Alpha Stable Human Visual System Models for Digital Halftoning A J Gonz´aleza , J Baccaa , G R Arcea and D L Laub a Department of Electrical and Computer Engineering University of Delaware, Newark, DE 19716 b Department of Electrical and Computer Engineering University of Kentucky, Lexington, KY 40506 ABSTRACT Human visual system (HVS) modeling has become a critical component in the design of digital halftoning algorithms Methods that exploit the characteristics of the HVS include the direct binary search (DBS) and optimized tone-dependent halftoning approaches The spatial sensitivity of the HVS is lowpass in nature, reflecting the physiological characteristics of the eye Several HVS models have been proposed in the literature, among them, as¨anen’s the broadly used N¨ as¨anen’s exponential model As shown experimentally by Kim and Allebach,1 N¨ model is constrained in shape and richer models are needed in order to attain better halftone attributes and to control the appearance of undesired patterns As an alternative, they proposed a class of HVS models based on mixtures of bivariate Gaussian density functions The mathematical characteristics of the HVS model thus play a key role in the synthesis of model-based halftoning In this work, alpha stable functions, an elegant class of models richer than mixed Gaussians, are exploited These are more efficient than Gaussian mixtures as they use less parameters to characterize the tails and bandwidth of the model It is shown that a decrease in the model’s bandwidth leads to homogeneous halftone patterns and conversely, models with heavier tails yield smoother textures These characteristics, added to their simplicity, make alpha stable models a powerful tool for HVS characterization Keywords: Digital Halftoning, Direct Binary Search, HVS models, Blue noise theory INTRODUCTION Digital halftoning focuses on the quantization of continuous tone images that minimizes the visibility of artifacts In order to apply this concept in practice there is the need for specifying a computational model of the human visual system (HVS), and to calculate visual error, which can be used to automatically rank halftone images in increasing order of quality Relatively simple models for the HVS have proved to be quite successful when applied to algorithms that search for the best possible halftones Digital halftoning techniques including screening algorithms, error diffusion algorithms and iterative halftone methods all use either implicitly or explicitly a model for the human visual system In fact, even those methods which cannot be classified as model based, because they not include an explicit HVS block within their block diagram (like for example Bayer’s screen2 and Void and Cluster3 ), nevertheless agree with a model that treats the HVS as a low-pass filter Kim and Allebach not only discovered that a HVS model has been crucial for the design of almost every halftoning technique, but also showed that the shape of any HVS model can be tuned to yield better texture quality in the obtained dither patterns Thus the performance of a halftoning algorithm can be maximized by properly designing improved HVS models HVS models have been proposed based on the experimental response of the eye to stimuli in both, frequency and spatial domains When the frequency domain is chosen, the model is called the “Modulation Transfer Function (MTF)” or the “Contrast Sensitivity Function (CSF)” of the human eye Associated experiments consist of a square wave grating of dark and light bars where the modulation or contrast of the grating can be measured as the ratio between the difference of the maximum and minimum amount of light reflected by the grating to the sum of the two The width of each bar defines the spatial frequency of the grating, which is varied from lower to higher frequencies in each experiment For equally spaced particular frequencies, a subjective measure of the contrast appreciated by the human viewer is taken, and the ratio between the real contrast of the image and the contrast resolved by the viewer is used as the contrast sensitivity of the eye for that exact frequency The sensitivity of the eye is high (near unity) for low frequencies, but as the frequency is increased, the eye starts to fail in detecting the real contrast of the grating, and the sensitivity decays to zero On the other hand, when a model for the HVS is proposed based on the eye’s response to stimuli in the spatial domain, the function obtained is called the “Point Spread Function (PSF)” The PSF and the CSF form a Fourier pair Whesteimer4 states that under no circumstances are point objects ever actually imaged as points; several physical and geometrical optical factors prevent this As a result, a point object gives rise to a retinal light distribution that is bell-shaped in cross-section This distribution is precisely the PSF of the eye and has significance not only when the object is a point but whenever it is necessary to know the light distribution for a target more complex than a point source of light, since any visual object can be thought of as made of points In applying a HVS model to halftoning, Kim and Allebach1 studied the patterns obtained with the Direct Binary Search (DBS) algorithm5 using four different models generated in the frequency domain, namely the models proposed by Campbell,6 Mannos,7 N¨as¨anen8 and Daly.9 Campbell and Mannos’ models are band-pass, with peak sensitivity around cycles/degree Mullen10 demonstrated