Annex A (INFORMATIVE) A DERIVATION OF THE GRAYSCALE STANDARD DISPLAY FUNCTION

A.1. Rationale for Selecting the Grayscale Standard Display Function

In choosing the Grayscale Standard Display Function, it was considered mandatory to have only one continuous, monotonically behaving mathematical function for the entire Luminance Range of interest. Correspondingly, for simplicity of implementing the Grayscale Standard Display Function, it was felt to be useful to define it by only one table of data pairs. As a secondary objective, it was considered desirable that the Grayscale Standard Display Function provide similarity in grayscale rendition on Display Systems of different Luminance Range and that good use of the available DDLs of a Display System was facilitated.

Perceptual linearization was thought to be a useful concept for arriving at a Grayscale Standard Display Function for meeting the above secondary objectives; however, it is not considered an objective by itself. Apart from the fact that is probably an elusive goal to perceptually linearize all types of medical images under various viewing conditions by one mathematical function, medical images are mostly presented by application-specific Display Functions that assign contrast non-uniformly according to clinical needs.

Intuitively, one would assume that perceptually linearized images on different Display Systems will be judged to be similar. To achieve perceptual linearization, a model of the human visual system response was required and the Barten model [A1] was chosen.

Early experiments showed that an appealing degree of contrast equalization and similarity could be obtained with a Display Function derived from Bartenís model of human visual system response. The employed images were square patterns, the SMPTE pattern, and the Briggsí pattern [A2].

It was wished to relate DDLs of a Display System to some perceptually linear scale, primarily, to gain efficient utilization of the available input levels. If digitization levels lead to luminance or optical density levels that are perceptually indistinguishable, they are wasted. If they are too far apart, the observer may see contours. Hence, the concept of perceptual linearization was retained, not as a goal for the Grayscale Standard Display Function, but to obtain a concept for a measure of how well these objectives have been met.

Perceptual linearization is realizable, in a strict sense, only for rather simple images like square patterns or gratings in a uniform surrounding. Nevertheless, the concept of a perceptually linearized Display Function derived from experiments with simple test patterns has been successfully applied to complex images as described in the literature [A3-A8]. While it was clearly recognized that perceptual linearization can never be achieved for all details or spatial frequencies and object sizes at once, perceptual linearization for frequencies and object sizes near the peak of human Contrast Sensitivity seemed to do a ìreasonable jobî also in complex images.

Limited (unpublished) experiments have indicated that perceptual linearization for a particular detail in a complex image with a wide Luminance Range and heterogeneous surround required Display Functions which are rather strongly bent in the dark regions of the image and that such Display Functions for a low-luminance and a high-luminance display system would not be part of a continuous, monotonic function. This experience may underly the considerations of the CIELab curve [A9] proposed by other standards groups.

Other experiments and observations with computed radiographs seemed to suggest that similarity could also be obtained between grayscale renditions on Display Systems of different Luminance when the same application-specific function is combined with log-linear Characteristic Curves of the Display Systems. Thus similarity, if not contrast equalization, could be gained by a straight, luminance-independent shape for the Display Function.

While it might have been equally sensible to choose the rather simple log-linear Display Function as a standard, this was not done for the following reason, among others.

For high-resolution Display Systems with high intrinsic video bandwidth, digitization resolution is limited to 8 or 10 bits because of technology and other constraints. The more a Grayscale Standard Display Function deviates from the Characteristic Curve of a Display System, the poorer the utilization of DDLs typically is from a perception point of view. The Characteristic Curve of CRT Display Systems has a convex curvature with respect to a log-linear straight line. It differs much less from Display Functions derived from human vision models and the concept of perceptual linearization than from a log-linear Display Function.

When using application-specific display processes which cause the resultant Display Function to deviate strongly from the Grayscale Standard Display Function, the function conceivably does not provide good similarity. In this case, other functions may yield better similarity.

