PNG (Portable Network Graphics) Specification, Version 1.1


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10. Recommendations for Decoders

This chapter gives some recommendations for decoder behavior. The only absolute requirement on a PNG decoder is that it successfully read any file conforming to the format specified in the preceding chapters. However, best results will usually be achieved by following these recommendations.

10.1. Error checking

To ensure early detection of common file-transfer problems, decoders should verify that all eight bytes of the PNG file signature are correct. (See Rationale: PNG file signature.) A decoder can have additional confidence in the file's integrity if the next eight bytes are an IHDR chunk header with the correct chunk length.

Unknown chunk types must be handled as described in Chunk naming conventions. An unknown chunk type is not to be treated as an error unless it is a critical chunk.

It is strongly recommended that decoders should verify the CRC on each chunk.

In some situations it is desirable to check chunk headers (length and type code) before reading the chunk data and CRC. The chunk type can be checked for plausibility by seeing whether all four bytes are ASCII letters (codes 65-90 and 97-122); note that this need be done only for unrecognized type codes. If the total file size is known (from file system information, HTTP protocol, etc), the chunk length can be checked for plausibility as well.

If CRCs are not checked, dropped/added data bytes or an erroneous chunk length can cause the decoder to get out of step and misinterpret subsequent data as a chunk header. Verifying that the chunk type contains letters is an inexpensive way of providing early error detection in this situation.

For known-length chunks such as IHDR, decoders should treat an unexpected chunk length as an error. Future extensions to this specification will not add new fields to existing chunks; instead, new chunk types will be added to carry new information.

Unexpected values in fields of known chunks (for example, an unexpected compression method in the IHDR chunk) must be checked for and treated as errors. However, it is recommended that unexpected field values be treated as fatal errors only in critical chunks. An unexpected value in an ancillary chunk can be handled by ignoring the whole chunk as though it were an unknown chunk type. (This recommendation assumes that the chunk's CRC has been verified. In decoders that do not check CRCs, it is safer to treat any unexpected value as indicating a corrupted file.)

10.2. Pixel dimensions

Non-square pixels can be represented (see the pHYs chunk), but viewers are not required to account for them; a viewer can present any PNG file as though its pixels are square.

Conversely, viewers running on display hardware with non-square pixels are strongly encouraged to rescale images for proper display.

10.3. Truecolor image handling

To achieve PNG's goal of universal interchangeability, decoders are required to accept all types of PNG image: indexed-color, truecolor, and grayscale. Viewers running on indexed-color display hardware need to be able to reduce truecolor images to indexed format for viewing. This process is usually called "color quantization".

A simple, fast way of doing this is to reduce the image to a fixed palette. Palettes with uniform color spacing ("color cubes") are usually used to minimize the per-pixel computation. For photograph-like images, dithering is recommended to avoid ugly contours in what should be smooth gradients; however, dithering introduces graininess that can be objectionable.

The quality of rendering can be improved substantially by using a palette chosen specifically for the image, since a color cube usually has numerous entries that are unused in any particular image. This approach requires more work, first in choosing the palette, and second in mapping individual pixels to the closest available color. PNG allows the encoder to supply suggested palettes, but not all encoders will do so, and the suggested palettes may be unsuitable in any case (they may have too many or too few colors). High-quality viewers will therefore need to have a palette selection routine at hand. A large lookup table is usually the most feasible way of mapping individual pixels to palette entries with adequate speed.

Numerous implementations of color quantization are available. The PNG reference implementation, libpng, includes code for the purpose.

10.4. Sample depth rescaling

Decoders may wish to scale PNG data to a lesser sample depth (data precision) for display. For example, 16-bit data will need to be reduced to 8-bit depth for use on most present-day display hardware. Reduction of 8-bit data to 5-bit depth is also common.