that their measurements had not extended to very low spatial frequencies correctly since very few bars had been displayed to the viewers at the lowest frequencies, and a number of bars below four or five is known to reduce sensitivity to this kind of gratings It is expected that these models are inadequate in representing the HVS Until recently, N¨ as¨anen’s model was the most extensively used model in the context of digital halftoning This model is an exponential function of the radial frequency ρ,8 as shown in Table 1, where L is the average luminance and c, d are constants that make the model fit the experimental data Although N¨ as¨anen’s model has proven to be an adequate approximation to the HVS model, Kim and Allebach1 proposed a new and richer class of HVS models that offer better halftoning results than that produced by the N¨ as¨anen’s model These models are based on mixed Gaussian functions whose functional form is shown in Table The advantage of these models is that their frequency response in terms of bandwidth and tail weight can be optimized by varying a set of parameters Such flexibility is not available in N¨ as¨anen’s exponential model The diversity attained with mixed Gaussians however, comes at a penalty: overparametrization A total of four parameters characterize the mixed Gaussian model, making the tuning process a rather delicate task This paper focuses on developing HVS models having flexibility in the design of the tails and bandwidth of the model without overparametrization The proposed approach is based on the functions describing alpha stable random variables.11 These models are richer than mixed Gaussian models (Gaussian random variables are a sub-family of alpha stable random variables) and they are also simple and succinct with few parameters needed for their characterization Notably, empirical approximations to the PSF (obtained by measuring the response of the eye to spatial stimuli) were found to have the analytical form of the characteristic function of alpha stable random variables.4 Thus, stable models fit well empirical measurements and at the same time, they offer unique mathematical characteristics that ultimately render visually pleasant halftones PRELIMINARIES 2.1 HVS model and the Scale Parameter S A HVS model is a linear shift invariant filter based on the PSF or the CSF of the human eye It is denoted as h(x, y) (x, y in inches) in the spatial domain and as H(u, v) (u, v in cycles/degree) in the frequency domain ¯ x, y¯) with x The inverse Fourier transform of H(u, v) yields h(¯ ¯, y¯ in degrees To convert these angular units to the units on the printed page, notice that a length x inches when viewed at a distance D inches will subtend an angle of x ¯ degrees satisfying x ¯ = (180/π)(arctan(x/D)) ≈ (180x)/(πD) for x D Therefore, assuming a printer with resolution R (in dpi), the discrete filter characterizing the HVS model in the spatial domain will be given by h [m, n] = 1802 ¯ h π D2 180m 180n , πRD πRD (1) The term S ≡ RD in (1) is called the Scale Parameter Kim and Allebach1 experimented with different values of S in HVS models that are used in DBS and demonstrated why this parameter, which in theory should be determined precisely by the intended viewing distance and printer resolution, in reality serves more as a free parameter that can be adjusted to yield halftone textures of the desired quality When the distance (or the resolution) is increased, the viewer’s eye will reduce its sensitivity at a fixed amount of cycles per degree, therefore the bandwidth of the HVS filter in the frequency domain is decreased by an amount proportional to the distance (or resolution) increase In this scenario, if the viewer observes a printed page from a greater distance (or if the resolution is larger), it is expected that the eye will perceive a better overall impression of the image (at the expense of loosing the ability to discern the details of the printed page), i.e the image will appear in the viewer’s mind more homogeneous than it was from the original distance The effect is the contrary if the distance (or the resolution) is decreased: the bandwidth of the HVS grows, and the textures lose consistency 2.2 Direct Binary Search (DBS) The goal of halftoning algorithms is to produce a distribution of printed dots that, when viewed by a human subject, creates an apparent image indiscernable from the continuous tone original As a means of generating visually optimal dot distributions, Allebach and Analoui5 introduced DBS as an iterative halftoning method which minimizes the error between the perceived continuous tone image and the perceived halftone image by changing the state of the halftone pixels In DBS, an initial halftone image g0 [m, n] is provided The algorithm evaluates the difference between the original continuous tone image f [m, n] and the initial halftone g0 [m, n] to produce the error image e[m, n], which is filtered by the HVS model The filtered error image e˜[m, n] is used to calculate the metrics (cost) of the algorithm, given by C = m n | e˜[m, n] |2 Once the cost C has been calculated, the algorithm starts to evaluate changes in the initial halftone g0 [m, n] that could lead to a decrease in C Specifically, for each pixel of g0 [m, n] the algorithm determines if a toggle (change the status of the current pixel) or a swap (swap the values of the current pixel and one of its nearest neighbor pixels that has a different value) causes an improvement in the error The toggle