In summary, a Display Function was derived from Bartenís model of the human visual system to gain a single continuous mathematical function which in its curvature falls between a log-linear response and a Display Function that may yield perceptual linearization in complex scenery with a wide luminance range within the image. Other models of human contrast sensitivity may potentially provide a better function, but were not evaluated. The notion of perceptual linearization was chosen to meet the secondary objectives of the Grayscale Standard Display Function, but not as an explicit goal of the Grayscale Standard Display Function itself. It is recognized that better functions may exist to meet these objectives. It is believed that almost any single mathematically defined Standard Function will greatly improve image presentations on Display Systems in communication networks.

A.2. Details of the Barten model

Barten’s model considers neural noise, lateral inhibition, photon noise, external noise, limited integration capability, the optical modulation transfer function, orientation, and temporal filtering. Neuron noise represents the upper limit of Contrast Sensitivity at high spatial frequencies. Low spatial frequencies appear to be attenuated by lateral inhibition in the ganglion cells which seems to be caused by the subtraction of a spatially low-pass filtered signal from the original. Photon noise is defined by the fluctuations of the photon flux h, the pupil diameter d, and quantum detection efficiency η of the eye. At low light levels, the Contrast Sensitivity is proportional to the square-root of Luminance according to the de Vries-Rose law. The temporal integration capability in the model used here is simply represented by a time constant of T = 0.1 sec. Temporal filtering effects are not included. Next to the temporal integration capability, the eye also has limited spatial integration capability: There is a maximum angular size XE x YE as well as a maximum number of cycles NE over which the eye can integrate information in the presence of various noise sources. The optical modulation transfer function

M opt (u) = e σ = (A1)

(u, spatial frequency in c/deg) is derived from a Gaussian point-spread function including the optical properties of the eye-lens, stray light from the optical media, diffusion in the retina, and the discrete nature of the receptor elements as well as from the spherical aberration, Csph , which is the main pupil-diameter-dependent component. σ0 is the value of σ at small pupil sizes. External noise may stem from Display System noise and image noise. Contrast sensitivity varies approximately sinusoidally with the orientation of the test pattern with equal maximum sensitivity at 0 and 90 deg and minimal sensitivity at 45 deg. The difference in Contrast Sensitivity is only present at high spatial frequencies. The effect is modeled by a variation in integration capability.

The combination of these effects yields the equation for contrast as a function of spatial frequency:

S(u) = (A2)

The effect of noise appears in the first parenthesis within the square-root as a noise contrast related to the variances of photon (first term), filtered neuron (second term), and external noise. The Illuminance, IL = π/4 d 2L, of the eye is expressed in trolands [td], d is the pupil diameter in mm, and L the Luminance of the Target in cd/m2. The pupil diameter is determined by the formula of de Groot and Gebhard:

d = 4.6 - 2.8 . tanh(0.4 . Log10(0.625 . L)) (A3)

The term (1 - F(u)) 2 = 1 - exp(-u 2/u02) describes the low frequency attenuation of neuron noise due to lateral inhibition (u0 = 8 c/deg). Equation (A2) represents the simplified case of square targets, X0 = Y0 [deg]. Φext is the contrast variance corresponding to external noise. k = 3.3, η = 0.025, h = 357 . 3600 photons/td sec deg2; the contrast variance corresponding to the neuron noise Φ0 = 3 . 10-8 sec deg2, XE = 12 deg, NE = 15 cycles (at 0 and 90 deg and NE = 7.5 cycles at 45 deg for frequencies above 2 c/deg), σ0 = 0.0133 deg, Csph = 0.0001 deg/mm3 [A1]. Equation (A2) provides a good fit of experimental data for 10-4 ≤ L ≤ 103 cd/m2, 0.5 ≤ X0 ≤ 60 deg, 0.2 ≤ u ≤ 50 c/deg.

After inserting all constants, Equation (A2) reduces to

S(L) = (A4)

with q1 = 0.1183034375, q2 = 3.962774805 . 10-5, and q3 = 1.356243499 . 10-7.

When viewed from 250 mm distance, the Standard Target has a size of about 8.7 mm x 8.7 mm and the spatial frequency of the grid equals about 0.92 line pairs per millimeter.

