Synthetic Surfaces

Beside functions for analysis of measured data, Gwyddion provides several generators of artificial surfaces that can be used for testing or simulations also outside Gwyddion.

All the surface generators share a certain set of parameters, determining the dimensions and scales of the created surface and the random number generator controls. These parameters are described below, the parameters specific to each generator are described in the corresponding subsections.

Image parameters:

Horizontal, Vertical size

The horizontal and vertical resolution of the generated surface in pixels.

Square image

This option, when enabled, forces the horizontal and vertical resolution to be identical.

Width, Height

The horizontal and vertical physical dimensions of the generated surface in selected units. Note square pixels are assumed so, changing one causes the other to be recalculated.

Dimension, Value units

Units of the lateral dimensions (Width, Height) and of the values (heights). The units chosen here also determine the units of non-dimensionless parameters of the individual generators.

Take Dimensions from Current Channel

Clicking this button fills all the above parameters according to the current channel.

Note that while the units of values are updated, the value scale is defined by generator-specific parameters that might not be directly derivable from the statistical properties of the current channel. Hence these parameters are not recalculated.

Replace the current channel

This option has two effects. First, it causes the dimensions and scales to be automatically set to those of the current channel. Second, it makes the generated surface replace the current channel instead of creating a new channel.

Start from the current channel

This option has two effects. First, it causes the dimensions and scales to be automatically set to those of the current channel. Second, it makes the generator to start from the surface contained in the current channel and modify it instead of starting from a flat surface. Note this does not affect whether the result actually goes to the current channel or a new channel is created.

Random generator controls:

Random seed

The random number generator seed. Choosing the same parameters and resolutions and the same random seed leads to the same surface, even on different computers. Different seeds lead to different surfaces with the same overall characteristics given by the generator parameters.

New

Replaces the seed with a random number.

Randomize

Enabling this option makes the seed to be chosen randomly every time the generator is run. This permits to conveniently re-run the generator with a new seed simply by pressing Ctrl-F (see keyboard shortcuts).

Spectral

Spectral synthesis module creates randomly rough surfaces by constructing the Fourier transform of the surface according to specified parameters and then performing the inverse Fourier transform to obtain the real surface. The generated surfaces are periodic (i.e. perfectly tilable).

The Fourier image parameters define the shape of the PSDF, i.e. the Fourier coefficient modulus, the phases are chosen randomly. At present, all generated surfaces are isotropic, i.e. the PSDF is radially symmetric.

RMS

The root mean square value of the heights (or of the differences from the mean plane which, however, always is the z = 0 plane). Button Like Current Channel sets the RMS value to those of the current channel.

Minimum, maximum frequency

The minimum and maximum spatial frequency. Increasing the minimum frequency leads to flattening of the image, i.e. to removal of large features. Decreasing the maximum frequency limits the sharpness of the features.

Enable Gaussian multiplier

Enables the multiplication of the Fourier coefficients by a Gaussian function that in the real space corresponds to the convolution with a Gaussian.

Enable Lorentzian multiplier

Enables the multiplication of the Fourier coefficients by a function proportional to 1/(1 + k2T2)3/4, where T is the autocorrelation length. So, the factor itself is not actually Lorentzian but it corresponds to Lorentzian one-dimensional power spectrum density which in turn corresponds to exponential autocorrelation function (see section Statistical Analysis for the discussion of autocorrelation functions). This factor decreases relatively slowly so the finite resolution plays usually a larger role than in the case of Gaussian.

Autocorrelation length

The autocorrelation length of the Gaussian or Lorentzian factors (see section Statistical Analysis for the discussion of autocorrelation functions).

Enable power multiplier

Enables multiplication of Fourier coefficients by factor proportional to 1/kp, where k is the spatial frequency and p is the power. This permits to generate various fractal surfaces.

Power

The power p.

Artificial surfaces generated by spectral synthesis: a narrow range of spatial frequencies (left), Gaussian random surface (centre) and a fractal surface generated with power multiplier and p equal to 1.5 (right).

Objects

The object placement method permits to create random surfaces composed of features of a specific shape. The algorithm is simple: the given number of objects is placed on random positions at the surface. For each object placed, the new heights are changed to max(z, z0 + h), where z is the current height at a specific pixel, h is the height of the object at this pixel (assuming a zero basis) and z0 is the current minimum height over the basis of the object being placed. The algorithm considers the horizontal plane to be filled with identical copies of the surface, hence, the generated surfaces are also periodic (i.e. perfectly tilable).

Shape

The shape (type) of placed objects. At present the possibilities include half-spheres, boxes, pyramids, tetrahedrons and some more weird shapes.

Coverage

The average number of times an object covers a pixel on the image. Coverage value of 1 means the surface would be exactly once covered by the objects assuming that they covered it uniformly. Larger values mean more layers of objects – and slower image generation.

Size

The lateral object size, usually the side of a containing square.