or swap that results in the greatest decrease in the squared error is accepted When the first iteration is complete (every pixel in the image has been visited), the process is repeated over the newly obtained halftone When no changes are accepted during an iteration, the algorithm has converged to a local minimum of the error metrics Notice the strong dependance of this algorithm on the HVS model The HVS filter determines what should be understood as a good image and what not When a toggle or a swap is being considered, the cost function of the algorithm is determined by the HVS model It is not surprising that different models for the HVS produce considerably different DBS halftone results 2.3 Blue noise model Blue noise is the statistical model describing the ideal spatial and spectral characteristics of dispersed-dot dither patterns,12 and in essence, the ideal blue noise halftoning scheme produces stochastic dither patterns of dots distributed as homogeneously as possible Specifically, the blue noise model states that the ideal spatial distribution of minority pixels representing a constant gray level g ∈ [0, 1] is one that is aperiodic, isotropic (radially symmetric) and contains only high frequency spectral energy.13 Given the low-pass nature of the HVS, blue noise characterizes Figure The RAPSD measure for an ideal blue patterns that are visually appealing simply because the spectral noise dither pattern components of the pattern lie in the regions least visible to the human viewer; furthermore, the stochastic distribution of dots creates a grid-defiance illusion where the structure of the underlying grid on which the pixels are aligned is no longer apparent to the viewer.14 The blue noise principal frequency fb is defined as15 : ⎧ ⎨ √ g 1/2 fb = ⎩ √ 1−g , for < g ≤ 1/4 , for 1/4 < g ≤ 3/4 , for 3/4 < g ≤ (2) 10 0.12 0.1 Mixed Gaussian −1 10 Mixed Gaussian H(ρ) H(ρ) 0.08 0.06 α-SG, α = 0.95 α-SG, α = 1.05 N¨as¨anen 0.04 −2 10 α-SG, α = 0.95 Mixed Gaussian 0.02 N¨as¨anen Mixed Gaussian 1/3 α-SG, α = 1.05 1/2 ρ √ 1/ −3 10 ρ 1/2 √ 1/ Figure Frequency response of N¨ as¨ anen, mixed Gaussian and α-SG filters (left) and their logarithmic view (right) Figure depicts the radially averaged power spectral density (RAPSD) of an ideal blue noise pattern as defined by Ulichney The RAPSD is calculated by taking the average value of the power spectrum of the patterns within an annular ring and plotting this average versus the radial frequency.13 The blue noise model places an increased emphasis on the need for maintaining radial symmetry and avoiding periodic textures by allowing certain amount of clustering for gray levels between 1/4 and 1/2 so that the frequency response of the patches of gray within this range remains constant In this way, the spectral energy of the halftones is not packed into the corners of the baseband, preventing the occurrence of checkerboard patterns In this paper, the metrics for the design of visually pleasing halftones will be based on a subjective qualification of pleasantness and also on the blue noise model, i.e the RAPSD of the obtained patterns will be compared to the ideal RAPSD in Figure HUMAN VISUAL SYSTEM MODELS Figure shows the tails of the frequency response of the five models to be studied in this paper These are radially symmetric and thus only two dimensional plots of their CSF are presented √ A cross-section of the 2-D CSF was taken along the diagonal so that the radial frequency goes from to 1/ The five filters were generated using a scale factor S = RD = 300dpi × 9.5in 3.1 N¨ as¨ anen’s (exponential) model N¨ as¨anen’s model in Table is a radially symmetric exponential function8 which has been used extensively in the context of digital halftoning The logarithmic view of this function shows the tails decaying linearly down to √ approximately 10−3 at the maximum radial frequency of ρ = 1/ This is a filter with a bandwidth∗ of 0.078 3.2 Mixed Gaussian models Kim and Allebach,1 in an attempt to reduce the computational cost of DBS, proposed the use of two-component Gaussian functions as models for the HVS Since the Gaussian filters are radially symmetric, separable, unimodal, smooth and have a simple closed form Fourier Transform, they are well suited for models of the HVS Their initial approach was to find a Gaussian model with parameters optimized to minimize the difference between this model and N¨ as¨anen’s model, but it was found that a single Gaussian function could not yield a sufficiently accurate approximation, so they chose a two-component alternative The functional form of these two-component Gaussian filters is shown in Table ∗ The bandwidth is defined as the frequency at which the filter has decayed to 50% of its maximum amplitude Table Contrast Sensitivity Function and Point Spread Function of HVS models Contrast Sensitivity Function Point Spread Function Constants k = c·log1L+d k N¨ as¨ anen H (ρ) = exp (−kρ) h (r) = (2π)2 k 3/2 L = 11cd/m2 ( 2π +r ) c = 0.525, d = 3.91 (κ1 , κ2 , σ1 , σ2 )I = ” “ ´ ` 2 2 r2 Mixed Gaussian H (ρ) = 2πκ1 σ1 exp −2π σ1 ρ (h ∗ h) (r) = κ1 exp − 2σ2 (43.2, 38.7, 0.02, 0.06) ” “ ´ ` 2 2 r2 +2πκ2 σ2 exp −2π σ2 ρ +κ2 exp − 2σ2 (κ1 , κ2 , σ1 , σ2 )II = Model (∗ means convolution) α-stable No closed form h (r) = K exp (γ rα ) (19.1, 42.7, 0.03, 0.06) I (α, γ ) = (0.95, 27) II (α, γ ) = (1.05, 27) K is a normalization constant Kim and Allebach could not find one single filter which works best for every level of gray, so they used two different filters