The Grayscale Standard Display Function is obtained by computing the Threshold Modulation Sj as a function of mean grating Luminance and then stacking these values on top of each other. The mean Luminance of the next higher level is calculated by adding the peak-to-peak modulation to the mean Luminance Lj of the previous level:

L j+1 = L j . (A5)

Thus, in PS 3.14, the peak-to-peak Threshold Modulation is called a just-noticeable Luminance difference.

When a Display System conforms with the Grayscale Standard Display Function, it is perceptually linearized when observing the Standard Target: If a Display System had infinitely fine digitization resolution, equal increments in P-Value would produce equally perceivable contrast steps and, under certain conditions, just-noticeable Luminance differences (displayed one at a time) for the Standard Target (the grating with sinusoidal modulation of 4 c/degree over a 2 degree x 2 degree area, embedded in a uniform background with a Luminance equal to the mean target Luminance).

The display of the Standard Target at different Luminance levels one at a time is an academic display situation. An image containing different Luminance levels with different targets and Luminance distributions at the same time is in general not perceptually linearized. It is once more emphasized that the concept of perceptual linearization of Display Systems for the Standard Target served as a logical means for deriving a continuous mathematical function and for meeting the secondary goals of the Grayscale Standard Display Function. The function may represent a compromise between perceptual linearization of complex images by strongly-bent Display Functions and gaining similarity of grayscale perception within an image on Display Systems of different Luminance by a log-linear Display Function.

The Characteristic Curve of the Display System is measured and represented by {Luminance, DDL}-pairs Lm = F(Dm). A discrete transformation may be performed that maps the previously used DDLs, Dinput , to Doutput according to Equations (A6) and (A7) such that the available ensemble of discrete Luminance levels is used to approximate the Grayscale Standard Display Function L = G(j). The transformation is illustrated in Fig. A1. By such an operation, conformance with the Grayscale Standard Display Function may be reached.

D output = s . F -1 [G(j)] (A6)

s is a scale factor for accommodating different input and output digitization resolutions.

The index j (which in general will be a non-integer number) of the Standard Luminance Levels is determined from the starting index j0 of the Standard Luminance level at the minimum Luminance of the Display System (including ambient light), the number of Standard JNDs, NJND , over the Luminance Range of the Display System, the digitization resolution DR, and the DDLs, Dinput , of the Display System:

I = I0 + . D input (A7)

A detailed example for executing such a transformation is given in Annex D.

A.3. REFERENCES

[A1] P.G.J. Barten: Physical model for the Contrast Sensitivity of the human eye. Proc. SPIE 1666 , 57-72 (1992) and Spatio-temporal model for the Contrast Sensitivity of the human eye and its temporal aspects. Proc. SPIE 1913 -01 (1993)

[A2] S.J. Briggs: Digital test target for display evaluation . Proc. SPIE 253 , 237-246 (1980)

[A3] S.J. Briggs: Photometric technique for deriving a "best gamma" for displays . Proc. SPIE 199 , Paper 26 (1979) and Opt. Eng. 20, 651-657 (1981)

[A4] S.M. Pizer: Intensity mappings: linearization, image-based, user-controlled . Proc. SPIE 271 , 21-27 (1981)

[A5] S.M. Pizer: Intensity mappings to linearize display devices . Comp. Graph. Image. Proc. 17 , 262-268 (1981)

[A6] R.E. Johnston, J.B. Zimmerman, D.C. Rogers, and S.M. Pizer: Perceptual standardization . Proc. SPIE 536 , 44-49 (1985)

[A7] R.C. Cromartie, R.E. Johnston, S.M. Pizer, D.C. Rogers: Standardization of electronic display devices based on human perception . University of North Carolina at Chapel Hill, Technical Report 88-002, Dec. 1987

[A8] B. M. Hemminger, R.E. Johnston, J.P. Rolland, K.E. Muller: Perceptual linearization of video display monitors for medical image presentation . Proc. SPIE 2164 , 222-241 (1994)

[A9] CIE 1976

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Fig. A-1. Illustration for determining the transform that changes the Characteristic Curve of a Display System to a Display Function that approximates the Grayscale Standard Display Function