Image Interpolation
		Image interpolation occurs in all digital photos at some stage-- whether this be in bayer demosaicing or in photo enlargement.
		Bicubic goes one step beyond bilinear by considering the closest 4x4 neighborhood of known pixels-- for a total of 16 pixels.
		Bicubic produces noticeably sharper images than the previous two methods, and is perhaps the ideal combination of processing time and output quality.
	www.cambridgeincolour.com /tutorials/image-interpolation.htm (1388 words)

	Interpolation: Digital Imaging: Glossary: Learn: Digital Photography Review (Site not responding. Last check: 2007-10-21)
		Interpolation (sometimes called resampling) is an imaging method to increase (or decrease) the number of pixels in a digital image.
		Bilinear Interpolation determines the value of a new pixel based on a weighted average of the 4 pixels in the nearest 2 x -2 neighborhood of the pixel in the original image.
		Fractal interpolation is mainly useful for extreme enlargements (for large prints) as it retains the shape of things more accurately with cleaner, sharper edges and less halos and blurring around the edges than bicubic interpolation would do.
	www.dpreview.com /learn/?/Glossary/Digital_Imaging/Interpolation_01.htm (446 words)

	Geometric Image Manipulation
		Bicubic interpolation reduces resampling artifacts even further by using the 16 nearest neighbors in the interpolation and by using bicubic waveforms rather than the linear waveforms used in bilinear interpolation.
		The bicubic interpolation routine assigns to point D in the destination image a value that is a bicubic function of the 16 pixels nearest S in the source image.
		When interpolations that require padding the source such as bilinear or bicubic interpolation are specified, the boundary of the source image needs to be extended such that it has the extra pixels needed to compute all the destination pixels.
	java.sun.com /products/java-media/jai/forDevelopers/jai1_0_1guide-unc/Geom-image-manip.doc.html (9090 words)

	Bicubic interpolation for image scaling (Site not responding. Last check: 2007-10-21)
		The standard approach is called bicubic interpolation, it estimates the colour at a pixel in the destination image by an average of 16 pixels surrounding the closest corresponding pixel in the source image.
		Another interpolation technique called bilinear interpolation will not be discussed here, it uses the value of 4 pixels in the source image.
		The nearest pixel coordinate (i,j) is the integer part of x and y, dx and dy in the diagram is the difference between these, dx = x - i, dy = y - j.
	astronomy.swin.edu.au /~pbourke/colour/bicubic (416 words)

	55:148 Dig. Image Proc. Chapter 4, Part 2
		The brightness interpolation problem is usually expressed in a dual way (by determining the brightness of the original point in the input image that corresponds to the point in the output image lying on the discrete raster).
		Bicubic interpolation does not suffer from the step-like boundary problem of nearest neighborhood interpolation, and copes with linear interpolation blurring as well.
		Bicubic interpolation is often used in raster displays that enable zooming with respect to an arbitrary point -- if the nearest neighborhood method were used, areas of the same brightness would increase.
	www.icaen.uiowa.edu /~dip/LECTURE/PreProcessing2.html (945 words)

	Spatial Transformations (Image Processing Toolbox) (Site not responding. Last check: 2007-10-21)
		Interpolation is the process by which we estimate an image value at a location in between image pixels.
		For bilinear interpolation, the output pixel value is a weighted average of pixels in the nearest 2-by-2 neighborhood.
		For bicubic interpolation, the output pixel value is a weighted average of pixels in the nearest 4-by-4 neighborhood.
	www-rohan.sdsu.edu /doc/matlab/toolbox/images/geom3.html (264 words)

	What's the point? Interpolation and extrapolation with a regular grid DEM
		Double linear interpolation attempts to minimise the likelihood of such errors by averaging the linear plane heights of the two triangles into which a grid cell can be divided by one of its diagonals.
		Bicubic interpolation is the lowest order 2-D interpolation procedure that maintains the continuity of the function and its first derivatives (both normal and tangential) across cell boundaries (Russell, 1995).
		The trend (or slopes) of the surface are important for interpolation, since we should not be constrained to using the nearest 3 or 4 vertices of the DEM cell.
	www.geocomputation.org /1999/082/gc_082.htm (6132 words)

	creativepro.com - Digital Photography How-To: The Ins and Outs of Interpolation
		Common interpolation algorithms can be grouped into two categories: adaptive and non-adaptive. Adaptive methods change depending on what they are interpolating (sharp edges vs. smooth texture), whereas non-adaptive methods treat all pixels equally.
		Bilinear interpolation considers the closest 2x2 neighborhood of known pixel values surrounding the unknown pixel. It then takes a weighted average of these 4 pixels to arrive at its final interpolated value. This results in much smoother looking images than nearest neighbor.
		On the other hand, some of these "artifacts" from adaptive interpolators may also be seen as benefits. Since the eye expects to see detail down to the smallest scales in fine-textured areas such as foliage, these patterns have been argued to trick the eye from a distance (for some subject matter).
	www.creativepro.com /story/feature/23593.html (811 words)