The constants that characterize the first (I) and the second (II) mixed Gaussian models are shown in Table The cross section of these filters along the diagonal is presented in Figure It can be seen that the second model has wider bandwidth than the first, and both present exponential order tails, which means that, as the radial frequency is increased, the CSF of the mixed Gaussian filters decays faster than the CSF of the exponential model 3.3 Alpha Stable models HVS models used in DBS are commonly derived from empirical approximations to the CSF That is the case in the N¨ as¨anen and the mixed Gaussian models described before However, modeling of the PSF from experimental observations is a full fledged discipline within the area of the physiological optics The experimental determination of the eye’s PSF is carried out with the use of instruments designed to measure the entire refractive error of the eye that causes the effect of blurred vision These instruments are called “wavefront sensors” or “aberrometers” A wavefront sensor measures the shape of the wavefronts of light (surfaces of constant phase) that exit the eye’s pupil If the eye were a perfect optical system, these wavefronts would be perfectly flat, and the eye’s PSF would be infinitely narrow Since the eye is not perfect, the wavefronts are not flat and have irregular curved shapes, and its PSF has a bandwidth greater than zero Interestingly enough, the work of Whesteimer4 models the shape of the PSF with h(r) = 0.952 exp −2.59 |r| 1.36 + 0.048 exp −2.43 |r| 1.74 (3) which is close in form to the characteristic function of the α-stable distributions If the PSF of the HVS model has the form of (3), its CSF will have algebraic tails,11 hence it might be a good idea to use models whose tails are heavier than that obtained with exponential and mixed Gaussian models This fact provides the physiological foundation to the use of α-stable models to characterize the HVS Stable distributions describe a rich class of processes that allow heavy tails and skewness in their functions.11 The class was characterized by L´evy in 1925.16 Stable distributions are described by four parameters: an index of stability α ∈ (0, 2], a dispersion parameter γ > 0, a skewness parameter δ ∈ [−1, 1], and a location parameter β ∈ R The stability parameter α measures the thickness of the tails of the distribution and provides this model with the flexibility needed to characterize a wide range of impulsive processes The dispersion γ is similar to the variance of the Gaussian distribution When the skewness parameter is set to δ = 0, the stable distribution is symmetric about the location parameter β Symmetric stable processes are also referred to as symmetric α-stable or simply SαS In HVS modelinig, bivariate SαS distributions will be used for the characterization In particular, the focus will be on a subclass of them, the so-called α-sub-Gaussian (α-SG(R)) random vectors,17 whose characteristic function is of the form φ(t) = exp − T t Rt α/2 where t = [x, y]T is a two dimensional column vector and the matrix R is positive definite This characteristic function will be used to model the PSF of the HVS In order to guarantee radial symmetry for the filters, the matrix R will be of the form γ 0 γ (5) where γ > is the dispersion parameter for the model With this form of the matrix R and expanding the index of the exponential, the PSF in (4) becomes φ(x, y) = exp − γ α/2 x2 + y 2 α/2 (6) r≥0 −1 10 α = 0.95 α = 1.05 α = 1.15 −2 10 α = 1.25 It is possible to simplify (6) one step further by transforming the rectangular coordinates into polar coordinates, resulting in φ(r) = exp − γ α/2 rα , 10 H(ρ) R= (4) (7) √ −1/ −1/2 ρ 1/2 √ 1/ Figure Logarithmic view of the frequency response of α-SG filters with r2 = x2 + y and where α ∈ (0, 2] is the index of stability that determines the heaviness of the model’s tails For simplicity in the notation the term 12 γ α/2 in (7) is replaced by one only term that is called γ Figure shows the CSF of α-SG HVS models with different values of α The CSF of these models is obtained by sampling the PSF in (7), truncating it and finding its inverse Discrete Fourier Transform (IDFT), a procedure that carries no computational burden Notice that this approximation based on the IDFT could not be applied if an α-stable probability density function were being sought ANALYSIS OF DBS HALFTONE PATTERNS WITH DIFFERENT HVS MODELS 4.1 N¨ as¨ anen’s (exponential) model The first row in Figure shows the DBS halftone results obtained using the N¨ as¨anen’s model over the image “Barbara” and two different patches of gray: g = 22% and g = 50% A frequency analysis of these signals sheds light on the capacity of these filters to generate halftones that are visually pleasant The first spectral measure evaluated is the two dimensional power spectral density (PSD) for each of these patterns, depicted in the right center column of Figure The superimposed black circles mark the location of the principal frequency Dark regions of the spectra are maxima, and light regions are minima These spectra suggest that the halftones generated with the N¨ as¨anen’s model satisfy the radial symmetry imposed by the blue noise model If this holds, one can analyze the RAPSD which gives a more quantitative notion of the frequency response of the halftones As mentioned before, the RAPSD is simply a radial average of the PSD It is important to note that if the power spectral densities are not radially symmetric, the