i found a shader to do a drop shadow from http://madebyevan.com/shaders/fast-rounded-rectangle-shadows/
// License: CC0 (http://creativecommons.org/publicdomain/zero/1.0/)
// This approximates the error function, needed for the gaussian integral
vec4 erf(vec4 x) {
vec4 s = sign(x), a = abs(x);
x = 1.0 + (0.278393 + (0.230389 + 0.078108 * (a * a)) * a) * a;
x *= x;
return s - s / (x * x);
}
// Return the mask for the shadow of a box from lower to upper
float boxShadow(vec2 lower, vec2 upper, vec2 point, float sigma) {
vec4 query = vec4(point - lower, upper - point);
vec4 integral = 0.5 + 0.5 * erf(query * (sqrt(0.5) / sigma));
return (integral.z - integral.x) * (integral.w - integral.y);
}
i thought that a shader need a main function and should return color.
My question is how to use the function boxShadow in c++ code with opengl given a box
thanks
A function can be defined in a shader just as you would do with a C function. I mean, the code for the function is in the same "unit" as the rest of the shader.
#version XX-YY
//MyFunc(.....)
whateverreturn MyFunc(.....)
{
do something and return a whateverreturn
}
void main(void)
{
//use MyFunc
whateverreturn var = MyFunc(....)
}
A bit different case is when you have a function that can be part of several shaders, but it isn't a "full" shader, it has no main() function. This function lives in a file or in string-array or something similar.
Say you have the function in a specific file:
#version XX-YY
//MyFunc(.....)
whateverreturn MyFunc(.....)
{
do something and return a whateverreturn
}
And the file with the shader where you want to use it:
#version XX-YY
//declare the function
whateverreturn MyFunc(.....);
void main(void)
{
//use MyFunc
whateverreturn var = MyFunc(....)
}
Like you do with any common GLSL code, use glShaderSource and glCompileShader.
Now the key step is how to integrate the code within a full (with 'main') shader: just use glAttachShader (again, like you do with VS or FS) before glLinkProgram and that'as all.
Related
I'm trying to import many transitions from GL Transitions into my video sequencer by converting GLSL to HLSL.
For example, this simple cross fade:
vec4 transition (vec2 uv) {
return mix(
getFromColor(uv),
getToColor(uv),
progress
);
}
is correctly translated in my HLSL code:
#define D2D_INPUT_COUNT 2
#define D2D_INPUT0_SIMPLE
#define D2D_INPUT1_SIMPLE
#define D2D_REQUIRES_SCENE_POSITION // The pixel shader requires the SCENE_POSITION input.
#include "d2d1effecthelpers.hlsli"
cbuffer constants : register(b0)
{
float progress : packoffset(c0.x);
...
}
float4 crossfade(float4 v1,float4 v2)
{
return lerp(v1, v2, progress);
}
D2D_PS_ENTRY(main)
{
float4 v1 = D2DGetInput(0);
float4 v2 = D2DGetInput(1);
return crossfade(v1,v2);
}
The same doesn't work for Wind effect:
// Custom parameters
uniform float size; // = 0.2
float rand (vec2 co) {
return fract(sin(dot(co.xy ,vec2(12.9898,78.233))) * 43758.5453);
}
vec4 transition (vec2 uv) {
float r = rand(vec2(0, uv.y));
float m = smoothstep(0.0, -size, uv.x*(1.0-size) + size*r - (progress * (1.0 + size)));
return mix(
getFromColor(uv),
getToColor(uv),
m
);
}
This time HLSL is this:
float fract(float x)
{
return x - floor(x);
}
float rand(float2 co)
{
return fract(sin(dot(co.xy, float2(12.9898, 78.233))) * 43758.5453);
}
float4 wind(float4 v1, float4 v2,float2 uv)
{
float r = rand(float2(0, uv.y));
p1 = 0.2f;
progress = 0.5f; // hardcoded variables for testing, they will be taken from the buffer
float m = smoothstep(0.0f, -p1, uv.x*(1.0f-p1) + p1*r - (progress * (1.0f + p1)));
return lerp(v1, v2, m);
}
D2D_PS_ENTRY(main)
{
float4 v1 = D2DGetInput(0);
float4 v2 = D2DGetInput(1);
return wind(v1,v2,D2DGetScenePosition().xy);
}
Have I misunderstood the OpenGL's mix and fract and rand stuff? I only get the second image pixels in my HLSL version without mixing.