	Graphics.com: The shared resource for creative design (Site not responding. Last check: 2007-10-21)
		Bilinear interpolation produces anti-aliased pixels within the new, enlarged image, because whole number values for new pixels are not possible.
		In this case, a pixel is assigned a new value based upon information taken in horizontal, vertical and diagonal directions, and then the sum is averaged with a preference for the predominating tones in the area, resulting in a weighted average.
		Bicubic interpolation is the most processor-intensive resizing method because it involves the most calculations, but the aesthetic results are also the most faithful to the original image.
	www.graphics.com /modules.php?name=Sections&op=viewarticle&artid=56 (1257 words)

	Digital Photo Interpolation - what is the best image resizing / resampling method? Genuine Fractals, S-Spline, Bicubic, ...
		Photo interpolation is the process by which the number of pixels comprising an image is increased to allow printing enlargements that are of higher quality than photos that are not interpolated.
		Interpolation is commonly needed to make quality large prints from digital photos and film-scanned images.
		Interpolation of test photos was performed by the tool author using the method of his choosing.
	www.americaswonderlands.com /digital_photo_interpolation.htm (1034 words)

	Interpolation -- Part III
		The Bicubic, Genuine Fractals, and S-Spline are slightly sharper in the area of the scales than the Bicubic Smoother and Stairstep.
		Other than the Bicubic Smoother being a little bit softer than the other interpolations, there is no discernable difference between the clock interpolations.
		As was the case on the 300% doll interpolation, at 400%, Genuine Fractals is again the clear winner.
	ronbigelow.com /articles/interpolation3/interpolation3.htm (515 words)

	Abstract (Site not responding. Last check: 2007-10-21)
		Those are the bilinear interpolation scheme and the bicubic interpolation scheme.
		The interpolation method is coded in program 'mtt.for', where the kind of the interpolation scheme may be selected.
		The difference between the computational time required for the bicubic interpolation and the computational time required for the bilinear interpolation is negligible with respect to the computational time required to run the whole program.
	seis.karlov.mff.cuni.cz /papers/r10pb2.htm (402 words)

	Polynomials and Interpolation (Mathematics) (Site not responding. Last check: 2007-10-21)
		The value of an interpolated point is a combination of the values of the four closest points.
		The value of an interpolated point is a combination of the values of the sixteen closest points.
		Use bicubic interpolation when the interpolated data and its derivative must be continuous.
	www.rdg.ac.uk /SerDepts/su/Topic/Graphics/GrGMatl601/Matlab6/techdoc/math_anal/ch_5_p15.html (255 words)

	Efficient Image Magnification by Bicubic Spline Interpolation
		For high-resolution bicubic spline interpolation, the continuous image consists of a number of square regions, each region bordered on its corners by input pixels.
		The continuous image in each region is defined by high-resolution cubic spline interpolation in both the horizontal and vertical directions.
		For high-resolution bicubic spline interpolation, the index of the lowest indexed input pixel on which an intermediate pixel depends is _
	members.bellatlantic.net /~vze2vrva/design.html (715 words)

	Bilinear & Bicubic Interpolation - GameDev.Net Discussion Forums
		You get the general idea: perform linear interpolation along X twice, and then perform linear interpolation along Y once (between the two "x-values).
		Trilinear interpolation is similar, except you interpolate between 8 points (resulting in 7 interpolation operations).
		Bicubic interpolation is also similar, but you use 16 points instead of 4 points (still 3 interpolation operations).
	www.gamedev.net /community/forums/topic.asp?topic_id=61956&whichpage=1 (322 words)

	Coding Horror: The myth of infinite detail: Bilinear vs. Bicubic (Site not responding. Last check: 2007-10-21)
		There's another image sample at Interpolate This with a writeup that implies that bicubic is flat-out superior, but I'm not sure that's the case.
		I think the people pointing out that bicubic interpolation works best on photos weren't trying to claim that this would enable you to expand 3 pixels into a sharp and legible number plate.
		I think they were just pointing out that bicubic interpolation is often significantly better than bilinear interpolation on photographic images.
	www.codinghorror.com /blog/archives/000367.html (1330 words)

