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[研究内容]

目前比较常用的几种插值算法

[正文]

目前比较常用的插值算法有这么几种：最邻近插值，双线性二次插值，三次插值，

Lanczos插值等等，今天我们来对比一下这几种插值效果的优劣。

1，最邻近插值

最邻近插值算法也叫做零阶插值算法，主要原理是让输出像素的像素值等于邻域内离它距离最近的像素值。例如下图中所示，P1距离0灰度值像素的距离小于100灰度值的距离，因此，P1位置的插值像素为0。这个算法的优点是计算简单方便，缺点是图像容易出现锯齿。

2，双线性二次插值

在介绍双线性插值前，我们先介绍一下拉格朗日插值多项式。本文参考引用均来自张铁的《数值分析》一书。

我们的方法是这样的，根据水平方向上的双线性二次插值，由$f(i,j)$和$f(i+1,j)$求取$f(x,j)$，由

$f(i,j+1)$和$f(i+1,j+1)$求取$f(x,j+1)$，然后再根据这两点的二次插值求取$f(x,y)$。

根据前面的例题，我们可以很容易的求取各点插值如下：

$$f(x,j)=(i+1-x)f(i,j)+(x-i)f(i+1,j)\text{\hspace{1em}(1-4)}$$

$$f(x,j+1)=(i+1-x)f(i,j+1)+(x-i)f(i+1,j+1)\text{\hspace{1em}(1-5)}$$

$$f(x,y)=(i+1-y)f(x,j)+(y-j)f(x,j+1)\text{\hspace{1em}(1-6)}$$ 以上三式综合可以得到：

$$\begin{array}{rl}f(x,y)=& (j+1-y)(i+1-x)f(i,j)+(j+1-y)(x-i)f(i+1,j)\\ & +(y-j)(i+1-x)f(i,j+1)+(y-j)(x-i)f(i+1,j+1)\end{array}\text{\hspace{1em}(1-7)}$$ 我们令$x=i+p,y=j+q$得：

$$\begin{array}{rl}f(i+p,j+q)=& (1-q)(1-p)f(I,j)+p(1-q)f(i+1,j)+q(1-p)f(i,j+1)\\ & +pqf(i+1,j+1)\end{array}\text{\hspace{1em}(1-8)}$$

上式即为数字图像处理中的双线性二次插值公式。

3，双线性三次插值

$$A=\left[\begin{array}{cccc}S(1+v)& S(v)& S(1-v)& S(2-v)\end{array}\right]\text{\hspace{1em}(1-12)}$$ $$B=\left[\begin{array}{cccc}f(i-1,j-1)& f(i-1,j)& f(i-1,j+1)& f(i-1,j+2)\\ f(i,j-1)& f(i,j)& f(i,j+1)& f(i,j+2)\\ f(i+1,j-1)& f(i+1,j)& f(i+1,j+1)& f(i+1,j+2)\\ f(i+2,j-1)& f(i+2,j)& f(i+2,j+1)& f(i+2,j+2)\end{array}\right]\text{\hspace{1em}(1-13)}$$ $$C=\left[\begin{array}{c}S(1+u)\\ S(u)\\ S(1-u)\\ S(2-u)\end{array}\right]\text{\hspace{1em}(1-14)}$$ 等价于（可自行推导）

$$f(i+v,j+u)=\sum _{row=-1}^2\sum _{col=-1}^{2}f(i+row,j+col)S(row-v)S(col-u)\text{\hspace{1em}(1-15)}$$ 提示: 一定要区分本文中$v$, $u$和$row$, $col$的对应关系，$v$代表行数偏差，$u$代表列数偏差（如果混淆了，会造成最终的图像偏差很大）

4，Lanczos插值算法

该算法的主要原理介绍地址：

这里我大概介绍一下算法的流程：

这个算法也是一个模板算法，主要内容是计算模板中的权重信息。

对于一维信息，假如我们输入点集为$X$，那么，Lanczos对应有个窗口模板Window，这个窗口中每个位置的权重计算如下：

Fig.6 Lanczos

通常，这个 $a$取2或者3， $a=2$时，该算法适用于图像缩小插值； $a=3$时，该算法适用于放大插值；对应不同 $a$值得Lanczos插值曲线如上图6所示；上述的公式分别为连续和离散的公式。我们根据输入点X的位置，确定对应window中不同位置的权重L(x)，然后对模板中的点值取加权平均，公式如下： (1-13) 这个 $S(x)$即为 $X$处的插值结果。