EDIT: I 've hardcoded size to 0.992 and multiplied progress by 4 in the HLSL. Now it seems to work, do I miss some bounds-related issues? Is the smoothstep function working as expected?
I found it.
It would need in main entry the usage of D2DGetInputCoordinate instead of D2DGetScenePosition
After doing that, the transitions run fine.
In webGL1 it was possible to test the availability of a GLSL extension from a fragment shader using (for instance) #ifdef GL_EXT_shader_texture_lod .
It seems to no longer be working in webGL2 (=GLSL-ES3.0): Extensions are not the same, but for instance #ifdef GL_EXT_color_buffer_float seems false despite https://webglreport.com/?v=2 tells that the extension is there.
Or what am I doing wrong ?
It's up to the extension whether or not it adds a flag to GLSL
EXT_shader_texture_lod is specifically an extension that effects GLSL. It's spec says it adds that macro
The GLSL macro GL_EXT_shader_texture_lod is defined as 1.
EXT_color_buffer_float is not an extension that affects GLSL. It's spec does not mention any GLSL macros. No change from WebGL1
Those flags though are mostly nonsense in WebGL anyway. You can trivially do your own string manipulation
const shaderTextureLodExt = gl.getExtension('EXT_shader_texture_lod');
const shader = `
#if ${shaderTextureLodExt ? 1 : 0}
... code if shader texture lod exists
#else
... code if shader texture lod does not exist
#endif
...
`;
Or a thousand other ways to manipulate shader strings.
Here's another
const colorBufferFloatExt = gl.getExtension('EXT_color_buffer_float');
function replaceIfDefs(s) {
return `
${colorBufferExtension ? '#define EXTENSION_color_buffer_float' : ''}
${s.replace(/GL_EXT_color_buffer_float/g 'EXTENSION_color_buffer_float')}
`;
}
const shader = replaceIfDefs(`
#ifdef GL_EXT_color_buffer_float
...
#endif
...
`);
etc...
Also since there never was a GL_EXT_color_buffer_float even in OpenGL there isn't much point in calling the macro GL_EXT_color_buffer_float. In fact it would arguably be a bad idea because it would end up looking like an official specified macro even though it's not. Best to chose your own name that doesn't start with GL_ .
Also consider that using #ifdef might not even be a good idea since you can just use string manipulation. For example
const colorBufferFloatExt = gl.getExtension('EXT_color_buffer_float');
const snippet = colorBufferFloatExt
? `
float decode_float(vec4 v) {
return v;
}
`
: `
float decode_float(vec4 v) {
vec4 bits = v * 255.0;
float sign = mix(-1.0, 1.0, step(bits[3], 128.0));
float expo = floor(mod(bits[3] + 0.1, 128.0)) * 2.0 +
floor((bits[2] + 0.1) / 128.0) - 127.0;
float sig = bits[0] +
bits[1] * 256.0 +
floor(mod(bits[2] + 0.1, 128.0)) * 256.0 * 256.0;
return sign * (1.0 + sig / 8388607.0) * pow(2.0, expo);
}
`;
const shader = `
precision highp float;
${snippet}
uniform sampler2D data;
uniform vec2 dataSize;
void main(
vec4 d = texture2D(data, gl_FragCoord.xy / dataSize);
vec4 v = decode_float(d) * 2.0;
gl_FragColor = v;
}
`;
...etc...
I have the a webgl blur shader:
precision mediump float;
precision mediump int;
uniform sampler2D u_image;
uniform float blur;
uniform int u_horizontalpass; // 0 or 1 to indicate vertical or horizontal pass
uniform float sigma; // The sigma value for the gaussian function: higher value means more blur
// A good value for 9x9 is around 3 to 5
// A good value for 7x7 is around 2.5 to 4
// A good value for 5x5 is around 2 to 3.5
// ... play around with this based on what you need :)
varying vec4 v_texCoord;
const vec2 texOffset = vec2(1.0, 1.0);
// uniform vec2 texOffset;
const float PI = 3.14159265;
void main() {
vec2 p = v_texCoord.st;
float numBlurPixelsPerSide = blur / 2.0;
// Incremental Gaussian Coefficent Calculation (See GPU Gems 3 pp. 877 - 889)
vec3 incrementalGaussian;
incrementalGaussian.x = 1.0 / (sqrt(2.0 * PI) * sigma);
incrementalGaussian.y = exp(-0.5 / (sigma * sigma));
incrementalGaussian.z = incrementalGaussian.y * incrementalGaussian.y;
vec4 avgValue = vec4(0.0, 0.0, 0.0, 0.0);
float coefficientSum = 0.0;
// Take the central sample first...
