Opencv 峰值信噪比|极客教程

离散余弦逆变换中如果不使用 $8$ 作为系数，而是使用 $4$ 作为系数的话，图像的画质会变差。来求输入图像和经过离散余弦逆变换之后的图像的峰值信噪比吧！再求出离散余弦逆变换的比特率吧！

峰值信噪比（Peak Signal to Noise Ratio）缩写为PSNR，用来表示信号最大可能功率和影响它的表示精度的破坏性噪声功率的比值，可以显示图像画质损失的程度。

峰值信噪比越大，表示画质损失越小。峰值信噪比通过下式定义。MAX表示图像点颜色的最大数值。如果取值范围是 $[0,255]$ 的话，那么MAX的值就为255。MSE表示均方误差（Mean Squared Error），用来表示两个图像各个像素点之间差值平方和的平均数：
$\text{PSNR}=10\ \log_{10}\ \frac{{v_{max}}^2}{\text{MSE}}\\ \text{MSE}=\frac{\sum\limits_{y=0}^{H-1}\ \sum\limits_{x=0}^{W-1}\ [I_1(x,y)-I_2(x,y)]^2}{H\ W}$
如果我们进行 $8\times8$ 的离散余弦变换，离散余弦逆变换的系数为 $KtimesK$ 的话，比特率按下式定义：
$\text{bit rate}=8\ \frac{K^2}{8^2}$

python实现：

import cv2
import numpy as np
import matplotlib.pyplot as plt

# DCT hyoer-parameter
T = 8
K = 4
channel = 3

# DCT weight
def w(x, y, u, v):
    cu = 1.
    cv = 1.
    if u == 0:
        cu /= np.sqrt(2)
    if v == 0:
        cv /= np.sqrt(2)
    theta = np.pi / (2 * T)
    return (( 2 * cu * cv / T) * np.cos((2*x+1)*u*theta) * np.cos((2*y+1)*v*theta))

# DCT
def dct(img):
    H, W, _ = img.shape

    F = np.zeros((H, W, channel), dtype=np.float32)

    for c in range(channel):
        for yi in range(0, H, T):
            for xi in range(0, W, T):
                for v in range(T):
                    for u in range(T):
                        for y in range(T):
                            for x in range(T):
                                F[v+yi, u+xi, c] += img[y+yi, x+xi, c] * w(x,y,u,v)

    return F


# IDCT
def idct(F):
    H, W, _ = F.shape

    out = np.zeros((H, W, channel), dtype=np.float32)

    for c in range(channel):
        for yi in range(0, H, T):
            for xi in range(0, W, T):
                for y in range(T):
                    for x in range(T):
                        for v in range(K):
                            for u in range(K):
                                out[y+yi, x+xi, c] += F[v+yi, u+xi, c] * w(x,y,u,v)

    out = np.clip(out, 0, 255)
    out = np.round(out).astype(np.uint8)

    return out


# MSE
def MSE(img1, img2):
    H, W, _ = img1.shape
    mse = np.sum((img1 - img2) ** 2) / (H * W * channel)
    return mse

# PSNR
def PSNR(mse, vmax=255):
    return 10 * np.log10(vmax * vmax / mse)

# bitrate
def BITRATE():
    return 1. * T * K * K / T / T


# Read image
img = cv2.imread("imori.jpg").astype(np.float32)

# DCT
F = dct(img)

# IDCT
out = idct(F)

# MSE
mse = MSE(img, out)

# PSNR
psnr = PSNR(mse)

# bitrate
bitrate = BITRATE()

print("MSE:", mse)
print("PSNR:", psnr)
print("bitrate:", bitrate)

# Save result
cv2.imshow("result", out)
cv2.waitKey(0)
cv2.imwrite("out.jpg", out)

c++实现：

#include <opencv2/core.hpp>
#include <opencv2/highgui.hpp>
#include <iostream>
#include <math.h>
#include <complex>


const int height = 128, width = 128, channel = 3;

// DCT hyper-parameter
int T = 8;
int K = 4;