根据上述一维插值，推广得到多维插值公式如下(这里以二维为例)：

(1-14) 上述内容是对不同插值算法简单的进行了介绍，如果不明白可以查找相关知识。

现在我们来看下相应的效果图：

上面的一组效果图均是先将原图缩小50%，然后使用不同算法放大到原图大小得到的。由上面这组图我们可以发现，效果最差的是最邻近插值算法，效果最好的是双线性三次插值，Lanczos算法跟三次插值大致一致；

由于编程语言不同，可能会造成耗时的差距，但是，对于同一种语言，统计得出：最邻近插值速度最快，三次插值速度最慢，而Lanczos算法与二次插值相仿。

综上，Lanczos插值具有速度快，效果好，性价比最高的优点，这也是目前此算法比较流行的原因。

最后，给出一个本人使用C#写的Lanczos代码，代码未经优化，仅供测试，这里的NP是对权重计算构建的映射表：

private Bitmap ZoomLanczos2Apply(Bitmap srcBitmap, double k) { Bitmap src = new Bitmap(srcBitmap); int width = src.Width; int height = src.Height; BitmapData srcData = src.LockBits(new Rectangle(0, 0, width, height), ImageLockMode.ReadWrite, PixelFormat.Format32bppArgb); byte* pS = (byte*)srcData.Scan0; int w = (int)((double)width * k); int h = (int)((double)height * k); Bitmap dst = new Bitmap(w, h, PixelFormat.Format32bppArgb); BitmapData dstData = dst.LockBits(new Rectangle(0, 0, w, h), ImageLockMode.ReadWrite, PixelFormat.Format32bppArgb); byte* d = (byte*)dstData.Scan0; int offset = dstData.Stride - w * 4; int x = 0, y = 0; double p = 0, q = 0; double n1 = 0, n2 = 0, n3 = 0, n4 = 0, nSum = 0; int p0 = 0, p1 = 0, p2 = 0, p3 = 0, p4 = 0, gray = 0; byte* temp = null; double[] NP1 = new double[] { -0.00623896218505032, -0.0122238956025722, -0.0179556611741633, -0.0234353793884155, -0.028664425422539, -0.0336444239814841, -0.0383772438642046, -0.0428649922670393, -0.0471100088344975, -0.0511148594680326, -0.0548823299036605, -0.0584154190695433, -0.0617173322348938, -0.0647914739617815, -0.0676414408716196, -0.0702710142382982, -0.0726841524200902, -0.0748849831426012, -0.0768777956451566, -0.0786670327031274, -0.0802572825387739, -0.0816532706332581, -0.0828598514525135, -0.0838820000996894, -0.0847248039068906, -0.0853934539789171, -0.0858932367016755, -0.0862295252278804, -0.0864077709525887, -0.0864334949910196, -0.086312279671003, -0.0860497600522689, -0.0856516154846428, -0.0851235612170509, -0.0844713400690516, -0.0837007141764163, -0.0828174568220629, -0.0818273443634177, -0.0807361482670294, -0.0795496272610035, -0.0782735196155422, -0.0769135355615895, -0.0754753498572731, -0.0739645945115212, -0.0723868516738999, -0.0707476466993797, -0.0690524413963838, -0.0673066274661114, -0.0655155201407567, -0.0636843520278618, -0.0618182671676555, -0.0599223153098279, -0.0580014464157926, -0.0560605053920726, -0.0541042270600343, -0.0521372313667693, -0.0501640188415028, -0.048188966301476, -0.0462163228108206, -0.0442502058955092, -0.0422945980170336, -0.0403533433070252, -0.0384301445645995, -0.0365285605177724, -0.0346520033498648, -0.0328037364913793, -0.0309868726774121, -0.0292043722702325, -0.0274590418462525, -0.0257535330461864, -0.0240903416868009, -0.0224718071322479, -0.0209001119225812, -0.0193772816566703, -0.0179051851263444, -0.0164855346982315, -0.015119886939392, -0.013809643482501, -0.0125560521259855, -0.0113602081641975, -0.0102230559423803, -0.00914539063088284, -0.00812786021277406, -0.00717096767873418, -0.00627507342282205, -0.00544039783246632, -0.00466702406578012, -0.00395490100907222, -0.00330384640720963, -0.00271355015928674, -0.00218357777186753, -0.00171337396189679, -0.00130226640121749, -0.000949469594490465, -0.00065408888218426, -0.000415124560192089, -0.000231476107535702, -0.000101946513534946, -0 }; double[] NP2 = new double[] { 0.999794398630316, 0.999177779156011, 0.998150695261436, 0.996714069021198, 0.994869189802256, 0.992617712728975, 0.989961656713271, 0.986903402052547, 0.983445687598761, 0.979591607502511, 0.975344607536654, 0.9707084810045, 0.965687364238256, 0.960285731693901, 0.954508390649264, 0.948360475512591, 0.941847441749449, 0.934975059436304, 0.927749406449645, 0.920176861299994, 0.912264095620641, 0.904018066321406, 0.895446007418168, 0.886555421549337, 0.877354071190877, 0.867849969581877, 0.858051371373022, 0.847966763010743, 0.83760485287009, 0.826974561149762, 0.816085009542993, 0.804945510698284, 0.793565557484247, 0.781954812073053, 0.770123094857198, 0.758080373214511, 0.745836750136481, 0.733402452735159, 0.720787820644003, 0.708003294328147, 0.695059403319663, 0.68196675439345, 0.668736019699406, 0.655377924866579, 0.641903237094975, 0.628322753250659, 0.614647287979759, 0.600887661856885, 0.587054689583387, 0.573159168250756, 0.559211865684328, 0.545223508882287, 0.531204772564777, 0.517166267847729, 0.503118531055783, 0.489072012688425, 0.47503706655321, 0.461023939079635, 0.447042758826945, 0.433103526198797, 0.419216103377392, 0.405390204489315, 0.391635386014941, 0.37796103745288, 0.364376372250524, 0.350890419011333, 0.33751201298905, 0.324249787878619, 0.311112167913061, 0.298107360275149, 0.285243347832182, 0.272527882201696, 0.259968477155437, 0.247572402368387, 0.235346677519141, 0.223298066747373, 0.211433073473617, 0.199757935586031, 0.188278620998268, 0.177000823582037, 0.165929959477376, 0.155071163783102, 0.144429287629353, 0.13400889563358, 0.123814263740785, 0.113849377448258, 0.104117930414501, 0.0946233234514916, 0.0853686638988765, 0.0763567653781721, 0.0675901479244855, 0.059071038492763, 0.050801371835042, 0.0427827917446759, 0.0350166526629909, 0.0275040216433488, 0.0202456806670952, 0.0132421293054104, 0.00649358772061002 }; double[] NP3 = new double[] { 0.00649358772061002, 0.0132421293054104, 0.0202456806670952, 0.0275040216433488, 0.0350166526629909, 0.0427827917446759, 0.0508013718350421, 0.059071038492763, 0.0675901479244855, 0.0763567653781721, 0.0853686638988765, 0.0946233234514916, 0.104117930414501, 0.113849377448258, 0.123814263740785, 0.13400889563358, 0.144429287629353, 0.155071163783102, 0.165929959477376, 0.177000823582037, 0.188278620998268, 0.199757935586031, 0.211433073473617, 0.223298066747373, 0.235346677519141, 0.247572402368387, 0.259968477155437, 0.272527882201696, 0.285243347832182, 0.298107360275149, 0.311112167913061, 0.324249787878619, 0.33751201298905, 0.350890419011333, 0.364376372250524, 0.37796103745288, 0.391635386014941, 0.405390204489315, 0.419216103377392, 0.433103526198797, 0.447042758826944, 0.461023939079634, 0.475037066553209, 0.489072012688425, 0.503118531055783, 