avgValue += texture2D(u_image, p) * incrementalGaussian.x;
coefficientSum += incrementalGaussian.x;
incrementalGaussian.xy *= incrementalGaussian.yz;
// Go through the remaining 8 vertical samples (4 on each side of the center)
for (float i = 1.0; i <= numBlurPixelsPerSide; i += 1.0) {
avgValue += texture2D(u_image, p - i * texOffset) * incrementalGaussian.x;
avgValue += texture2D(u_image, p + i * texOffset) * incrementalGaussian.x;
coefficientSum += 2.0 * incrementalGaussian.x;
incrementalGaussian.xy *= incrementalGaussian.yz;
}
gl_FragColor = avgValue / coefficientSum;
}
When I build, I get the following error message:
webgl-renderer.js?2eb3:137 Uncaught could not compile shader:ERROR:
0:38: 'i' : Loop index cannot be compared with non-constant expression
I have also tried to use just the uniform float blur to compare i to. Is there any way to fix this?
The problem is further detailed here: https://www.khronos.org/webgl/public-mailing-list/archives/1012/msg00063.php
The solution that I've found looking around is to only use a constant expression when comparing a loop var. This doesn't fit with what I need to do which is vary how many times I'm looping based on the blur radius.
Any thoughts on this?
This happens because on some hardware, GLSL loops are un-rolled into native GPU instructions. This means there needs to be a hard upper limit to the number of passes through the for loop, that governs how many copies of the loop's inner code will be generated. If you replace numBlurPixelsPerSide with a const float or even a #define directive, and the shader compiler can then determine the number of passes at compile time, and generate the code accordingly. But with a uniform there, the upper limit is not known at compile time.
There's an interesting wrinkle in this rule: You're allowed to break or call an early return out of a for loop, even though the max iterations must be discernible at compile time. For example, consider this tiny Mandelbrot shader. This is hardly the prettiest fractal on GLSL Sandbox, but I chose it for its small size:
precision mediump float;
uniform float time;
uniform vec2 mouse;
uniform vec2 resolution;
varying vec2 surfacePosition;
const float max_its = 100.;
float mandelbrot(vec2 z){
vec2 c = z;
for(float i=0.;i<max_its;i++){ // for loop is here.
if(dot(z,z)>4.) return i; // conditional early return here.
z = vec2(z.x*z.x-z.y*z.y,2.*z.x*z.y)+c;
}
return max_its;
}
void main( void ) {
vec2 p = surfacePosition;
gl_FragColor = vec4(mandelbrot(p)/max_its);
}
In this example, max_its is a const so the compiler knows the upper limit and can un-roll this loop if it needs to. Inside the loop, a return statement offers a way to leave the loop early for pixels that are outside of the Mandelbrot set.
You still don't want to set the max iterations too high, as this can produce a lot of GPU instructions and possibly hurt performance.
Try something like this:
const float MAX_ITERATIONS = 100.0;
// Go through the remaining 8 vertical samples (4 on each side of the center)
for (float i = 1.0; i <= MAX_ITERATIONS; i += 1.0) {
if (i >= numBlurPixelsPerSide){break;}
avgValue += texture2D(u_image, p - i * texOffset) * incrementalGaussian.x;
avgValue += texture2D(u_image, p + i * texOffset) * incrementalGaussian.x;
coefficientSum += 2.0 * incrementalGaussian.x;
incrementalGaussian.xy *= incrementalGaussian.yz;
}
Sometimes you can use my very simple solving of issue.
My fragment of the shader source code:
const int cloudPointsWidth = %s;
for ( int i = 0; i < cloudPointsWidth; i++ ) {
//TO DO something
}
You can see '%' : syntax error above. But I am replace %s to a number in my javascript code before use my shader. For example:
vertexCode = vertexCode.replace( '%s', 10 );
vertexCode is my shader source code.