Aspect Ratio

The ratio between the x and y dimensions of an object – with respect to some default proportions.

Changing the aspect ratio does not always imply mere geometrical scaling, e.g. objects called nuggets change between half-spheres and rods when the ratio is changed.

Height

A quantity proportional to the height of the object, normally the height of the highest point.

Checking Scales with size makes unperturbed heights to scale proportionally with object size. Otherwise the height is independent on size.

Button Like Current Channel sets the height value to a value based on the RMS of the current channel.

Orientation

The rotation of objects with respect to some base orientation, measured counterclockwise.

Truncate

The shapes can be truncated at a certain height, enabling creation of truncated cones, pyramids, etc. The truncation height is given as a proportion to the total object height. Unity means the shape is not truncated, zero would mean complete removal of the object.

Each parameter can be randomized for individual objects, this is controlled by Variance. For multiplicative quantities (all except orientation and truncation), the distribution is log-normal with the RMS value of the logarithmed quantity given by Variance.

Artificial surfaces generated by object placement: spheres of varied size (left), narrow thatches of varied direction (centre), nuggets of varied aspect ratio (right).

Noise

Random uncorrelated point noise is generated independently in each pixel. Several distributions are available.

Distribution

The distribution of the noise value. The possibilities include Gaussian, exponential, uniform and triangular distributions.

Direction

The noise can be generated as symmetrical or one-sided. The mean value of the distribution of a symmetrical noise is zero, i.e. the mean value of data does not change when a symmetrical noise is added. One-sided noise only increases (if positive) or decreases (if negative) the data values.

RMS

Root mean square value of the noise distribution. More precisely, it is the RMS of the corresponding symmetrical distribution in the case the distribution is one-sided.

Different artificial noise applied to the same surface: symmetrical Gaussian noise (left); positive exponential noise (centre); negative exponential noise (right). All images have the same false colour scale and all noises have the same RMS.

Line Noise

Line noise represents noise with non-negligible duration that leads to typical steps or scars (also called strokes) in the direction of the fast scanning axis. Parameters Distribution, Direction and RMS have the same meaning as in Point noise. Other parameters control the lateral characteristics of the noise.

Two basic line defect types are available: steps and scars. Steps represent abrupt changes in the value that continue to the end of the scan (or until another step occurs). Scars are changes of the value with a finite duration, i.e. the values return to the original level after some time.

Steps have the following parameters:

Density

Average number of defects per scan line, including any dead time (as determined by parameter Within line).

Within line

Fraction of the time to scan one line that corresponds to actual data acquisition. The rest of time is a dead time. Value 1 means there is no dead time, i.e. all steps occur within the image. Value 0 means the data acquisition time is negligible to the total line scan time, consequently, steps only occur between lines.

Cumulative

For cumulative steps the random step value is always added to the current value offset; for non-cumulative steps the new value offset is directly equal to the random step value.

Scars have the following parameters:

Coverage

The fraction of the the image covered by defect if they did not overlap. Since the defect may overlap coverage value of 1.0 does not mean the image is covered completely.

Length

Scar length in pixels.

Variance

Variance of the scar length, see Objects for description of variances.

Different types of line noise added to an artificial pyramidal surface: unmodified surface (left); with relatively unfrequent non-cumulative steps (centre); with scars of mean length of 16 px and high coverage (right).

Pattern

Regular geometrical patterns represent surfaces often encountered in microscopy as standards or testing samples such as ridges, steps or holes. Each type of pattern has its own set of geometrical parameters determining the shape and dimensions of various part of the pattern. Each parameter has a variance control, similar to Object synthesis, that permits to make the pattern irregular in some aspects.

The placement of the pattern in the horizontal plane is controlled by parameters in tab Placement, common to all pattern types:

Orientation

The rotation of the pattern with respect to some base orientation, measured counterclockwise.

This tab also contains the deformation parameters. While enabling the variation of geometrical parameters makes the generated surface somewhat irregular the shape of its features is maintained. Deformation is a complementary method to introduce irregularity, specifically by distorting the pattern in the xy plane. It has two parameters:

Amplitude

The magnitude of the lateral deformation. It is a relative numerical quantity essentially determining how far the deformation can reach.

Lateral scale

The characteristic size of the deformations. It describes not how far the features are moved but how sharply or slowly the deformation itself changes within the horizontal plane.

Artificial pattern surfaces: sharp steps oriented at 30 deg, deformed with a deformation with a large autocorrelation length (left); non-uniformly spaced ridges with moderate slopes between the top and bottom planes, deformed with a deformation with a small autocorrelation length (centre); unperturbed pattern of rounded holes with a large variation in size and depth (right).

