# Threshold Schemes with Minimum Pixel Expansion

### secret image vss white

**Definition:** In visual secrete sharing schemes, attempts have been made to restrict the pixel expansion while images with high contrast are produced.

Although results obtained via a *(k,n)* visual secret sharing (VSS) scheme can be easily revealed by a human observer without any computations, the size of shares is usually much larger compared to the size of the original input image. By encrypting the secret pixel into *m 1 m 2* blocks of share pixels, the expansion is determined by the so-called expansion factor *m 1 m 2* which denotes the number of columns of the *n* × *m 1 m 2* basis matrices. In the recent past, attempts have been made to restrict the pixel expansion while images with high contrast are produced. The designer most often has to trade pixel expansion constraints to the recovered image quality and/or the order of *(k,n)* threshold schemes.

A probabilistic VSS scheme which offers no pixel expansion ( *m 1 m 2* =1) which can be regarded as a VSS scheme with the minimum pixel expansion is briefly reviewed in the sequence. The reconstruction of the secret image is probabilistic. In a deterministic scheme, any given secret pixel is encrypted into *m 1 m 2* subpixels. Hence the reconstructed image is *m 1 m 2* times larger in size compared to the original binary input. To obtain no pixel expansion, each secret pixel is reconstructed with a single pixel in the probabilistic scheme. The binary secret pixel is correctly reconstructed with a given probability. However the quality of the reconstructed binary image depends on how large the image areas of pixels showing the same binary value are.

A probabilistic VSS scheme can be constructed using the two sets, white set *P 1* and black set *P o* , each consisting of *n* x 1 matrices, respectively. When sharing a white (or black) pixel, the dealer first randomly chooses one *n* x 1 column matrix in *P 1* (or *P 0* ), and then randomly selects one row of this column matrix to a relative shadow. The chosen matrix defines the pixel color in every one of the *n* shadows. A probabilistic VSS scheme is considered valid if the contrast and security conditions are met.

An image secret sharing (ISS) scheme in can be viewed as an alternative to minimal pixel expansion based VSS schemes. The scheme is a (2,2) -ISS solution suitable for cost-effective encryption of the natural images. It encrypts the decomposed ‘black’ bit of the original secret pixel into a black (or white) bit in each of the two shares. To differentiate between the ‘black’ and ‘white’ bits of the original secret pixels, the decomposed ‘white’ bits are encrypted into black and white (or white and black) pixels in the corresponding shares. Such an approach satisfies the essential perfect reconstruction property and can serve as the private-key cryptosystem.

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