# ssim

Structural similarity (SSIM) index for measuring image quality

## Description

example

ssimval = ssim(A,ref) calculates the structural similarity (SSIM) index for grayscale image or volume A using ref as the reference image or volume. A value closer to 1 indicates better image quality.

ssimval = ssim(A,ref,Name,Value) calculates the SSIM, using name-value pairs to control aspects of the computation.

[ssimval,ssimmap] = ssim(___) also returns the local SSIM value for each pixel or voxel in A.

## Examples

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Read an image into the workspace. Create another version of the image, applying a blurring filter.

H = fspecial("Gaussian",[11 11],1.5);
A = imfilter(ref,H,"replicate");

Display both images as a montage. The images differ most along sharp high-contrast regions, such as the edges of the trellis.

montage({ref,A})
title("Reference Image (Left) vs. Blurred Image (Right)")

Calculate the global SSIM value for the image and local SSIM values for each pixel.

[ssimval,ssimmap] = ssim(A,ref);

Display the local SSIM map. Include the global SSIM value in the figure title. Small values of local SSIM appear as dark pixels in the local SSIM map. Regions with small local SSIM value correspond to areas where the blurred image noticeably differs from the reference image. Large values of local SSIM value appear as bright pixels. Regions with large local SSIM correspond to uniform regions of the reference image, where blurring has less of an impact on the image.

imshow(ssimmap,[])
title("Local SSIM Map with Global SSIM Value: "+num2str(ssimval))

Read an image into the workspace. Create another version of the image, applying a blurring filter.

Display both images as a montage.

montage({ref A})
title("Reference Image (Left) vs. Blurred Image (Right)")

Simulate batches of images by replicating the reference image and the blurred image 16 times along the 4th dimension.

A = repmat(A,[1 1 1 16]);
ref = repmat(ref,[1 1 1 16]);

Create formatted dlarray objects for the reference image batch and the blurred image batch. The format is "SSCB", for spatial-spatial-channel-batch.

A = dlarray(single(A),"SSCB");
ref = dlarray(single(ref),"SSCB");

Calculate the global SSIM value for the image and local SSIM values for each pixel. ssimVal returns a scalar SSIM value for each image in the batch. ssimMap returns a map of SSIM values, the same size as the image, for each image in the batch.

[ssimVal,ssimMap] = ssim(A,ref);
size(ssimVal)
ans = 1×4

1     1     1    16

size(ssimMap)
ans = 1×4

291   240     1    16

## Input Arguments

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Image for quality measurement, specified as a numeric array or a dlarray (Deep Learning Toolbox) object. If A is not a 2-D grayscale image or 3-D grayscale volume, such as an RGB image or stack of grayscale images, specify the DataFormat name-value argument. Do not specify the DataFormat name-value argument if A is a formatted dlarray object.

Data Types: single | double | int16 | uint8 | uint16

Reference image against which to measure quality, specified as a numeric array or a dlarray (Deep Learning Toolbox) object of the same size and data type as A. If ref is not a 2-D grayscale image or 3-D grayscale volume, such as an RGB image or stack of grayscale images, specify the DataFormat name-value argument. Do not specify the DataFormat name-value argument if ref is a formatted dlarray object.

Data Types: single | double | int16 | uint8 | uint16

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: ssim(A,ref,"DynamicRange",100)

Dimension labels of the input images A and ref, specified as a string scalar or character vector. Each character in DataFormat must be one of these labels:

• S — Spatial

• C — Channel

• B — Batch observations

The format cannot include more than one channel label or batch label. Do not specify the DataFormat name-value argument when the input images are formatted dlarray objects.

Example: "SSC" indicates that the array has two spatial dimensions and one channel dimension, appropriate for 2-D RGB image data.

Example: "SSCB" indicates that the array has two spatial dimensions, one channel dimension, and one batch dimension, appropriate for a sequence of 2-D RGB image data.

Data Types: char | string

Dynamic range of the input image, specified as a positive scalar. The default value of "DynamicRange" depends on the data type of image A, and is calculated as diff(getrangefromclass(A)). For example, the default dynamic range is 255 for images of data type uint8, and the default is 1 for images of data type double or single with pixel values in the range [0, 1].

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

Exponents for the luminance, contrast, and structural terms, specified as a 3-element vector of nonnegative numbers of the form [alpha beta gamma].

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

Standard deviation of isotropic Gaussian function, specified as a positive number. This value is used for weighting the neighborhood pixels around a pixel for estimating local statistics. This weighting is used to avoid blocking artifacts in estimating local statistics.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

Regularization constants for the luminance, contrast, and structural terms, specified as a 3-element vector of nonnegative numbers of the form [c1 c2 c3]. The ssim function uses these regularization constants to avoid instability for image regions where the local mean or standard deviation is close to zero. Therefore, small non-zero values should be used for these constants.

