幾種插值演算法對比研究
阿新 • • 發佈：2019-02-05
[研究內容]
目前比較常用的幾種插值演算法
[正文]
目前比較常用的插值演算法有這麼幾種：最鄰近插值，雙線性二次插值，三次插值，
Lanczos插值等等，今天我們來對比一下這幾種插值效果的優劣。
1，最鄰近插值
最鄰近插值演算法也叫做零階插值演算法，主要原理是讓輸出畫素的畫素值等於鄰域內
離它距離最近的畫素值。例如下圖中所示，P1距離0灰度值畫素的距離小於100灰度值的
距離，因此，P1位置的插值畫素為0。這個演算法的優點是計算簡單方便，缺點是影象容
易出現鋸齒。
2，雙線性二次插值
在介紹雙線性插值前，我們先介紹一下拉格朗日插值多項式。本文參考引用均來自
張鐵的《數值分析》一書。
我們的方法是這樣的，根據水平方向上的雙線性二次插值，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)，然後再根據這兩點的二次插值求取f(x,y)。
根據前面的例題，我們可以很容易的求取各點插值如下：
f(x,j)=(i+1-x)f(I,j)+(x-i)f(i+1,j) 公式1-(4)
f(x,j+1)=(i+1-x)f(I,j+1)+(x-i)f(i+1,j+1) 公式1-(5)
f(x,y)=(i+1-y)f(x,j)+(y-j)f(x,j+1) 公式1 -(6)
以上三式綜合可以得到：
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) 公式1-(7)
我們令x=i+p,y=j+q得：
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) 公式1-(8)
上式即為數字影象處理中的雙線性二次插值公式。
3，雙線性三次插值
4，Lanczos插值演算法
這裡我大概介紹一下演算法的流程：
這個演算法也是一個模板演算法，主要內容是計算模板中的權重資訊。
對於一維資訊，假如我們輸入點集為X，那麼，Lanczos對應有個視窗模板Window，這個視窗中每個位置的權重計算如下：
Fig.6 Lanczos
通常，這個a取2或者3，a=2時，該演算法適用於影象縮小插值；a=3時，該演算法適用於放大插值；對應不同a值得Lanczos插值曲線如上圖6所示；上述的公式分別為連續和離散的公式。我們根據輸入點X的位置，確定對應window中不同位置的權重L(x)，然後對模板中的點值取加權平均，公式如下：
這個S(x)即為X處的插值結果。
根據上述一維插值，推廣得到多維插值公式如下(這裡以二維為例)：
上述內容是對不同插值演算法簡單的進行了介紹，如果不明白可以查詢相關知識。
現在我們來看下相應的效果圖：
上面的一組效果圖均是先將原圖縮小50%，然後使用不同演算法放大到原圖大小得到的。由上面這組圖我們可以發現，效果最差的是最鄰近插值演算法，效果最好的是雙線性三次插值，Lanczos演算法跟三次插值大致一致；
由於程式語言不同，可能會造成耗時的差距，但是，對於同一種語言，統計得出：最鄰近插值速度最快，三次插值速度最慢，而Lanczos演算法與二次插值相仿。
綜上，Lanczos插值具有速度快，效果好，價效比最高的優點，這也是目前此演算法比較流行的原因。
最後，給出一個本人使用C#寫的Lanczos程式碼，程式碼未經優化，僅供測試，這裡的NP是對權重計算構建的對映表：
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;
}最後，分享一個專業的影象處理網站（微畫素），裡面有很多原始碼下載： http://www.zealpixel.com/portal.php 點選開啟連結