0.517166267847729, 0.531204772564777, 0.545223508882287, 0.559211865684328, 0.573159168250756, 0.587054689583387, 0.600887661856885, 0.614647287979759, 0.628322753250659, 0.641903237094975, 0.655377924866579, 0.668736019699406, 0.68196675439345, 0.695059403319663, 0.708003294328147, 0.720787820644003, 0.733402452735159, 0.745836750136481, 0.758080373214511, 0.770123094857198, 0.781954812073053, 0.793565557484247, 0.804945510698284, 0.816085009542993, 0.826974561149762, 0.837604852870089, 0.847966763010743, 0.858051371373022, 0.867849969581877, 0.877354071190877, 0.886555421549337, 0.895446007418168, 0.904018066321406, 0.912264095620641, 0.920176861299994, 0.927749406449646, 0.934975059436304, 0.941847441749449, 0.948360475512591, 0.954508390649264, 0.960285731693901, 0.965687364238256, 0.9707084810045, 0.975344607536654, 0.979591607502512, 0.983445687598761, 0.986903402052547, 0.989961656713271, 0.992617712728975, 0.994869189802256, 0.996714069021198, 0.998150695261436, 0.999177779156011, 0.999794398630316 }; double[] NP4 = new double[] { -0, -0.000101946513534946, -0.000231476107535702, -0.000415124560192089, -0.00065408888218426, -0.000949469594490465, -0.0013022664012175, -0.00171337396189679, -0.00218357777186754, -0.00271355015928674, -0.00330384640720965, -0.00395490100907222, -0.00466702406578012, -0.00544039783246632, -0.00627507342282205, -0.00717096767873415, -0.00812786021277406, -0.00914539063088281, -0.0102230559423803, -0.0113602081641975, -0.0125560521259855, -0.013809643482501, -0.015119886939392, -0.0164855346982315, -0.0179051851263444, -0.0193772816566703, -0.0209001119225812, -0.0224718071322479, -0.0240903416868009, -0.0257535330461864, -0.0274590418462525, -0.0292043722702326, -0.0309868726774121, -0.0328037364913794, -0.0346520033498648, -0.0365285605177724, -0.0384301445645995, -0.0403533433070252, -0.0422945980170336, -0.0442502058955092, -0.0462163228108205, -0.048188966301476, -0.0501640188415028, -0.0521372313667693, -0.0541042270600343, -0.0560605053920726, -0.0580014464157926, -0.0599223153098279, -0.0618182671676555, -0.0636843520278618, -0.0655155201407567, -0.0673066274661114, -0.0690524413963838, -0.0707476466993797, -0.0723868516738999, -0.0739645945115212, -0.0754753498572731, -0.0769135355615895, -0.0782735196155421, -0.0795496272610035, -0.0807361482670294, -0.0818273443634177, -0.0828174568220629, -0.0837007141764163, -0.0844713400690516, -0.0851235612170509, -0.0856516154846428, -0.0860497600522689, -0.086312279671003, -0.0864334949910196, -0.0864077709525887, -0.0862295252278804, -0.0858932367016755, -0.0853934539789171, -0.0847248039068906, -0.0838820000996894, -0.0828598514525135, -0.0816532706332581, -0.0802572825387739, -0.0786670327031274, -0.0768777956451566, -0.0748849831426012, -0.0726841524200902, -0.0702710142382982, -0.0676414408716196, -0.0647914739617815, -0.0617173322348938, -0.0584154190695433, -0.0548823299036604, -0.0511148594680326, -0.0471100088344975, -0.0428649922670393, -0.0383772438642045, -0.0336444239814841, -0.028664425422539, -0.0234353793884155, -0.0179556611741633, -0.0122238956025722, -0.00623896218505032 }; for (int j = 0; j < h; j++) { q = (double)j / (double)k; y = (int)q; q = Math.Abs(q - (double)y); p0 = y * srcData.Stride; y = y >= height ? height - 1 : y; for (int i = 0; i < w; i++) { p = (double)i / (double)k; x = (int)p; p = Math.Abs(p - (double)x); temp = d + i * 4 + j * dstData.Stride; if (p != 0) { x = (x >= width - 3 ? width - 3 : x); x = x < 1 ? 1 : x; gray = (int)(p * 100.0) - 1; gray = Math.Max(0, gray); n1 = NP1[gray]; n2 = NP2[gray]; n3 = NP3[gray]; n4 = 1.0 - n1 - n2 - n3;// NP4[gray]; p2 = x * 4 + p0; p1 = p2 - 4; p3 = p2 + 4; p4 = p2 + 8; nSum = n1 + n2 + n3 + n4; gray = (int)((n1 * (double)((pS + p1)[0]) + n2 * (double)((pS + p2)[0]) + n3 * (double)((pS + p3)[0]) + n4 * (double)((pS + p4)[0]))); gray = Math.Max(0, Math.Min(255, gray)); temp[0] = (byte)gray; gray = (int)((n1 * (double)((pS + p1)[1]) + n2 * (double)((pS + p2)[1]) + n3 * (double)((pS + p3)[1]) + n4 * (double)((pS + p4)[1]))); gray = Math.Max(0, Math.Min(255, gray)); temp[1] = (byte)gray; gray = (int)((n1 * (double)((pS + p1)[2]) + n2 * (double)((pS + p2)[2]) + n3 * (double)((pS + p3)[2]) + n4 * (double)((pS + p4)[2]))); gray = Math.Max(0, Math.Min(255, gray)); temp[2] = (byte)gray; } else { x = x >= width ? width - 1 : x; gray = x * 4 + y * srcData.Stride; temp[0] = (byte)(pS + gray)[0]; temp[1] = (byte)(pS + gray)[1]; temp[2] = (byte)(pS + gray)[2]; } temp[3] = (byte)255; } } for (int i = 0; i < w; i++) { p = (double)i / (double)k; x = (int)p; p = Math.Abs(p - (double)x); x = x >= width ? width - 1 : x; for (int j = 0; j < h; j++) { q = (double)j / (double)k; y = (int)q; q = Math.Abs(q - (double)y); p0 = y * srcData.Stride; temp = d + i * 4 + j * dstData.Stride; if (q != 0) { y = y >= height - 3 ? height - 3 : y; y = y < 1 ? 1 : y; gray = (int)(q * 100.0) - 1; gray = Math.Max(0, gray); n1 = NP1[gray]; n2 = NP2[gray]; n3 = NP3[gray]; n4 = 1.0 - n1 - n2 - n3;// NP4[gray]; nSum = n1 + n2 + n3 + n4; p2 = x * 4 + y * srcData.Stride; p1 = p2 - srcData.Stride; p3 = p2 + srcData.Stride; p4 = p3 + srcData.Stride; gray = (int)((n1 * (double)((pS + p1)[0]) + n2 * (double)((pS + p2)[0]) + n3 * (double)((pS + p3)[0]) + n4 * (double)((pS + p4)[0]))); gray = Math.Max(0, Math.Min(255, gray)); temp[0] = (byte)gray; gray = (int)((n1 * (double)((pS + p1)[1]) + n2 * (double)((pS + p2)[1]) + n3 * (double)((pS + p3)[1]) + n4 * (double)((pS + p4)[1]))); gray = Math.Max(0, Math.Min(255, gray)); temp[1] = (byte)gray; gray = (int)((n1 * (double)((pS + p1)[2]) + n2 * (double)((pS + p2)[2]) + n3 * (double)((pS + p3)[2]) + n4 * (double)((pS + p4)[2]))); gray = Math.Max(0, Math.Min(255, gray)); temp[2] = (byte)gray; } else { y = y >= height ? height - 1 : y; gray = x * 4 + y * srcData.Stride; temp[0] = (byte)(pS + gray)[0]; temp[1] = (byte)(pS + gray)[1]; temp[2] = (byte)(pS + gray)[2]; } temp[3] = (byte)255; } } src.UnlockBits(srcData); dst.UnlockBits(dstData); return dst; }

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