// DCT coefficient
struct dct_str {
  double coef[height][width][channel];
};

// Discrete Cosine transformation
dct_str dct(cv::Mat img, dct_str dct_s){
  double I;
  double F;
  double Cu, Cv;

  for(int ys = 0; ys < height; ys += T){
    for(int xs = 0; xs < width; xs += T){
      for(int c = 0; c < channel; c++){
        for(int v = 0; v < T; v ++){
          for(int u = 0; u < T; u ++){
            F = 0;

            if (u == 0){
              Cu = 1. / sqrt(2);
            } else{
              Cu = 1;
            }

            if (v == 0){
              Cv = 1. / sqrt(2);
            }else {
              Cv = 1;
            }

            for (int y = 0; y < T; y++){
              for(int x = 0; x < T; x++){
                I = (double)img.at<cv::Vec3b>(ys + y, xs + x)[c];
                F += 2. / T * Cu * Cv * I * cos((2. * x + 1) * u * M_PI / 2. / T) * cos((2. * y + 1) * v * M_PI / 2. / T);
              }
            }

            dct_s.coef[ys + v][xs + u][c] = F;
          }
        }
      }
    }
  }

  return dct_s;
}

// Inverse Discrete Cosine transformation
cv::Mat idct(cv::Mat out, dct_str dct_s){
  double f;
  double Cu, Cv;

  for (int ys = 0; ys < height; ys += T){
    for (int xs = 0; xs < width; xs += T){
      for(int c = 0; c < channel; c++){
        for (int y = 0; y < T; y++){
          for (int x = 0; x < T; x++){
            f = 0;

            for (int v = 0; v < K; v++){
              for (int u = 0; u < K; u++){
                if (u == 0){
                  Cu = 1. / sqrt(2);
                } else {
                  Cu = 1;
                }

                if (v == 0){
                  Cv = 1. / sqrt(2);
                } else { 
                  Cv = 1;
                }

                f += 2. / T * Cu * Cv * dct_s.coef[ys + v][xs + u][c] * cos((2. * x + 1) * u * M_PI / 2. / T) * cos((2. * y + 1) * v * M_PI / 2. / T);
              }
            }

            f = fmin(fmax(f, 0), 255);
            out.at<cv::Vec3b>(ys + y, xs + x)[c] = (uchar)f;
          }
        }
      }
    }
  }

  return out;
}

// Compute MSE
double MSE(cv::Mat img1, cv::Mat img2){
  double mse = 0;

  for(int y = 0; y < height; y++){
    for(int x = 0; x < width; x++){
      for(int c = 0; c < channel; c++){
        mse += pow(((double)img1.at<cv::Vec3b>(y, x)[c] - (double)img2.at<cv::Vec3b>(y, x)[c]), 2);
      }
    }
  }

  mse /= (height * width);
  return mse;
}

// Compute PSNR
double PSNR(double mse, double v_max){
  return 10 * log10(v_max * v_max / mse);
}

// Compute bitrate
double BITRATE(){
  return T * K * K / T * T;
}

// Main
int main(int argc, const char* argv[]){

  double mse;
  double psnr;
  double bitrate;

  // read original image
  cv::Mat img = cv::imread("imori.jpg", cv::IMREAD_COLOR);

  // DCT coefficient
  dct_str dct_s;

  // output image
  cv::Mat out = cv::Mat::zeros(height, width, CV_8UC3);

  // DCT
  dct_s = dct(img, dct_s);

  // IDCT
  out = idct(out, dct_s);

  // MSE, PSNR
  mse = MSE(img, out);
  psnr = PSNR(mse, 255);
  bitrate = BITRATE();

  std::cout << "MSE: " << mse << std::endl;
  std::cout << "PSNR: " << psnr << std::endl;
  std::cout << "bitrate: " << bitrate << std::endl;

  cv::imwrite("out.jpg", out);
  //cv::imshow("answer", out);
  //cv::waitKey(0);
  cv::destroyAllWindows();

  return 0;
}

输入：

Opencv 峰值信噪比