RAPSD will not be accurate The right column of Figure displays the corresponding RAPSD of the dither patterns obtained with the N¨ as¨anen’s model The vertical axis in each plot is normalized to σg2 = g(1 − g), the variance of a single pixel, as defined by Ulichney.12 The principal frequency according to the blue noise model is indicated with a vertical line This model’s RAPSD fits well the blue noise model for gray level g = 22%, with little or no low frequency spectral components; a flat, high frequency spectral region of amplitude σg2 and a spectral peak at cutoff frequency fb , the blue noise principal frequency At gray level g = 50%, the high frequencies exhibit a normalized amplitude bigger than unity (an excess of high frequency energy) For this level of gray, the highest value of the RAPSD is clearly above f = 1/2, which means that the energy has been packed into the corners of the baseband In the spatial 2.5 22% N a ¨ s a ¨ n e n M G a u s s i a n M G a u s s i a n 2 1.5 0.5 50% 0 ρ 1/2 √ 1/ ρ 1/2 √ 1/ ρ 1/2 √ 1/ ρ 1/2 √ 1/ ρ 1/2 √ 1/ 2.5 22% 1.5 0.5 50% 0 2.5 22% 1.5 0.5 50% 0 2.5 22% 1.5 α S G 1 0.5 50% 0 2.5 22% 1.5 α S G 0.5 50% 0 Figure Spectral analysis of DBS halftones A portion of the “Barbara” image (left), halftone patches for g = 22% and g = 50% using different HVS models (left center), PSD (right center) and RAPSD (right) of corresponding gray patches In the RAPSD, the dashed line corresponds to g = 22%, fb = 0.47, and the solid line corresponds to g = 50%, fb = 0.5; the vertical line indicates the position of the principal frequency domain, this packing of energy is achieved by adding correlation between minority pixels along the diagonal, creating an unpleasant checkerboard pattern that can be observed in the patch of the left center column for the corresponding gray level (g = 50%) These checkerboard patterns can also be seen in the gray ramp obtained with this model (top of Figure 5) in the zone of the middle gray tones 4.2 Mixed Gaussian models The halftones generated with the two mixed Gaussian models are shown in the second and third rows of Figure 4, and their gray ramps are also the second and the third of Figure They show that the first model produces a very smooth halftone result overall, but generates patchy textures with checkerboard artifacts in midtone areas The second model, in contrast, generates acceptable results in midtone areas and in areas of extreme tone, but around the quarter tones, especially at g = 75%, the arrangement of dots is granular, giving the texture an undesirable rough aspect The first model presents clipping at extreme tone levels (see how the minority pixels not migrate far away in the extremes of the second ramp in Figure 5) while, on the other hand, the second filter does a better job in this regards, producing a more homogeneous halftone gray ramp (third ramp of Figure 5) This phenomenon has to with the bandwidth of the filter The second mixed Gaussian filter, having a narrower bandwidth than the first, presents a more homogeneous response, as was explained in section The right center column of Figure verifies the condition of radial symmetry in the PSD of the gray patches obtained with these models The two filters exhibit a RAPSD that complies with the blue noise model for low levels of gray (not shown) The interesting (and revealing) phenomena start to happen at g = 22% The first mixed Gaussian model shows an acceptable response using this filter for gray level g = 22%, with cutoff frequency accurately located However, exactly the same phenomenon that was observed with the N¨as¨anen’s model occurs here for gray level g = 50%: the cutoff frequency of the pattern goes beyond f = 1/2 and the peak goes past the limits of the plot This high amount of energy is concentrated in the corners of the baseband, implying that checkerboard patterns must have appeared in the spatial pattern for g = 50% The right side of the second ramp of Figure shows that this is exactly the case: checkerboard patterns give this region of the ramp a disturbing appearance as a consequence of the discontinuities in texture that they cause The phenomenon observed with the second mixed Gaussian model is the opposite: for gray levels g = 22% and g = 50% the RAPSD remains unchanged —which is desirable— but the cutoff frequency falls below f = 1/2 This excess of low spectral energy inside the radial frequency f = 1/2 is introduced by a disproportionate clustering of minority pixels that gives the patterns the coarse look that Kim and Allebach had observed Although Lau et.al.18, 19 demonstrate why the existence of some spectral energy inside the principal frequency ring achieves radial symmetry, it is clear that the clustering of minority pixels will have desirable properties for halftoning only if it is not done to an extreme These phenomena are discernable also in the halftones of the image “Barbara” in Figure The two filters are identical within the region between DC to radial frequency f = 0.12, at which point the tails of the filters start to deviate from one another; the first filter being the one that exhibits heavier tails (see the logarithmic view of the cross-sections, Figure 2) It is not surprising that the two filters expose similar halftone results for gray levels below g = 25% Beyond gray level g = 25%, the energy in the second mixed Gaussian filter decreases (lower tails) causing the pattern to feel less force moving energy to higher radial frequencies and hence, preventing