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# Peak signal-to-noise ratio

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### Peak signal-to-noise ratio

Peak signal-to-noise ratio, often abbreviated PSNR, is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic range, PSNR is usually expressed in terms of the logarithmic decibel scale.

PSNR is most commonly used to measure the quality of reconstruction of lossy compression codecs (e.g., for image compression). The signal in this case is the original data, and the noise is the error introduced by compression. When comparing compression codecs, PSNR is an approximation to human perception of reconstruction quality. Although a higher PSNR generally indicates that the reconstruction is of higher quality, in some cases it may not. One has to be extremely careful with the range of validity of this metric; it is only conclusively valid when it is used to compare results from the same codec (or codec type) and same content.[1][2]

PSNR is most easily defined via the mean squared error (MSE). Given a noise-free m×n monochrome image I and its noisy approximation K, MSE is defined as:

\mathit{MSE} = \frac{1}{m\,n}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1} [I(i,j) - K(i,j)]^2

The PSNR (in dB) is defined as:

\begin{align}\mathit{PSNR} &= 10 \cdot \log_{10} \left( \frac{\mathit{MAX}_I^2}{\mathit{MSE}} \right)\\ &= 20 \cdot \log_{10} \left( \frac{\mathit{MAX}_I}{\sqrt{\mathit{MSE}}} \right)\\ &= 20 \cdot \log_{10} \left( {\mathit{MAX}_I} \right) - 10 \cdot \log_{10} \left( } \right)\end{align}

Here, MAXI is the maximum possible pixel value of the image. When the pixels are represented using 8 bits per sample, this is 255. More generally, when samples are represented using linear PCM with B bits per sample, MAXI is 2B−1. For color images with three RGB values per pixel, the definition of PSNR is the same except the MSE is the sum over all squared value differences divided by image size and by three. Alternately, for color images the image is converted to a different color space and PSNR is reported against each channel of that color space, e.g., YCbCr or HSL.[3][4]

Typical values for the PSNR in lossy image and video compression are between 30 and 50 dB, provided the bit depth is 8 bits, where higher is better. For 16-bit data typical values for the PSNR are between 60 and 80 dB.[5][6] Acceptable values for wireless transmission quality loss are considered to be about 20 dB to 25 dB.[7][8]

In the absence of noise, the two images I and K are identical, and thus the MSE is zero. In this case the PSNR is infinite (or undefined, see Division by zero).[9]

Q=10, PSNR 31.45dB (full resolution is here.)
Example luma PSNR values for a cjpeg compressed image at various quality levels.

## References

1. ^ Huynh-Thu, Q.; Ghanbari, M. (2008). "Scope of validity of PSNR in image/video quality assessment". Electronics Letters 44 (13): 800.
2. ^ MIT.edu
3. ^ Oriani, Emanuele. "qpsnr: A quick PSNR/SSIM analyzer for Linux". Retrieved 6 April 2011.
4. ^ "pnmpsnr User Manual". Retrieved 6 April 2011.
5. ^ Welstead, Stephen T. (1999). Fractal and wavelet image compression techniques. SPIE Publication. pp. 155–156.
6. ^ Raouf Hamzaoui, Dietmar Saupe (May 2006). Barni, Mauro, ed. "Fractal Image Compression". Document and image compression (CRC Press) 968: 168–169.
7. ^ Thomos, N., Boulgouris, N. V., & Strintzis, M. G. (2006, January). Optimized Transmission of JPEG2000 Streams Over Wireless Channels. IEEE Transactions on Image Processing , 15 (1).
8. ^ Xiangjun, L., & Jianfei, C. Robust transmission of JPEG2000 encoded images over packet loss channels. ICME 2007 (pp. 947-950). School of Computer Engineering, Nanyang Technological University.
9. ^ Salomon, David (2007). Data Compression: The Complete Reference (4 ed.). Springer. p. 281.
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