Everytime if I want to change cloudPointsWidth, I am destroying my old shader and creating new shader with new cloudPointsWidth .
Hope sometimes my solving can to help you.
You can just do a for loop with large constant number and use a break.
for(int i = 0; i < 1000000; ++i)
{
// your code here
if(i >= n){
break;
}
}
I've had similar problem with image downsampling shader. The code is basically the same:
for (int dx = -2 * SCALE_FACTOR; dx < 2 * SCALE_FACTOR; dx += 2) {
for (int dy = -2 * SCALE_FACTOR; dy < 2 * SCALE_FACTOR; dy += 2) {
/* accumulate fragment's color */
}
}
What I've ended up doing is using preprocessor and creating separate shader programs for every SCALE_FACTOR used (luckily, only 4 was needed). To achieve that, a small helper function was implemented to add #define ... statements to shader code:
function insertDefines (shaderCode, defines) {
var defineString = '';
for (var define in defines) {
if (defines.hasOwnProperty(define)) {
defineString +=
'#define ' + define + ' ' + defines[define] + '\n';
}
}
var versionIdx = shaderCode.indexOf('#version');
if (versionIdx == -1) {
return defineString + shaderCode;
}
var nextLineIdx = shaderCode.indexOf('\n', versionIdx) + 1;
return shaderCode.slice(0, nextLineIdx) +
defineString +
shaderCode.slice(nextLineIdx);
}
The implementation is a bit tricky because if the code already has #version preprocessor statement in it, all other statements have to follow it.
Then I've added a check for SCALE_FACROR being defined:
#ifndef SCALE_FACTOR
# error SCALE_FACTOR is undefined
#endif
And in my javascript code I've done something like this:
var SCALE_FACTORS = [4, 8, 16, 32],
shaderCode, // the code of my shader
shaderPrograms = SCALE_FACTORS.map(function (factor) {
var codeWithDefines = insertDefines(shaderCode, { SCALE_FACTOR: factor });
/* compile shaders, link program, return */
});
I use opengl es3 on android and solve this problem by using extension above the beginning of program like this:
#extension GL_EXT_gpu_shader5 : require
I don't know whether it work on webGL, but you can try it.
Hope it can help.
You can also use template litterals to set the length of the loop
onBeforeCompile(shader) {
const array = [1,2,3,4,5];
shader.uniforms.myArray = { value: array };
let token = "#include <begin_vertex>";
const insert = `
uniform float myArray[${array.length}];
for ( int i = 0; i < ${array.length}; i++ ) {
float test = myArray[ i ];
}
`;
shader.vertexShader = shader.vertexShader.replace(token, token + insert);
}
I have the a webgl blur shader:
precision mediump float;
precision mediump int;
uniform sampler2D u_image;
uniform float blur;
uniform int u_horizontalpass; // 0 or 1 to indicate vertical or horizontal pass
uniform float sigma; // The sigma value for the gaussian function: higher value means more blur
// A good value for 9x9 is around 3 to 5
// A good value for 7x7 is around 2.5 to 4
// A good value for 5x5 is around 2 to 3.5
// ... play around with this based on what you need :)
varying vec4 v_texCoord;
const vec2 texOffset = vec2(1.0, 1.0);
// uniform vec2 texOffset;
const float PI = 3.14159265;
void main() {
vec2 p = v_texCoord.st;
float numBlurPixelsPerSide = blur / 2.0;
// Incremental Gaussian Coefficent Calculation (See GPU Gems 3 pp. 877 - 889)
vec3 incrementalGaussian;
incrementalGaussian.x = 1.0 / (sqrt(2.0 * PI) * sigma);
incrementalGaussian.y = exp(-0.5 / (sigma * sigma));
incrementalGaussian.z = incrementalGaussian.y * incrementalGaussian.y;
vec4 avgValue = vec4(0.0, 0.0, 0.0, 0.0);
float coefficientSum = 0.0;
// Take the central sample first...
avgValue += texture2D(u_image, p) * incrementalGaussian.x;
coefficientSum += incrementalGaussian.x;
incrementalGaussian.xy *= incrementalGaussian.yz;
// Go through the remaining 8 vertical samples (4 on each side of the center)
for (float i = 1.0; i <= numBlurPixelsPerSide; i += 1.0) {
avgValue += texture2D(u_image, p - i * texOffset) * incrementalGaussian.x;
avgValue += texture2D(u_image, p + i * texOffset) * incrementalGaussian.x;
coefficientSum += 2.0 * incrementalGaussian.x;
incrementalGaussian.xy *= incrementalGaussian.yz;
}
gl_FragColor = avgValue / coefficientSum;
}
When I build, I get the following error message:
webgl-renderer.js?2eb3:137 Uncaught could not compile shader:ERROR:
0:38: 'i' : Loop index cannot be compared with non-constant expression
I have also tried to use just the uniform float blur to compare i to. Is there any way to fix this?