it from achieving a cutoff frequency higher than fb = 1/2 In the spatial domain this low cutoff frequency is achieved by allowing certain amount of clustering On the other hand, the first mixed Gaussian model, which has heavier tails —and hence is capable of pushing spectral energy to the high frequency bands— produces generally smooth results (does not allow clustering), but generates periodic textures in midtone areas, as a consequence of energy packed into the corners of the baseband An ideal HVS model must lay somewhere in the middle of these two mixed Gaussian models First, this ideal model needs to have a sufficiently narrow bandwidth to guarantee the homogeneity of the halftones Second, the model must be versatile enough so that the designer can easily play with the heaviness of its tails without varying its bandwidth, making possible an optimization to achieve smoothness without compromising homogeneity N¨ as¨anen’s (exponential) model will not satisfy the first condition since in order to change the heaviness of its tails, the bandwidth must be varied The mixed Gaussian models can allow this tradeoff, but the fact that four constants need to be specified in order to completely define the model makes it a rather intricate filter to design A simpler approach is given by an alpha stable HVS filter, as described next Figure DBS gray ramps using the the N¨ as¨ anen’s model (first), the first mixed Gaussian model (second), the second mixed Gaussian model (third), the α-SG model with α = 0.95 (fourth) and the α-SG model with α = 1.05 (fifth) 4.3 Alpha Stable models In α-stable models, the parameter γ in (7) (as well as the scale parameter S = RD) determines the bandwidth of the filter and consequently the homogeneity of the halftones The parameter α controls the heaviness of the tails of the filter hence the smoothness/coarseness of the dither patterns This relationship Figure Relationship between the parameter between α and the characteristics of the halftones is illustrated α and the quality of the halftones in Figure It was found experimentally that a model with a bandwidth around 0.08 produces the homogeneity that is being sought It was also found that a good point to start in terms of heaviness of tails is to have an initial model that satisfies H(ρ)|ρ= √1 ≈ 10−2 The proposed starting model is labeled in Figure as “α-SG, α = 0.95” It was generated with constants α = 0.95, γ = 27, D = 9.5in, R = 300dpi The initial size of the model was N = 101, although it was possible to truncate it further to obtain a final N = 31 Notice how the shape of the α-SG model being proposed dramatically differs from the other three models, especially in the logarithmic view (Figure 2) The fourth ramp of Figure corresponds to the gray ramp obtained with this model; notice that the extreme tones of the ramp were effectively improved with respect to the mixed Gaussian models However, the RAPSD for gray level g = 50% (fourth row of Figure 4) demonstrates that the cutoff frequency is going beyond its ideal position, hence and improvement of the model is due The procedure to follow at this point is to start generating models where the heaviness of the tails is slowly decreased, i.e the α value is slightly increased, so that the spectral energy in the RAPSD starts to be pulled to lower frequencies The value of α is not increased anymore when gray patterns with a RAPSD that satisfies the blue noise model are found, and their subjective appearance is highly qualified It was found that a model with α = 1.05 would yield the spectral response that is being sought This new model is shown in Figure Notice that the second filter has lower tails than the initial filter This function must be normalized, but it cannot be multiplied by the inverse of its maximum value because that would affect the heaviness of its tails and consequently the quality in the halftone patterns that is being pursued What is proposed is to pass this filter through a rectifying block that can be easily implemented with the hyperbolic tangent function tanh(x) = (exp(x) − exp(−x))/(exp(x) + exp(−x)) The final rectified filter is plotted in Figure 2, labeled as “α-SG, α = 1.05” This filter provides all the characteristics of an ideal dither pattern for all tones of gray in both the frequency and the spatial domains The fifth ramp of Figure is the one generated with this model The halftone shows that the extreme zones maintain the good characteristics of the initial α-SG model, and the quarter tones are smooth and pleasant Moreover, the middle tones are free from periodic artifacts These same characteristics can be verified in the gray patches of the fifth row of Figure 4, and in their radially averaged frequency responses These RAPSDs show no accumulation of energy below f = 1/2 (no exaggerated clustering) nor beyond (no packing of energy into the corners of the baseband) This model generates dither arrays that comply, from all points of view, with the blue noise model proposed by Lau and Ulichney HVS MODELS FOR DBS SCREENS DESIGN In order to obtain halftone images with quality similar to that yielded by DBS but without its computational burden, it is possible to use the DBS algorithm to design a dither matrix The halftoning process using screens reduces to a thresholding operation at each pixel Allebach and Lin20 have shown that these screens in fact are able to maintain halftone image quality while significantly reducing the required computation To design a dither matrix, a halftone