## The Fx Special Effects Image Operator

Use the Fx special effects image operator to apply a mathematical expression to an image or image channels. Use Fx to:

- create canvases, gradients, mathematical colormaps
- move color values between images and channels
- translate, flip, mirror, rotate, scale, shear and generally distort images
- merge or composite multiple images together
- convolve or merge neighboring pixels together
- generate image metrics or 'fingerprints'

The expression can be simple--

      convert -size 64x64 xc:black -channel blue -fx "1/2" fx_navy.png

Here, we convert a black to a navy blue image:

-- or complex:

      convert rose.jpg \
              -fx "(1.0/(1.0+exp(10.0*(0.5-u)))-0.006693)*1.0092503" \
              rose-sigmoidal.png

This expression results in a high contrast version of the image:

The expression can include variable assignments. Assignments, in most cases, reduce the complexity of an expression and permits some operations that might not be possible any other way. For example, lets create a radial gradient with statements:

      convert -size 70x70 xc: \
              -fx "Xi=i-w/2; Yj=j-h/2; 1.2*(0.5-hypot(Xi,Yj)/70.0)+0.5" \
              radial-gradient.png

These statements return this image:

See Using FX, The Special Effects Image Operator for more examples on how to apply a mathematical expression to an image or image channels.

The next section discusses the Fx expression language.

## The Anatomy of an Fx Expression

##### The Fx Expression Language

The formal Fx expression language is defined in this table:

      numbers: integer, floating point, or scientfic notation (+/- required, e.g. 3.81469e-06)
      constants: e, Epsilon, QuantumRange, QuantumScale, Opaque, Pi, Transparent
      Fx operators (in order of precedence):
         ^ (power), *, /, % (modulo), +, -,
         <<, >>, <, <=, >, >=, ==, !=,
         & (bitwise AND),   | (bitwise OR),
         && (logical AND),  || (logical OR),
         ~ (logical NOT),  ?: (ternary conditional)
      math functions:
         abs(), acos(), alt(), asin(), atan(), atan2(), ceil(), cos(), cosh(), debug(), exp(), floor(),
         hypot(), int(), ln(), log(), logtwo(), max(), min(), mod(), pow(), rand(), round(),
         sign(), sin(), sinh(), sqrt(), tan(), tanh()
      channel functions: channel(r,g,b,a), channel(c,m,y,k,a)
      color names: red, cyan, black, etc.
      color functions: rgb(), rgba(), cmyk(), cmyka(), hsl(), hsla()
      color hex values: #ccc, #cbfed0, #b9e1cc00
      symbols:
         s = current image in sequence
         t = scene number of current image in sequence
         u = first image in sequence
         v = second image in sequence
         n = number of images in sequence

         i = column offset
         j = row offset
         p = pixel to use (absolute or relative to current pixel)

         w = width of this image
         h = height of this image
         z = channel depth

         r = red value (from RGBA), of a specific or current pixel
         g = green    ''
         b = blue     ''
         a = alpha    ''
         o = opacity  ''

         c = cyan value of CMYK color of pixel
         y = yellow   ''
         m = magenta  ''
         k = black    ''

         intensity = pixel intensity

         hue = pixel hue
         saturation = pixel saturation
         lightness = pixel lightness
         luminance = pixel luminance

         page.width = page width
         page.height = page height
         page.x = page x offset
         page.y = page y offset

         resolution.x = x resolution
         resolution.y = y resolution

         depth = image depth
         minima = image minima
         maxima = image maxima
         mean = image mean
         standard_deviation = image standard deviation
           (add a channel specifier to compute a statistic for that channel, e.g. depth.r)

##### The Fx Expression

The Fx expressions includes any combination of the following:

- x `^` y: exponentiation (xy)
- `(` ... `)`: grouping
- x `*` y: multiplication (the asterisk `*` is optional, for example, `2u` or `2(x+y)` are acceptable)
- x `/` y: division
- x `%` y: modulo
- x `+` y: addition
- x `-` y: subtraction
- x `<<` y: left shift
- x `>>` y: right shift
- x `<` y: boolean relation, return value 1.0 if x < y, otherwise 0.0
- x `<=` y: boolean relation, return value 1.0 if x <= y, otherwise 0.0
- x `>` y: boolean relation, return value 1.0 if x > y, otherwise 0.0
- x `>=` y: boolean relation, return value 1.0 if x >= y, otherwise 0.0
- x `==` y: boolean relation, return value 1.0 if x == y, otherwise 0.0
- x `!=` y: boolean relation, return value 1.0 if x != y, otherwise 0.0
- x `&` y: binary AND
- x `|` y: binary OR
- x `&&` y: logical AND connective, return value 1.0 if x > 0 and y > 0, otherwise 0.0
- x `||` y: logical OR connective (inclusive), return value 1.0 if x > 0 or y > 0 (or both), otherwise 0.0
- `~`x: logical NOT operator, return value 1.0 if not x > 0, otherwise 0.0
- `+`x: unary plus, return 1.0*value
- `-`x: unary minus, return -1.0*value
- x `?` y : z: ternary conditional expression, return value y if x != 0, otherwise z; only one ternary conditional permitted per statement
- x `=` y: assignment; assignment variables are restricted to letter combinations only (e.g. Xi not X1)
- x `;` y: statement separator
- `pi`: constant (3.141659...)
- `e`: constant (2.71828...)
- `QuantumRange`: constant maximum pixel value (255 for Q8, 65535 for Q16)
- `QuantumScale`: constant 1.0/`QuantumRange`
- `intensity`: pixel intensity; equivalent to 0.299*red+0.587*green+0.114*blue
- `hue`: pixel hue
- `saturation`: pixel saturation
- `lightness`: pixel lightness; equivalent to 0.5*max(red,green,blue) + 0.5*min(red,green,blue)
- `luminance`: pixel luminance; equivalent to 0.2126*red + 0.7152*green + 0.0722*blue
- `red, green, blue`, etc.: color names
- `#ccc, #cbfed0, #b9e1cc00`, etc.: color hex values
- `rgb(), rgba(), cmyk(), cmyka(), hsl(), hsla()`: color functions
- `s, t, u, v, n, i, j, w, h, z, r, g, b, a, o, c, y, m, k`: symbols
- `abs(`x`)`: absolute value function
- `acos(`x`)`: arc cosine function
- `alt(`x`)`: sign alternation function (return 1.0 if `int(`x`)` is even, -1.0 if `int(`x`)` is odd)
- `asin(`x`)`: arc sine function
- `atan(`x`)`: arc tangent function
- `atan2(`y,x`)`: arc tangent function of two variables
- `ceil(`x`)`: smallest integral value not less than argument
- `channel(`r,g,b,a`)`: select numeric argument based on current channel
- `channel(`c,m,y,k,a`)`: select numeric argument based on current channel
- `cos(`x`)`: cosine function
- `cosh(`x`)`: hyperbolic cosine function
- `debug(`x`)`: print x (useful for debugging your expression)
- `exp(`x`)`: natural exponential function (ex)
- `floor(`x`)`: largest integral value not greater than argument
- `hypot(`x,y`)`: the square root of x2+y2
- `int(`x`)`: greatest integer function (return greatest integer less than or equal to x)
- `ln(`x`)`: natural logarithm function
- `log(`x`)`: logarithm base 10
- `logtwo(`x`)`: logarithm base 2
- `ln(`x`)`: natural logarithm
- `max(`x, y`)`: maximum of x and y
- `min(`x, y`)`: minimum of x and y
- `mod(`x, y`)`: floating-point remainder function
- `pow(`x,y`)`: power function (xy)
- `rand()`: value uniformly distributed over the interval [0.0, 1.0) with a 2 to the 128th-1 period
- `round()`: round to integral value, regardless of rounding direction
- `sign(`x`)`: return -1.0 if x is less than 0.0 otherwise 1.0
- `sin(`x`)`: sine function
- `sinh(`x`)`: hyperbolic sine function
- `sqrt(`x`)`: square root function
- `tan(`x`)`: tangent function
- `tanh(`x`)`: hyperbolic tangent function

The expression semantics include these rules:

- symbols are case insensitive
- only one ternary conditional (e.g. x ? y : z) per statement
- statements are assignments or the final expression to return
- an assignment starts a statement, it is not an operator
- assignments to built-ins do not throw errors and have no effect; e.g. `r=3.0; r` returns the pixel red color value, not 3.0

##### Source Image

The default image for `p`, `r`, `g`, `b`, `a`, etc. is `s` which refers to the current image being looked at in %[fx: ] and %[pixel: ] escapes. The `-fx` image `s` defaults to the first (zeroth) image `u`. The index of `s` is given by `t`.

The symbol `u` refers to the first (zeroth) image in the current image sequence, `v` the second image. Refer to a particular image in a sequence by appending its index to any image reference (usually `u`). A negative index counts from the end. For example, `u[2]` is the third image the sequence, `u[-1]` is the last image, and `u[t]` is `s`.

Here we reduce the intensity of the red channel by 50%:

      convert image.png -channel red -fx "u/2.0" image.jpg

##### Accessing Pixels

All color values are in the range of 0.0 to 1.0. The alpha channel ranges from 0.0 (fully transparent) to 1.0 (fully opaque).

The pixels are processed one at a time, but a different pixel of a image can be specified with a pixel index represented by `p`. For example,

       p[-1].g      green value of pixel to the immediate left of current
       p[-1,-1].r   red value, diagonally left and up from current pixel

To specify an absolute position, use braces, rather than brackets.

      p{12,34}.b   blue pixel value at the 12th column and 34th row of the image

The other symbols specify the value you wish to retrieve.

Integer pixel references retrieve the color of the pixel referred to, whereas non-integer references return a blended color according to the current -interpolate setting.

A pixel outside the boundary of the image has a value dictated by the -virtual-pixel option setting.

##### Apply an Expression to Select Image Channels

Use the -channel setting to specify the output channel of the result. If no output channel is given, the result is set over all channels except the opacity channel. For example, suppose you want to replace the red channel of `alpha.png` with the average of the green channels from the images `alpha.png` and `beta.png`, use:

      convert alpha.png beta.png -channel red -fx "(u.g+v.g)/2" gamma.png

##### Results

The `-fx` operator evaluates the given expression once for each pixel in the first image (u) in the sequence for each channel. The computed value is placed in a copy (clone) of that first image. This new image replaces all images in the current image sequence. As such in the previous example, the updated `alpha.png` replaces both the original `alpha.png` and `beta.png` images, before being saved as `gamma.png`.

The current image `s` is initially set to the first image in the sequence (`u`), and `t` to its index, 0. The symbols `i` and `j` define the current pixel being processed.

The value escape `%[fx: ]` is evaluated once only for each image in the current image sequence, with `s` and `t` specifying which image is being referred to `i` and `j` is set to zero and the current channel is set to red (-channel is ignored).

The color escape `%[pixel: ]` is evaluated once per image and per color channel in that image (-channel is ignored), The values generated are then converted into a color string (named color or hex color value). The symbols `i` and `j` are set to zero and `s` and `t` define the current image.

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