By default,

• C1 = (0.01*L).^2, where L is the specified DynamicRange value.

• C2 = (0.03*L).^2, where L is the specified DynamicRange value.

• C3 = C2/2

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

## Output Arguments

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SSIM index, returned as one of these values.

Input Image TypeSSIM Value

• Unformatted numeric arrays

• Formatted numeric arrays with neither a channel ("C") nor batch ("B") dimension

Numeric scalar with a single SSIM measurement.
• Unformatted dlarray (Deep Learning Toolbox) objects

Scalar dlarray object with a single SSIM measurement.

• Numeric arrays with a channel or batch dimension specified using the DataFormat name-value argument

Numeric array of the same dimensionality as the input images. The spatial dimensions of ssimval are singleton dimensions. There is one SSIM measurement for each element along any channel or batch dimension.
• Formatted dlarray objects with a channel or batch dimension

• Unformatted dlarray objects with a channel or batch dimension specified using the DataFormat name-value argument

dlarray object of the same dimensionality as the input images. The spatial dimensions of ssimval are singleton dimensions. There is one SSIM measurement for each element along any channel or batch dimension.

ssimval is of data type double except when A is of data type single, in which case ssimval is of data type single.

The value of ssimval is typically in the range [0, 1]. The value 1 indicates the highest quality and occurs when A and ref are equivalent. Smaller values correspond to poorer quality. For some combinations of inputs and name-value pair arguments, ssimval can be negative.

Local values of the SSIM index, returned as one of these values.

Input Image TypeSSIM Value

• Unformatted numeric arrays

• Formatted numeric arrays with neither a channel ("C") nor batch ("B") dimension

Numeric array the same size as the input images. There is one SSIM measurement for each element in the input image.
• Unformatted dlarray (Deep Learning Toolbox) objects

dlarray object the same size as the input images. There is one SSIM measurement for each element in the input image.
• Numeric arrays with a channel or batch dimension specified using the DataFormat name-value argument

Numeric array the same size as the input images. Each spatial element in the input image has an SSIM measurement along any channel or batch dimension.
• Formatted dlarray objects with a channel or batch dimension

• Unformatted dlarray objects with a channel or batch dimension specified using the DataFormat name-value argument

dlarray object the same size as the input images. Each spatial element in the input image has an SSIM measurement along any channel or batch dimension.

ssimmap is of data type double except when A is of data type single, in which case ssimmap is of data type single.

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### Structural Similarity Index

An image quality metric that assesses the visual impact of three characteristics of an image: luminance, contrast and structure.

## Tips

• If A and ref specify RGB image data, use the "DataFormat" name-value argument to label the channel dimension, "C". You can then apply the mean function along the channel dimension of ssimval and ssimmap to approximate the SSIM index for the overall image.

## Algorithms

The SSIM Index quality assessment index is based on the computation of three terms, namely the luminance term, the contrast term and the structural term. The overall index is a multiplicative combination of the three terms.

$SSIM\left(x,y\right)={\left[l\left(x,y\right)\right]}^{\alpha }\cdot {\left[c\left(x,y\right)\right]}^{\beta }\cdot {\left[s\left(x,y\right)\right]}^{\gamma }$

where

$\begin{array}{l}l\left(x,y\right)=\frac{2{\mu }_{x}{\mu }_{y}+{C}_{1}}{{\mu }_{x}^{2}+{\mu }_{y}^{2}+{C}_{1}},\\ c\left(x,y\right)=\frac{2{\sigma }_{x}{\sigma }_{y}+{C}_{2}}{{\sigma }_{x}^{2}+{\sigma }_{y}^{2}+{C}_{2}},\\ s\left(x,y\right)=\frac{{\sigma }_{xy}+{C}_{3}}{{\sigma }_{x}{\sigma }_{y}+{C}_{3}}\end{array}$

where μx, μy, σxy, and σxy are the local means, standard deviations, and cross-covariance for images x, y. If α = β = γ = 1 (the default for Exponents), and C3 = C2/2 (default selection of C3) the index simplifies to:

$SSIM\left(x,y\right)=\frac{\left(2{\mu }_{x}{\mu }_{y}+{C}_{1}\right)\left(2{\sigma }_{xy}+{C}_{2}\right)}{\left({\mu }_{x}^{2}+{\mu }_{y}^{2}+{C}_{1}\right)\left({\sigma }_{x}^{2}+{\sigma }_{y}^{2}+{C}_{2}\right)}$

When you specify a noninteger value for "Exponents", the ssim function prevents complex valued outputs by clamping the intermediate luminance, contrast, and structural terms to the range [0, inf].

## References

[1] Zhou, W., A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. "Image Quality Assessment: From Error Visibility to Structural Similarity." IEEE Transactions on Image Processing. Vol. 13, Issue 4, April 2004, pp. 600–612.

## Version History

Introduced in R2014a

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