pattern for each gray level must be generated subject to the stacking constraint Initially a pattern for an intermediate gray level is generated, and then lighter patterns are produced by gradually removing dots; and darker patterns by gradually adding dots The initial intermediate pattern is generated with the DBS algorithm described in section 2, with the difference that only swaps are considered; toggles are discarded since using them would alter the average gray level of the initial pattern After the initial pattern has been designed, the lighter and darker patterns are generated using a similar process Suppose a lighter pattern is to be designed A given number of dots corresponding to the Figure Gray ramps obtained with DBS screens of size 128 × 128 generated using the α-SG model with α = 0.95 (first) and the α-SG model with α = 1.05 (second) difference in adjacent gray levels are deleted randomly from the initial pattern In order to satisfy the stacking property of the dither matrix patterns, only the positions of the newly deleted dots are allowed to be moved The metrics of DBS is again minimized using the swap and toggle operations It is important to realize that if it is needed to keep the exact dot number in the refined pattern, the toggling should be disabled To generate the darker patterns, a similar process is performed starting with the addition of dots to the previously designed pattern The final dither matrices can be obtained by summing the binary patterns for all gray levels Different DBS screens can be obtained if different HVS models are used in the process As expected, the same appreciations described in the sections on the results obtained by DBS with the different HVS models also hold for the results produced using the screens Figure shows the gray ramps generated with the DBS screens of size 128 × 128 corresponding to the α-SG HVS models The similarity with the ramps of Figure is easily verified CONCLUSIONS The characteristics of the HVS model in a digital halftoning algorithm play a dramatic role in the performance of the algorithm In the particular case of DBS, the frequency response of the filter determines the quality of the output in terms of homogeneity, coarseness, smoothness and appearance of artifacts Blue noise theory, which gives strong importance to the radial symmetry of the frequency response of the dither patterns, provides a quantitative approach to qualify the characteristics of the halftones The characteristic function and the probability density function of bivariate α-SG random variables provide a powerful tool for modeling the PSF and the CSF of the HVS, respectively These functions can be easily tuned to yield a HVS model which, when used in DBS, produces results that comply with the blue noise model for all levels of gray When these filters are used to generate DBS masks, the characteristics of the screening outputs agree with the characteristics of the halftone results yielded by DBS using each corresponding HVS model ACKNOWLEDGEMENT In completing this paper, we would like to acknowledge the financial support of Agere Systems and the assistance and professional guidance received from Natalya Lyubashevskaya, Dr Steven Pinault and Dr Anatoly Moskalev We would also like to thank Dr Sang Ho Kim at Samsung Electronics and Dr Jan P Allebach at Purdue University for their valuable contributions during the first stages of this research work Finally, a great deal of thanks is also directed to Cory Budischak at University of Delaware for his contributions in the development of part of the software used in this project REFERENCES S H Kim and J P Allebach, “Impact of HVS models on model-based halftoning,” IEEE Transactions on Image Processing 11, pp 258–269, March 1995 B E Bayer, “An optimum method for two-level rendition of cotinuous-tone pictures,” Proceedings of the IEEE International Conference on Communications 26, pp 2611–2615, June 1973 R A Ulichney, “The void-and-cluster method for dither array generation,” Proceedings SPIE, Human Vision, Visual Processing, Digital Displays IV 1913, pp 332–343, 1993 G Whesteimer, “The eye as an optical instrument,” Handbook of Human Perception and Performance , pp 325–345, 1986 M Analoui and J P Allebach, “Model-based halftoning using direct binary search,” Proceedings IS&T/SPIE International Symposium on Electronic Imaging Science and Technology 1666, pp 96–108, February 1992 F W Campbell, R H S Carpenter, and J Z Levinson, “Visibility of aperiodic patterns compared with that of sinusiodal gratings,” Journal of Physiology 204, pp 283–292, 1969 J L Mannos and D J Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Transactions on Information Theory IT-20, pp 525–536, July 1974 R N¨ as¨anen, “Visibility of halftone dot textures,” IEEE Transactions on Systems, Man and Cybernetics 14, pp 920–924, November 1984 S Daly, “Subroutine for the generation of a two dimensional human visual contrast sensitivity function,” Eastman Kodak, Tech Rep 233203y , 1987 10 K T Mullen, “The contrast sensitivity of human colour vision to red-green and blue-yellow chromatic gratings,” Journal of Physiology 359, pp 381–400, March 1985 11 G R Arce, Nonlinear Signal Processing, Wiley and Sons, Hoboken, NJ, 2004 12 R A Ulichney, “Dithering with blue noise,” Proceedings of the IEEE 76, pp 56–79, January 1988 13 D L Lau and G R Arce, Modern Digital Halftoning, Marcel Dekker, New York, NY, 2001 14 D L Lau, R Ulichney, and G R Arce, “Blue- and green-noise halftoning