Columnar films

The columnar film growth simulation utilises a simple Monte Carlo deposition algorithm in which small particles are incident on the surface from directions generated according to given paramerters, and they stick to the surface around the point where they hit it, increasing the height there locally. The shadowing effect then causes more particles to stick to higher parts of the surface and less particles to the lower parts. This positive local height feedback leads to the formation of columns. The algorithm considers the horizontal plane to be filled with identical copies of the surface, hence, the generated surfaces are also periodic (i.e. perfectly tilable). The surface generator has the following parameters:

Coverage

The average number of times a particle is generated above each surface pixel.

Height

Local height increase occurring when the particle sticks to a pixel. Since the horizontal size of the particle is always one pixel the height is measured in pixels. A height of one pixel essentially means cubical particles, as far as growth is concerned. From the point of view of collision detection the particles are considered infinitely small.

Inclination

Central inclination angle with which the particles are generated (angle of incidence). The value of zero means very small angles of incidence have the highest probability. Large values mean that the particles are more likely to impinge at large angles than at small angles. But for large direction variance, the distribution is still isotropic in the horizontal plane.

Direction

Central direction in the horizontal plane with which the particles are generated. Large variance means the distribution is isotropic in the horizontal plane; for small variances the growth is anisotopic.

Relaxation

Method for determination of the pixel the particle will finally stick to. Two options exist at this moment. Weak relaxation, in which only the two pixels just immediately before collision and after collision are considered and the particle sticks to the lower of them. In the case of strong relaxation, a 3×3 neighbourhood is considered in addition. The particle can move to a lower neighbour pixel with a certain probability before sticking permanently.

Artificial columnar surfaces: loosely packed columns (left); tightly packed columns (centre); directional growth with strong relaxation (right).

Waves

The wave synthesis method composes the image using interference of waves from a number of point sources. Beside the normal cosine wave, a few other wave types are available, each having a cosine and sine form that differ by phase shift π/2 in all frequency components. When the wave is treated as complex, the cosine form is its real part and its sine form is its imaginary part.

Sine and cosine wave forms for the available wave types. Only the cosine form is used for displacement images; the entire complex wave is used for intensity and phase images.

The generator has the following options:

Quantity

Quantity to display in the image. Displacement is the sum of the real parts. Amplitude is the absolute value of the complex wave. Phase is the phase angle of the complex wave.

Number of waves

The number of point sources the waves propagate from.

Wave form

One of the wave forms described above.

Amplitude

Approximate amplitude (rms) of heights in the generated image. Note it differs from the amplitude of individual waves: the amplitude of heights in the generated image would grow with the square root of the number of waves in such case, whereas it in fact stays approxiately constant unless the amplitude changes.

Frequency

Spatial frequency of the waves. It is relative to the image size, i.e. the value of 1.0 means wavelength equal to the lenght of the image side.

X center, Y center

Locations of the point sources. Zero corresponds to the image centre. The positions are also measured in image sizes. Generally, at least one of the corresponding variances should be non-zero; otherwise all the point sources coincide (some interesting patterns can be still generated with varying frequency though).

Artificial wave surfaces: displacement image generated with fairly large frequency variation (left); amplitude image for inverse cosh wave form (centre); phase image for a small cluster of far-away sources (right).

Domains

The simulation implements a hybrid non-equilibrium Ising model [1], combining a discrete short-scale discrete Ising model with continuous slow inhibitor.

The discrete variable u has two possible values, between which it can flip probabilistically with probability

where ΔE is the change of energy resulting from flipping and T is the temperature. The energy is given by the number neighbours in the opposite state in the directions along the coordinate axes, in the diagonals and bias caused by the inhibitor field:

The continuous inhibitor field v is governed by a local differential equation coupled to the variable u:

where μ is inhibitor coupling and ν is bias. The boundary conditions of the simulation are periodic, hence, the generated images are also periodic (i.e. perfectly tilable).

The calculation has the following parameters:

Number of iterations

One iteration consists of four steps: A Monte Carlo update of u, a time step of solution of the differential equation for v, another Monte Carlo step and another differential euqation time step.

Temperature

Temperature determines the probability with which the two-state variable u can flip to the other value if it results in a configurations with higher energy or similar (flips that lower the energy a lot occur unconditonally). A larger temperature means less separation between the two u domains.

Inhibitor strength

Strength with which the continuous variable biases the energy of each pixel. For large values the inhibitor has larger influence compared to the surface tension.

Inhibitor coupling

Coupling factor between u and v in the differential equation for v.

Bias

Bias in the differential equation for v towards larger or smaller values.

Monte Carlo time step

Time step in the differential equation corresponding to one Monte Carlo step. So this parameter determines the relative speed of the two processes.

Height

Range of values of the created images.

Artificial domain surfaces: spiral waves with long-range order (left); short-range ordered spiral waves at a higher temperature (centre); the continous variable for a low-temperature situation with small strength and coupling (right).

References

[1] L. M. Pismen, M. I. Monine, G. V. Tchernikov: Physica D, (2004) 199 p 82