Bit-depth quantization and reconstruction error in digital images
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ORIGINAL PAPER
Bit-depth quantization and reconstruction error in digital images Isidora Stankovi´c1,2 · Miloš Brajovi´c1 · Miloš Dakovi´c1 · Cornel Ioana2 · Ljubiša Stankovi´c1 Received: 15 January 2020 / Revised: 3 March 2020 / Accepted: 15 April 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract Digital images can be considered as sparse or approximately sparse in the two-dimensional discrete cosine transform domain. According to the compressive sensing theory, these images can be recovered from a reduced set of pixels. Such reconstructions are influenced by the noise and the non-reconstructed coefficients for approximately sparse images. In the literature, this kind of reconstruction error is described by the error bound relations. In some hardware implementations, the pixels may be sensed with a low number of bits, causing quantization error. In this paper, we derive exact formula for the expected error energy in the reconstructed images, caused by the pixels quantization. It is validated trough numerical examples. Keywords Compressed sensing · Image processing · Bit depth · Quantization · Reconstruction
1 Introduction Signals are sparse in one of their representation domains if they are characterized with a few nonzero coefficients in the respective transformation domain [1–10]. The sparse signals can be reconstructed using a small set of linear combinations of the sparsity domain coefficients, which are referred to as the measurement [1–24]. The theory under which these reconstructions are considered is known as compressive sensing (CS). In image processing, the standard sparsity domain is the 2D-DCT, since most of the images can be defined, with a high accuracy, using a small number of the 2D-DCT coefficients within the analyzed image blocks [7,20–24]. The reduced number of pixels can appear as part of sampling strategy, aiming to reduce the signal acquisition time, equipment
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Miloš Brajovi´c [email protected] Isidora Stankovi´c [email protected] ; [email protected] Miloš Dakovi´c [email protected] Cornel Ioana [email protected] Ljubiša Stankovi´c [email protected]
1
Faculty of Electrical Engineering, University of Montenegro, 81000 Podgorica, Montenegro
2
GIPSA Lab, INP Grenoble, 38400 Saint-Martin-d’Hères, France
load, or memory storage, [1–14]. In many applications, physical constraints or strong disturbances (noise) can corrupt some pixels, resulting in a reduced set of available measurements. Regardless of the cause of the pixels unavailability, the previous cases can be considered within the same theoretical framework. Ideally, the pixels should be taken accurately, assuming a large number of bits. It is common to assume that a digital format with B = 8 is sufficient for accurate image analysis and reconstruction. Note that the pixels quantization is an inseparable part of the common image compression algorithms, such as the JPEG. This number of bits could be demanding for some hardware implementations, when the pixels could be sensed using a
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