Why do I find different SSIM values depending on whether the images

 Why do I find different SSIM values depending on whether the images' intensity information is stored in the range [0 1] or [0 255]?

 

I import an image using the im2gray() function, which stores it as an array with a range of values between 0 and 255. I convert it into a double using double(), which does not alter the values of the elements but just changes the file format. I then alter the values and compare the altered image to the original in two ways. Firstly, I use mat2gray() to convert the intensity information in both images to the range 0-1 instead of 0-255. Secondly, I compare the 0-255 matrices. The SSIM values I get are quite different (the first value is higher than the second).

 

Why is this? Does the mat2gray() function lose image information? Which is the "correct" value of SSIM between the original and distorted images?

 

NB you can use any image for this. The one I used it attached to this answer.

 

% Import an image using im2gray
im = im2gray(imread('peppers.png'));

% Convert this a double (without rescaling the values from the range 0-255)
% and then modify the values slightly to create a distorted image.

imdouble = double(im);
imdouble_new = imdouble * 0.5;
im_new = mat2gray(imdouble_new, [0 255]);

% NB im_new is a grayscale image with values 0-1 rather than 0-255. If we
% want to compare this image to the original using the SSIM metric we can
% do so in 2 ways.

% 1) Convert im into a 0-1 image matrix and use the ssim function.

ssim_values_1 = ssim(mat2gray(imdouble, [0 255]), im_new);
ssim_values_1(1)

% 2) Just compare the double matrices using the ssim function.

ssim_values_2 = ssim(imdouble, imdouble_new);
ssim_values_2(1)

Ans = 0.7508

% 2) Just compare the double matrices using the ssim function.
ssim_values_2 = ssim(imdouble, imdouble_new);
ssim_values_2(1)

ans = 0.6400

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You're passing images to ssim() which are improperly-scaled for their class. Many functions make assumptions about the expected levels of an image based on its class, and ssim() is one. It uses the class to implicitly select the dynamic range parameter. The unit-scale ([0 1]) images work fine, since that's what ssim() expects a floating point image to be. It's the uint8-scale ([0 255]) images that it doesn't expect.

 

You can either cast the uint8-scale images as uint8, or to avoid the rounding loss on a working image, you can explicitly set the dynamic range parameter:

 

% Import an image using im2gray
im = rgb2gray(imread('peppers.png'));

% Convert this a double (without rescaling the values from the range 0-255)
% and then modify the values slightly to create a distorted image.
imuint8 = double(im);
imuint8_half = imuint8 * 0.5;

imdouble = mat2gray(imuint8, [0 255]);
imdouble_half = mat2gray(imuint8_half, [0 255]);

% NB im_new is a grayscale image with values 0-1 rather than 0-255. If we
% want to compare this image to the original using the SSIM metric we can
% do so in 2 ways.

% 1) Convert im into a 0-1 image matrix and use the ssim function.
ssim_values_1 = ssim(imdouble, imdouble_half,'dynamicrange',1);
ssim_values_1(1)

% 2) Just compare the uint8-scale matrices using the ssim function.
ssim_values_2 = ssim(imuint8, imuint8_half,'dynamicrange',255);
ssim_values_2(1)

ans = 0.7508

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