How to modify the last 3 bits of signed numbers

Question

When I apply the function dwt2() on an image, I get the four subband coefficients. By choosing any of the four subbands, I work with a 2D matrix of signed numbers.

In each value of this matrix I want to embed 3 bits of information, i.e., the numbers 0 to 7 in decimal, in the last 3 least significant bits. However, I don't know how to do that when I deal with negative numbers. How can I modify the coefficients?

Welcome to Stack Overflow. While we appreciate English may not be your first language, your question is unclear at the moment. Could you please reword it to address this issue? You use a lot of question marks to end your sentences, which is quite confusing, but my understanding of your question is this: "I have a coefficient value, e.g., the number 37, and I want to embed the bit values 010 in it. How can I do it?" Am I correct to assume this is what you're really asking? — Reti43, Nov 14 '16 at 16:44
sorry if i dont write correctly.yes it is Almost right. i want to embed bit values in lsb of one of subband coefficient. but they are negative or decimal. i do not know how to do it. i want to do exatly this:Select one sub-band for embedding the secret message. If we donate ‘f ’ as coefficients matrix ofthe selected sub-band, then using the following equation, the 3 least significant bits of wavelet coefficients is replaced by the 3 bits of bit stream . — mersana, Nov 14 '16 at 16:59
Your last sentence, "If we donate 'f'...", is unclear. If you copied this instruction from somewhere else, it'd be really helpful if you could share the source to help us understand the context. But, if your question is that you want to embed bits in negative or decimal values, compared to positive integers from my hypothetical question, that can be answered straightforwardly. Is that what you want? — Reti43, Nov 14 '16 at 17:03
yes sorry i copy it imcomplete..generally i want to replace the 3 least significant bits of wavelet coefficients by the 3 bits of bit stream. — mersana, Nov 14 '16 at 17:12
I have edited your question to make it more clear. If I have misrepresented your meaning, feel free to add your own clarifications or rollback my change. — Reti43, Nov 14 '16 at 18:57

Reti43 · Accepted Answer · 2016-11-14T20:04:16.630

First of all, you want to use an Integer Wavelet Transform, so you only have to deal with integers. This will allow you a lossless transformation between the two spaces without having to round float numbers.

Embedding bits in integers is a straightforward problem for binary operations. Generally, you want to use the pattern

(number AND mask) OR bits

The bitwise AND operation clears out the desired bits of number, which are specified by mask. For example, if number is an 8-bit number and we want to zero out the last 3 bits, we'll use the mask 11111000. After the desired bits of our number have been cleared, we can substitute them for the bits we want to embed using the bitwise OR operation.

Next, you need to know how signed numbers are represented in binary. Make sure you read the two's complement section. We can see that if we want to clear out the last 3 bits, we want to use the mask ...11111000, which is always -8. This is regardless of whether we're using 8, 16, 32 or 64 bits to represent our signed numbers. Generally, if you want to clear the last k bits of a signed number, your mask must be -2^k.

Let's put everything together with a simple example. First, we generate some numbers for our coefficient subband and embedding bitstream. Since the coefficient values can take any value in [-510, 510], we'll use 'int16' for the operations. The bitstream is an array of numbers in the range [0, 7], since that's the range of [000, 111] in decimal.

>> rng(4)
>> coeffs = randi(1021, [4 4]) - 511

coeffs =

   477   202  -252   371
    48  -290   -67   494
   483   486   285  -343
   219  -504  -309    99

>> bitstream = randi(8, [1 10]) - 1

bitstream =

     0     3     0     7     3     7     6     6     1     0

We embed our bitstream by overwriting the necessary coefficients.

>> coeffs(1:numel(bitstream)) = bitor(bitand(coeffs(1:numel(bitstream)), -8, 'int16'), bitstream, 'int16')

coeffs =

   472   203  -255   371
    51  -289   -72   494
   480   486   285  -343
   223  -498  -309    99

We can then extract our bitstream by using the simple mask ...00000111 = 7.

>> bitand(coeffs(1:numel(bitstream)), 7, 'int16')

ans =

     0     3     0     7     3     7     6     6     1     0

How to modify the last 3 bits of signed numbers

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