	Paul Butzi Photography - Digital Myth #1 - Staircase Interpolation
		One particular algorithm that's generated a lot of interest is 'stair interpolation' where the image is resampled upward in size, in small increments (typically 10%) until it's the size we want.
		To make comparing this version to the 'bicubic' version easy, I've arranged it so that if you 'hover' over the image with your mouse pointer, you get the 'bicubic' version, and when you move the mouse pointer away, you get the 'bicubic smoother' version.
		Which one you use might well be a matter of preference and your goals with the particular image you're working on, and perhaps the noise level or grain in the image.
	www.butzi.net /articles/resample.htm (1154 words)

	Designorati : Photoshop Jargon: Bicubic Interpolation (Site not responding. Last check: 2007-10-21)
		“Bicubic interpolation” is a method available in Photoshop for adding pixels to an image when an increase in size or resolution is made.
		Interpolation happens when Photoshop adds pixels to an image according to the color of its neighboring pixels.
		Bicubic interpolation is the highest-quality interpolation option but also the slowest.
	designorati.com /photoshop/2005/photoshop-jargon-bicubic-interpolation (260 words)

	3D Interpolation (Site not responding. Last check: 2007-10-21)
		BicubicCoefficients - Calculates the coefficients for the bicubic interpolation polynomial.
		BicubicInterpolation(double[],...) - Evaluates the value of the bicubic interpolation polynomial at a point and the derivatives in the directions of the grid.
		BicubicInterpolation(double[][],...) - Evaluates the values of the bicubic interpolation polynomial at a point and the derivatives in the directions of the grid when its coefficients are known.
	www.webcabcomponents.com /support/index.php?topic=339.msg1686 (231 words)

	CameraHobby - Genuine Fractals Review
		Scanned with the Epson 1200U at 1200 dpi and edited through Photoshop, I increased the file size using the Bicubic interpolation re-sizing and GF for a 360 dpi print output since the original scan file would only be large enough to output a 3x5 inch print at the same output resolution of 360 dpi.
		I expected GF to walk all over Bicubic when using larger file sizes but if I had not’t marked the prints to identify which was which I wouldn’t have been able to tell the difference.
		However, the Bicubic image seems to have a touch more clarity that may give it the appearance of sharpness in a print.
	www.camerahobby.com /Digital_GenuineFractals.htm (1451 words)

	Stitcher interpolation method: Why the Current Approach Isn't Good Enough (Site not responding. Last check: 2007-10-21)
		The discovery is that Stitcher appears to be using a bilinear (or similar) interpolation (a.k.a.
		Aside from the lightness of the image (which is affected by weighting of the interpolation algorithm) the essence of the image is the same, including quality limitations which produce the jagginess, blurriness, etc. This is why Photoshop DOES NOT USE bilinear interpolation by default.
		The rollover image is scaled in Photoshop using bicubic interpolation.
	www.carbonltd.com /interpolation (1117 words)

	Bigger digital images through interpolation - Jun. 10, 2003 (Site not responding. Last check: 2007-10-21)
		There are many technical definitions of interpolation, but itís just the way a software fills up the gaps when a pictureís dimension is increased.
		One is Fred Mirandaís (of www.fredmiranda.com) Stair Interpolation (SI).
		It uses the usual bicubic interpolation on the smoother parts of the pictures, but then employs proprietary interpolation techniques in sharper areas.
	www.inq7.net /inf/2003/jun/10/inf_37-2.htm (699 words)

	Interpolation - making your images bigger
		Image editing software uses interpolation whenever we rotate, resample or resize an image, but it is only when using these programs that we have some control over the type of interpolation being used.
		Bicubic is the slowest of the three methods as it averages a wider area of pixels but produces the best estimation of new values, and is generally the recommended choice for photos.
		When it is not possible to capture at a appropriate resolution due to the limitations in the capture device, then capture at the highest available native resolution and keep interpolation to a minimum.
	www.ephotozine.com /articles/viewarticle.cfm?id=52 (1559 words)

	photo.net Forum: Question on resampling & bicubic interpolation
		If bicubic interpolation produces similar results to Genuine Fractals for you, then that's more a matter of good luck at having the right image.
		After wasting a lot of time with stepping bicubic interpolation in 5, 10, 20 and 30-ish % steps all I managed to produce was a very obvious and unacceptable fractal pattern across the entire image which varied in size depending on the percentage steps I tried to up-sample in.
		They presented some comparisons between their results and those obtained with bicubic, Lanczos and other algorithms, and I must say that in some cases their method was noticeably superior (in some other cases performance was about the same as bicubic).
	www.photo.net /bboard/q-and-a-fetch-msg?msg_id=004MMk (2605 words)

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