Halftoning is an Ill- . . . Inverse Halftoning: POCS Wavelet Inverse . . . Fast Inverse Halftoning What We Are . . . On Inverse Halftoning: A General Problem General Problem: . . . Our First Result: . . . Computational Complexity Reduction Proof and Interval Computations Interval Computations Towards New Interval- . . . New Algorithm: . . . S. D. Cabrera and K. Iyer New Algorithm: Cont-d Department of Electrical and Computer Engineering New Algorithm: Details University of Texas at El Paso, El Paso, Texas 79968 A POCS Iterative . . . Results and Future Work e-mail: cabrera@ece.utep.edu, kish 199@yahoo.com Acknowledgments G. Xiang and V. Kreinovich Title Page Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso, El Paso, Texas 79968 e-mail: gxiang@utep.edu, vladik@cs.utep.edu ◭ ◮ March 17, 2005 Page 1 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 1. Need for Halftoning Wavelet Inverse . . . Fast Inverse Halftoning • Inside the computer, a gray-scale image is represented by assigning, to every What We Are . . . pixel ( n 1 , n 2 ), the intensity f ( n 1 , n 2 ) of the color at this pixel. A General Problem • Usually, 8 bits are used to store the intensity, so we have 2 8 = 256 possible General Problem: . . . intensity levels for each pixel. Our First Result: . . . Reduction • For color images, we must represent the intensity of each color component. Proof • A laser printer either prints a black (or a colored) dot, or it does not print Interval Computations anything at all. Towards New Interval- . . . New Algorithm: . . . • Therefore, when we print an image, we must first transform it into the New Algorithm: Cont-d “halftone” form b ( n 1 , n 2 ) ∈ { 0 , 1 } . New Algorithm: Details • Crudely speaking, the level of intensity at a pixel is represented by the relative A POCS Iterative . . . frequency of black spots around it. Results and Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 2. Halftoning Techniques: In Brief Wavelet Inverse . . . Fast Inverse Halftoning • There exist many halftoning algorithms. What We Are . . . • Most widely used – error diffusion : A General Problem General Problem: . . . – we start with the original image u ( n 1 , n 2 ) := f ( n 1 , n 2 ) Our First Result: . . . – we sequentially update the processed image u ( n 1 , n 2 ) and quantize the Reduction processed value u ( n 1 , n 2 ) into b ( n 1 , n 2 ) = Q ( u ( n 1 , n 2 )), where: Proof Q ( u ) = 0 for u < 0 . 5 and Q ( u ) = 1 for u ≥ 0 . 5. Interval Computations Towards New Interval- . . . def – Once the pixel is quantized, the quantization error e ( n 1 , n 2 ) = b ( n 1 , n 2 ) − New Algorithm: . . . u ( n 1 , n 2 ) is spread out (“diffused”) to the neighboring pixels: New Algorithm: Cont-d � New Algorithm: Details u ( n 1 , n 2 ) = f ( n 1 , n 2 ) − h ( m 1 , m 2 ) · e ( n 1 − m 1 , n 2 − m 2 ) . A POCS Iterative . . . m 1 ,m 2 Results and Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 3. Need for Reverse Halftoning Wavelet Inverse . . . Fast Inverse Halftoning • Visually, the printed halftone image b ( n 1 , n 2 ) looks identical to the original What We Are . . . gray-scale image f ( n 1 , n 2 ). A General Problem • So, from the halftone values b ( n 1 , n 2 ), it is possible to reconstruct the original General Problem: . . . image. Our First Result: . . . Reduction • Why we need it: we know how to rotate or zoom the original image but not Proof the halftone image. Interval Computations • So, to go from a printed image to a printed zoomed and/or rotated image, Towards New Interval- . . . we can: New Algorithm: . . . New Algorithm: Cont-d – use b ( n 1 , n 2 ) to reconstruct f ( n 1 , n 2 ); New Algorithm: Details – then, we apply the zoom and/or rotation to f ( n 1 , n 2 ), resulting in A POCS Iterative . . . f ∗ ( n 1 , n 2 ); Results and Future Work – finally, halftone f ∗ ( n 1 , n 2 ), and print the resulting halftone image b ∗ ( n 1 , n 2 ). Acknowledgments • For that, we must reverse the halftoning procedure. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 4. Halftoning is an Ill-Posed Problem: A Reminder Wavelet Inverse . . . Fast Inverse Halftoning • Our objective is to reverse the halftoning operation. What We Are . . . • By definition, halftoning transforms: A General Problem General Problem: . . . – the original gray-scale image in which we stored at least 8 bits per pixel, Our First Result: . . . – into a black-and-white image in which we store only one bit per pixel. Reduction Proof • Thus, halftoning loses information. Interval Computations • Therefore, halftoning is a lossy compression. Towards New Interval- . . . New Algorithm: . . . • Hence, there may be several different images that lead to the same halftoned New Algorithm: Cont-d image. New Algorithm: Details A POCS Iterative . . . Results and Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 5. Inverse Halftoning: POCS Wavelet Inverse . . . Fast Inverse Halftoning • Main idea: each value b ( n 1 , n 2 ) of a halftone image represents a (convex) What We Are . . . constraint on the original image f ( n 1 , n 2 ). A General Problem • In geometric terms: General Problem: . . . Our First Result: . . . – we have a point b ( n 1 , n 2 ) in the function space, Reduction – we want to find the closest element to this point in the convex set S . Proof Interval Computations • It is known that to get this closest element, we can: Towards New Interval- . . . – first minimally modify the original halftone image so that it satisfies the New Algorithm: . . . first constraint, New Algorithm: Cont-d – then minimally modify the modification so that it satisfies the second New Algorithm: Details constraint, etc. A POCS Iterative . . . Results and Future Work • Result: we get a good quality inverse halftoning. Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 6. Wavelet Inverse Halftoning Wavelet Inverse . . . Fast Inverse Halftoning • Idea: use wavelet transform. What We Are . . . • Motivation: A General Problem General Problem: . . . – halftoning is an example of lossy compression; Our First Result: . . . – the experience of JPEG2000 has shown that wavelets best captures the Reduction visual quality of images uncompressed after a lossy compression. Proof Interval Computations • Results: wavelet-based inverse halftoning techniques lead to visually the best Towards New Interval- . . . reconstruction among all known inverse halftoning methods. New Algorithm: . . . • Comment: this empirical result is in good accordance with the JPEG2000 New Algorithm: Cont-d experience, New Algorithm: Details A POCS Iterative . . . • Problem: wavelet methods require a lot of computation time. Results and Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 7. Fast Inverse Halftoning Wavelet Inverse . . . Fast Inverse Halftoning • Idea: the value f ( n 1 , n 2 ) can be reconstructed from the density of black pixels What We Are . . . around ( n 1 , n 2 ). A General Problem • In engineering terms: f ( n 1 , n 2 ) can be obtained from b ( n 1 , n 2 ) by low-pass General Problem: . . . filtering. Our First Result: . . . Reduction • Problem: a low-pass filter blurs the edges. Proof • Solution: Interval Computations Towards New Interval- . . . – detect the edges, and New Algorithm: . . . – apply different filters (with different spatial radius) at different parts of New Algorithm: Cont-d the image. New Algorithm: Details A POCS Iterative . . . • This idea has been successfully implemented in inverse halftoning, by Kite et al. Results and Future Work Acknowledgments • Result: the new method is much faster than the wavelet-based, while the Title Page visual quality is almost as good as for the wavelet-based reconstruction. ◭◭ ◮◮ ◭ ◮ Page 8 of 22 Go Back Full Screen
Halftoning is an Ill- . . . Inverse Halftoning: POCS 8. What We Are Planning to Do Wavelet Inverse . . . Fast Inverse Halftoning • Remaining problem: the existing inverse haltoning methods are still not op- What We Are . . . timal, A General Problem • especially low-computations methods implementable within printing devices. General Problem: . . . Our First Result: . . . • In this talk: Reduction – we show that the problem of inverse half-toning is a particular case of a Proof class of difficult-to-solve problems: Interval Computations Towards New Interval- . . . – inverse problems for reconstructing piece-wise smooth images. New Algorithm: . . . – We show that this general problem is NP-hard. New Algorithm: Cont-d – We also propose a new idea for solving problems of this type, including New Algorithm: Details the inverse halftoning problem. A POCS Iterative . . . Results and Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 22 Go Back Full Screen
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