The problem is further detailed here: https://www.khronos.org/webgl/public-mailing-list/archives/1012/msg00063.php
The solution that I've found looking around is to only use a constant expression when comparing a loop var. This doesn't fit with what I need to do which is vary how many times I'm looping based on the blur radius.
Any thoughts on this?
This happens because on some hardware, GLSL loops are un-rolled into native GPU instructions. This means there needs to be a hard upper limit to the number of passes through the for loop, that governs how many copies of the loop's inner code will be generated. If you replace numBlurPixelsPerSide with a const float or even a #define directive, and the shader compiler can then determine the number of passes at compile time, and generate the code accordingly. But with a uniform there, the upper limit is not known at compile time.
There's an interesting wrinkle in this rule: You're allowed to break or call an early return out of a for loop, even though the max iterations must be discernible at compile time. For example, consider this tiny Mandelbrot shader. This is hardly the prettiest fractal on GLSL Sandbox, but I chose it for its small size:
precision mediump float;
uniform float time;
uniform vec2 mouse;
uniform vec2 resolution;
varying vec2 surfacePosition;
const float max_its = 100.;
float mandelbrot(vec2 z){
vec2 c = z;
for(float i=0.;i<max_its;i++){ // for loop is here.
if(dot(z,z)>4.) return i; // conditional early return here.
z = vec2(z.x*z.x-z.y*z.y,2.*z.x*z.y)+c;
}
return max_its;
}
void main( void ) {
vec2 p = surfacePosition;
gl_FragColor = vec4(mandelbrot(p)/max_its);
}
In this example, max_its is a const so the compiler knows the upper limit and can un-roll this loop if it needs to. Inside the loop, a return statement offers a way to leave the loop early for pixels that are outside of the Mandelbrot set.
You still don't want to set the max iterations too high, as this can produce a lot of GPU instructions and possibly hurt performance.
Try something like this:
const float MAX_ITERATIONS = 100.0;
// Go through the remaining 8 vertical samples (4 on each side of the center)
for (float i = 1.0; i <= MAX_ITERATIONS; i += 1.0) {
if (i >= numBlurPixelsPerSide){break;}
avgValue += texture2D(u_image, p - i * texOffset) * incrementalGaussian.x;
avgValue += texture2D(u_image, p + i * texOffset) * incrementalGaussian.x;
coefficientSum += 2.0 * incrementalGaussian.x;
incrementalGaussian.xy *= incrementalGaussian.yz;
}
Sometimes you can use my very simple solving of issue.
My fragment of the shader source code:
const int cloudPointsWidth = %s;
for ( int i = 0; i < cloudPointsWidth; i++ ) {
//TO DO something
}
You can see '%' : syntax error above. But I am replace %s to a number in my javascript code before use my shader. For example:
vertexCode = vertexCode.replace( '%s', 10 );
vertexCode is my shader source code.
Everytime if I want to change cloudPointsWidth, I am destroying my old shader and creating new shader with new cloudPointsWidth .
Hope sometimes my solving can to help you.
You can just do a for loop with large constant number and use a break.
for(int i = 0; i < 1000000; ++i)
{
// your code here
if(i >= n){
break;
}
}
I've had similar problem with image downsampling shader. The code is basically the same:
for (int dx = -2 * SCALE_FACTOR; dx < 2 * SCALE_FACTOR; dx += 2) {
for (int dy = -2 * SCALE_FACTOR; dy < 2 * SCALE_FACTOR; dy += 2) {
/* accumulate fragment's color */
}
}
What I've ended up doing is using preprocessor and creating separate shader programs for every SCALE_FACTOR used (luckily, only 4 was needed). To achieve that, a small helper function was implemented to add #define ... statements to shader code:
function insertDefines (shaderCode, defines) {
var defineString = '';
for (var define in defines) {
if (defines.hasOwnProperty(define)) {
defineString +=
'#define ' + define + ' ' + defines[define] + '\n';
}
}
var versionIdx = shaderCode.indexOf('#version');
if (versionIdx == -1) {
return defineString + shaderCode;
}
var nextLineIdx = shaderCode.indexOf('\n', versionIdx) + 1;
return shaderCode.slice(0, nextLineIdx) +
defineString +
shaderCode.slice(nextLineIdx);
}
The implementation is a bit tricky because if the code already has #version preprocessor statement in it, all other statements have to follow it.
