Alpha Stable Human Visual System Models for Digital Halftoning

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

Alpha Stable Human Visual System Models for Digital

Halftoning

A J Gonz´ aleza, J Baccaa, G R Arcea and D L Laub

aDepartment of Electrical and Computer Engineering

University of Delaware, Newark, DE 19716

bDepartment 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 algo-rithms 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, the broadly used N¨as¨anen’s exponential model As shown experimentally by Kim and Allebach,1 N¨as¨anen’s 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

1 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 do not include an explicit HVS block within their block diagram (like for example Bayer’s screen2and 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 Whesteimer4states 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¨anen8and Daly.9 Campbell and Mannos’ models are band-pass, with peak sensitivity around 7 cycles/degree Mullen10demonstrated 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 1 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

2 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 The inverse Fourier transform of H(u, v) yields ¯ h(¯ x, ¯ y) with ¯ x, ¯ 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] = 180

2

π2D2¯h



180m

πRD ,

180n

πRD



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 8 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

Figure 1 The RAPSD measure for an ideal blue

noise dither pattern

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

patterns that are visually appealing simply because the spectral

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 f b is defined as15:

fb=

⎧

⎨

⎩

√

g , for 0 < g ≤ 1/4

1/2 , for 1/4 < g ≤ 3/4

√

0.02

0.04

0.06

0.08

0.1

0.12

1/ √

2

1/2 1/3

ρ

Mixed Gaussian 1

Mixed Gaussian 2

N¨ as¨ anen

α-SG, α = 0.95

α-SG, α = 1.05

10 10 10

1/ √

2

1/2

0

ρ

Mixed Gaussian 1

Mixed Gaussian 2

N¨ as¨ anen

α-SG, α = 0.95 α-SG, α = 1.05

Figure 2 Frequency response of N¨as¨anen, mixed Gaussian and α-SG filters (left) and their logarithmic view (right).

Figure 1 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

1

3 HUMAN VISUAL SYSTEM MODELS

Figure 2 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 0 to 1/ √

2 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 1 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/ √

2 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 1

∗The bandwidth is defined as the frequency at which the filter has decayed to 50% of its maximum amplitude.

Table 1 Contrast Sensitivity Function and Point Spread Function of HVS models.

Model Contrast Sensitivity Function Point Spread Function Constants

(2π)2 1

(2π k +r2)3/2 L = 11cd/m

2

c = 0.525, d = 3.91 (κ1, κ2, σ1, σ2)I= Mixed Gaussian H2(ρ) = 2πκ1σ2exp`

−2π2

σ2ρ2´

(h ∗ h) (r) = κ1exp

“

− r2

2σ2

”

(43.2, 38.7, 0.02, 0.06) +2πκ2σ2exp`

−2π2σ2ρ2´

+κ2exp

“

− r2

2σ2

”

(κ1, κ2, σ1, σ2)II= (∗ means convolution) (19.1, 42.7, 0.03, 0.06)

(α, γ )I = (0.95, 27) α-stable No closed form h (r) = K exp (γ  r α) (α, γ )II = (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 1 The cross section of these filters along the diagonal is presented in Figure 2 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 Whesteimer4models the shape of the PSF with

h(r) = 0.952 exp



−2.59 |r| 1.36 + 0.048 exp



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



−

 1

2 tTRtα/2

(4)

where t = [x, y] T is a two dimensional column vector and the

10 10 10

1/ √2

ρ

α = 0.95

α = 1.05

α = 1.15

α = 1.25

Figure 3 Logarithmic view of the frequency

re-sponse of α-SG filters.

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

R =

where γ > 0 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



−1

2γ

α/2 x2+ y2α/2

(6)

It is possible to simplify (6) one step further by transforming

the rectangular coordinates into polar coordinates, resulting in

φ(r) = exp



−1

2γ

α/2 r α



with r2= x2+ y2and 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 12γ α/2 in (7) is replaced by one only term that is called γ  Figure 3 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

4 ANALYSIS OF DBS HALFTONE PATTERNS WITH DIFFERENT HVS MODELS 4.1 N¨ as¨ anen’s (exponential) model

The first row in Figure 4 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 4 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 4 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 σ g2= 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 σ2g and a spectral peak at cutoff frequency f b, 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

¨

s

¨

n

e

n

22%

0.5 1 1.5 2 2.5

1/ √

2

1/2 ρ

M

G

a

u

s

s

i

a

n

1

22%

0.5 1 1.5 2 2.5 3

1/ √

2

1/2 ρ

M

G

a

u

s

s

i

a

n

2

22%

0.5 1 1.5 2 2.5 3

1/ √

2

1/2 ρ

α

S

G

1

22%

0.5 1 1.5 2 2.5 3

1/ √

2

1/2 ρ

α

S

G

2

22%

0.5 1 1.5 2 2.5 3

1/ √

2

1/2 ρ

Figure 4 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%, f b = 0.47, and the solid line corresponds to g = 50%, f b = 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 5 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

do 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 do 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 2 The right center column of Figure 4 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 5 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, 19demonstrate 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 4

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 f b = 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 5 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

Figure 6 Relationship between the parameter

α and the quality of the halftones.

scale parameter S = RD) determines the bandwidth of the filter

and consequently the homogeneity of the halftones The

param-eter α controls the heaviness of the tails of the filter hence the

smoothness/coarseness of the dither patterns This relationship

between α and the characteristics of the halftones is illustrated

in Figure 6 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

2 ≈ 10 −2 The proposed starting model is labeled in Figure 2 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 5 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 3 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 5 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

5 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

4.3 Alpha Stable models< /b>

In α -stable models, the parameter γ  in (7) (as well as the

Figure... 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),... could not be applied if an α -stable< /i>

probability density function were being sought

4 ANALYSIS OF DBS HALFTONE PATTERNS WITH DIFFERENT HVS MODELS 4.1 Nă asă anens (exponential)