models,” IEEE Signal Processing Magazine , pp 28–38, July 2003 15 D L Lau and R A Ulichney, “Blue-noise halftoning for hexagonal grids,” IEEE Transactions on Image Processing , To appear 16 P L´evy, Calcul des probabilit´es, Gauthier-Villars, Paris, FR, 1925 17 S Cambanis and G Miller, “Linear problems in pth order and stable processes,” Journal of Applied Math 41, pp 43–69, August 1981 18 D L Lau, G R Arce, and N C Gallagher, “Green-noise digital halftoning,” Proceedings of the IEEE 86, pp 2424–2444, December 1998 19 D L Lau, G R Arce, and N C Gallagher, “Digital halftoning by means of gree-noise masks,” Journal of the Optical Society of America 16, pp 1575–1586, July 1999 20 J P Allebach and Q Lin, “Fm screen design using dbs algorithm,” Proceedings of the IEEE International Conference on Image Processing 1, pp 549–552, September 1996 [...]... 2611–2615, June 1973 3 R A Ulichney, “The void-and-cluster method for dither array generation,” Proceedings SPIE, Human Vision, Visual Processing, Digital Displays IV 1913, pp 332–343, 1993 4 G Whesteimer, “The eye as an optical instrument,” Handbook of Human Perception and Performance , pp 325–345, 1986 5 M Analoui and J P Allebach, “Model-based halftoning using direct binary search,” Proceedings IS&T/SPIE... Mannos and D J Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Transactions on Information Theory IT-20, pp 525–536, July 1974 8 R N¨ as¨anen, “Visibility of halftone dot textures,” IEEE Transactions on Systems, Man and Cybernetics 14, pp 920–924, November 1984 9 S Daly, “Subroutine for the generation of a two dimensional human visual contrast sensitivity function,”... process is performed starting with the addition of dots to the previously designed pattern The final dither matrices can be obtained by summing the binary patterns for all gray levels Different DBS screens can be obtained if different HVS models are used in the process As expected, the same appreciations described in the sections on the results obtained by DBS with the different HVS models also hold for the... and G R Arce, “Blue- and green-noise halftoning models, ” IEEE Signal Processing Magazine , pp 28–38, July 2003 15 D L Lau and R A Ulichney, “Blue-noise halftoning for hexagonal grids,” IEEE Transactions on Image Processing , To appear 16 P L´evy, Calcul des probabilit´es, Gauthier-Villars, Paris, FR, 1925 17 S Cambanis and G Miller, “Linear problems in pth order and stable processes,” Journal of Applied... thanks is also directed to Cory Budischak at University of Delaware for his contributions in the development of part of the software used in this project REFERENCES 1 S H Kim and J P Allebach, “Impact of HVS models on model-based halftoning, ” IEEE Transactions on Image Processing 11, pp 258–269, March 1995 2 B E Bayer, “An optimum method for two-level rendition of cotinuous-tone pictures,” Proceedings... Figure 7 shows the gray ramps generated with the DBS screens of size 128 × 128 corresponding to the α-SG HVS models The similarity with the ramps of Figure 5 is easily verified 6 CONCLUSIONS The characteristics of the HVS model in a digital halftoning algorithm play a dramatic role in the performance of the algorithm In the particular case of DBS, the frequency response of the filter determines the quality... pth order and stable processes,” Journal of Applied Math 41, pp 43–69, August 1981 18 D L Lau, G R Arce, and N C Gallagher, “Green-noise digital halftoning, ” Proceedings of the IEEE 86, pp 2424–2444, December 1998 19 D L Lau, G R Arce, and N C Gallagher, Digital halftoning by means of gree-noise masks,” Journal of the Optical Society of America 16, pp 1575–1586, July 1999 20 J P Allebach and Q Lin,... Mullen, “The contrast sensitivity of human colour vision to red-green and blue-yellow chromatic gratings,” Journal of Physiology 359, pp 381–400, March 1985 11 G R Arce, Nonlinear Signal Processing, Wiley and Sons, Hoboken, NJ, 2004 12 R A Ulichney, “Dithering with blue noise,” Proceedings of the IEEE 76, pp 56–79, January 1988 13 D L Lau and G R Arce, Modern Digital Halftoning, Marcel Dekker, New York,... function and the probability density function of bivariate α-SG random variables provide a powerful tool for modeling the PSF and the CSF of the HVS, respectively These functions can be easily tuned to yield a HVS model which, when used in DBS, produces results that comply with the blue noise model for all levels of gray When these filters are used to generate DBS masks, the characteristics of the screening... completing this paper, we would like to acknowledge the financial support of Agere Systems and the assistance and professional guidance received from Natalya Lyubashevskaya, Dr Steven Pinault and Dr Anatoly Moskalev We would also like to thank Dr Sang Ho Kim at Samsung Electronics and Dr Jan P Allebach at Purdue University for their valuable contributions during the first stages of this research work Finally, ... 3.3 Alpha Stable models HVS models used in DBS are commonly derived from empirical approximations to the CSF That is the case in the N¨ as¨anen and the mixed Gaussian models described before... to spatial stimuli) were found to have the analytical form of the characteristic function of alpha stable random variables.4 Thus, stable models fit well empirical measurements and at the same... patterns will be compared to the ideal RAPSD in Figure HUMAN VISUAL SYSTEM MODELS Figure shows the tails of the frequency response of the five models to be studied in this paper These are radially