Then I've added a check for SCALE_FACROR being defined:
#ifndef SCALE_FACTOR
# error SCALE_FACTOR is undefined
#endif
And in my javascript code I've done something like this:
var SCALE_FACTORS = [4, 8, 16, 32],
shaderCode, // the code of my shader
shaderPrograms = SCALE_FACTORS.map(function (factor) {
var codeWithDefines = insertDefines(shaderCode, { SCALE_FACTOR: factor });
/* compile shaders, link program, return */
});
I use opengl es3 on android and solve this problem by using extension above the beginning of program like this:
#extension GL_EXT_gpu_shader5 : require
I don't know whether it work on webGL, but you can try it.
Hope it can help.
You can also use template litterals to set the length of the loop
onBeforeCompile(shader) {
const array = [1,2,3,4,5];
shader.uniforms.myArray = { value: array };
let token = "#include <begin_vertex>";
const insert = `
uniform float myArray[${array.length}];
for ( int i = 0; i < ${array.length}; i++ ) {
float test = myArray[ i ];
}
`;
shader.vertexShader = shader.vertexShader.replace(token, token + insert);
}
I'm trying to implement Sketchy Drawings. I'm at the part of the process which calls for the use of the noise texture to derive uncertainty values that will provide an offset into the edge map.
Here is a picture of my edge map for a torus:
And here is the noise texture I've gotten using the Perlin function as suggested:
I have these saved as textures in edgeTexture and noiseTexture respectively.
Now I'm stuck on the section where you have to offset the texture coordinates of the edge map by uncertainty values derived from the noise texture. This image is from the book:
offs = turbulence(s, t);
offt = turbulence(1 - s, 1 - t);
I'm ignoring the 2x2 matrix for the time being. Here is my current fragment shader attempt and the result it produces:
#version 330
out vec4 vFragColor;
uniform sampler2D edgeTexture;
uniform sampler2D noiseTexture;
smooth in vec2 vTexCoords;
float turbulence(float s, float t)
{
float sum = 0;
float scale = 1;
float s1 = 1;
vec2 coords = vec2(s,t);
for (int i=0; i < 10; i++)
{
vec4 noise = texture(noiseTexture, 0.25 * s1 * coords);
sum += scale * noise.x;
scale = scale / 2;
s1 = s1 * 2;
}
return sum;
}
void main( void )
{
float off_s = turbulence(vTexCoords.s, vTexCoords.t);
float off_t = turbulence(1 - vTexCoords.s, 1 - vTexCoords.t);
vFragColor = texture(edgeTexture, vTexCoords + vec2(off_s, off_t));
}
Clearly my addition to the vTexCoords is way off, but I can't see why. I have tried several other turbulence function definitions but none were close to the desired output so I'm thinking my overall approach is flawed somewhere. Any help here is greatly appreciated, and please comment if I haven't been clear. The desired output for a torus would just look like a roughly drawn circle I would imagine.
Your turbulence function will return values in the range (0,1). Firstly you need to change this to get values centered on 0. This should be done inside the loop in the function or you'll end up with a strange distribution. So firstly, I think you should change the line:
vec4 noise = texture(noiseTexture, 0.25 * s1 * coords);
to
vec4 noise = texture(noiseTexture, 0.25 * s1 * coords) * 2.0 - 1.0;
You then need to scale the offset so that you're not sampling the edge texture too far away from the fragment being drawn. Change:
vFragColor = texture(edgeTexture, vTexCoords + vec2(off_s, off_t));
to
vFragColor = texture(edgeTexture, vTexCoords + vec2(off_s, off_t) * off_scale);
where off_scale is some small value (perhaps around 0.05) chosen by experimentation.