将float转换为整数

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如何缩放以及通过哪个因子缩放float的dctmtx系数以获得以下整数值：

浮动dctmtx：

( (0.3536    0.3536    0.3536    0.3536    0.3536    0.3536    0.3536    0.3536),
    (0.4904    0.4157    0.2778    0.0975   -0.0975   -0.2778   -0.4157   -0.4904),
    (0.4619    0.1913   -0.1913   -0.4619   -0.4619   -0.1913    0.1913    0.4619),
    (0.4157   -0.0975   -0.4904   -0.2778    0.2778    0.4904    0.0975   -0.4157),
    (0.3536   -0.3536   -0.3536    0.3536    0.3536   -0.3536   -0.3536    0.3536),
    (0.2778   -0.4904    0.0975    0.4157   -0.4157   -0.0975    0.4904   -0.2778),
    (0.1913   -0.4619    0.4619   -0.1913   -0.1913    0.4619   -0.4619    0.1913),
    (0.0975   -0.2778    0.4157   -0.4904    0.4904   -0.4157    0.2778   -0.0975)
)

整数dctmtx：

(( 125,   122,   115,    103,    88,     69,     47,      24  ),
        ( 125,   103,    47,    -24,   -88,   -122,   -115,     -69  ),
        ( 125,    69,   -47,   -122,   -88,     24,    115,     103  ),
        ( 125,    24,  -115,    -69,    88,    103,    -47,    -122  ),
        ( 125,   -24,  -115,     69,    88,   -103,    -47,     122  ),
        ( 125,   -69,   -47,    122,   -88,    -24,    115,    -103  ),
        ( 125,  -103,    47,     24,   -88,    122,   -115,      69  ),
        ( 125,  -122,   115,   -103,    88,    -69,     47,     -24  )
    );

鲁兹·莱曼（Lutz Lehmann）

如果您阅读离散余弦变换，将会发现基本系数为

cos(pi*i*(2*j+1)/16),  i,j=0..7

然后，第一个表由这些值按0.5缩放，第一行/列除以0.25 * sqrt（2）= 1 / sqrt（8）缩放外。这是获得正交矩阵的正确方法。第一列的平方和为8，其他列为4。

第二张表是将余弦值均匀乘以125时的舍入结果。在使用转置矩阵计算逆变换时，这里必须注意正确地重新缩放向量。

复制第一张表，但第一列除外：

> [[ Cos(pi*i*(2*j+1)/16)/2 : i in [0..7] ]: j in [0..7] ];        
[
    [ 0.5, 0.49039264, 0.46193977, 0.41573481, 0.35355339, 0.27778512, 0.19134172, 0.09754516 ],
    [ 0.5, 0.41573481, 0.19134172, -0.09754516, -0.35355339, -0.49039264, -0.46193977, -0.27778512 ],
    [ 0.5, 0.27778512, -0.19134172, -0.49039264, -0.35355339, 0.09754516, 0.46193977, 0.41573481 ],
    [ 0.5, 0.09754516, -0.46193977, -0.27778512, 0.35355339, 0.41573481, -0.19134172, -0.49039264 ],
    [ 0.5, -0.09754516, -0.46193977, 0.27778512, 0.35355339, -0.41573481, -0.19134172, 0.49039264 ],
    [ 0.5, -0.27778512, -0.19134172, 0.49039264, -0.35355339, -0.09754516, 0.46193977, -0.41573481 ],
    [ 0.5, -0.41573481, 0.19134172, 0.09754516, -0.35355339, 0.49039264, -0.46193977, 0.27778512 ],
    [ 0.5, -0.49039264, 0.46193977, -0.41573481, 0.35355339, -0.27778512, 0.19134172, -0.09754516 ]
]

第二个表，整数舍入之前

> [[ Cos( pi*i*(2*j+1)/16 ) *125 : i in [0..7] ]: j in [0..7] ];       
[
    [ 125, 122.5982, 115.4849, 103.9337, 88.3883, 69.4463, 47.8354, 24.3863 ],
    [ 125, 103.9337, 47.8354, -24.3863, -88.3883, -122.5982, -115.4849, -69.4463 ],
    [ 125, 69.4463, -47.8354, -122.5982, -88.3883, 24.3863, 115.4849, 103.9337 ],
    [ 125, 24.3863, -115.4849, -69.4463, 88.3883, 103.9337, -47.8354, -122.5982 ],
    [ 125, -24.3863, -115.4849, 69.4463, 88.3883, -103.9337, -47.8354, 122.5982 ],
    [ 125, -69.4463, -47.8354, 122.5982, -88.3883, -24.3863, 115.4849, -103.9337 ],
    [ 125, -103.9337, 47.8354, 24.3863, -88.3883, 122.5982, -115.4849, 69.4463 ],
    [ 125, -122.5982, 115.4849, -103.9337, 88.3883, -69.4463, 47.8354, -24.3863 ]
]

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