The most accurate scaling is achieved by the linear equation

   output = ROUND(input * MAXOUTSAMPLE / MAXINSAMPLE)

where

   MAXINSAMPLE = (2^sampledepth)-1
   MAXOUTSAMPLE = (2^desired_sampledepth)-1

A slightly less accurate conversion is achieved by simply shifting right by sampledepth - desired_sampledepth places. For example, to reduce 16-bit samples to 8-bit, one need only discard the low-order byte. In many situations the shift method is sufficiently accurate for display purposes, and it is certainly much faster. (But if gamma correction is being done, sample rescaling can be merged into the gamma correction lookup table, as is illustrated in Decoder gamma handling.)

When an sBIT chunk is present, the original pre-PNG data can be recovered by shifting right to the sample depth specified by sBIT. Note that linear scaling will not necessarily reproduce the original data, because the encoder is not required to have used linear scaling to scale the data up. However, the encoder is required to have used a method that preserves the high-order bits, so shifting always works. This is the only case in which shifting might be said to be more accurate than linear scaling.

When comparing pixel values to tRNS chunk values to detect transparent pixels, it is necessary to do the comparison exactly. Therefore, transparent pixel detection must be done before reducing sample precision.

10.5. Decoder gamma handling

See Gamma Tutorial if you aren't already familiar with gamma issues.

Decoders capable of full-fledged color management [ICC] will perform more sophisticated calculations than what is described here. Otherwise, this section applies.

For an image display program to produce correct tone reproduction, it is necessary to take into account the relationship between file samples and display output, and the transfer function of the display system. This can be done by calculating

   sample = integer_sample / (2^bitdepth - 1.0)
   display_output = sample ^ (1.0 / gamma)
   display_input = inverse_display_transfer(display_output)
   framebuf_sample = ROUND(display_input * MAX_FRAMEBUF_SAMPLE)

where integer_sample is the sample value from the file, framebuf_sample is the value to write into the frame buffer, and MAX_FRAMEBUF_SAMPLE is the maximum value of a frame buffer sample (255 for 8-bit, 31 for 5-bit, etc). The first line converts an integer sample into a normalized 0-to-1 floating-point value, the second converts to a value proportional to the desired display output intensity, the third accounts for the display system's transfer function, and the fourth converts to an integer frame buffer sample.

A step could be inserted between the second and third to adjust display_output to account for the difference between the actual viewing conditions and the reference viewing conditions. However, this adjustment requires accounting for veiling glare, black mapping, and color appearance models, none of which can be well approximated by power functions. The calculations are not described here. If viewing conditions are ignored, the error will usually be small.

Typically, the display transfer function can be approximated by a power function with exponent display_exponent, in which case the second and third lines can be merged into

   display_input = sample ^ (1.0 / (gamma * display_exponent))
                 = sample ^ decoding_exponent

so as to perform only one power calculation. For color images, the entire calculation is performed separately for R, G, and B values.

The value of gamma can be taken directly from the gAMA chunk. Alternatively, an application may wish to allow the user to adjust the appearance of the displayed image by influencing the value of gamma. For example, the user could manually set a parameter called user_exponent that defaults to 1.0, and the application could set

   gamma = gamma_from_file / user_exponent
   decoding_exponent = 1.0 / (gamma * display_exponent)
         = user_exponent / (gamma_from_file * display_exponent)

The user would set user_exponent greater than 1 to darken the mid-level tones, or less than 1 to lighten them.

It is not necessary to perform transcendental math for every pixel. Instead, compute a lookup table that gives the correct output value for every possible sample value. This requires only 256 calculations per image (for 8-bit accuracy), not one or three calculations per pixel. For an indexed-color image, a one-time correction of the palette is sufficient, unless the image uses transparency and is being displayed against a nonuniform background.

If floating-point calculations are not possible, gamma correction tables can be computed using integer arithmetic and a precomputed table of logarithms. Example code appears in the "Extensions to the PNG Specification" document [PNG-EXTENSIONS].

When the incoming image has unknown gamma (gAMA, sRGB, and iCCP all absent), choose a likely default gamma value, but allow the user to select a new one if the result proves too dark or too light. The default gamma can depend on other knowledge about the image, like whether it came from the Internet or from the local system.

In practice, it is often difficult to determine what value of the display exponent should be used. In systems with no built-in gamma correction, the display exponent is determined entirely by the CRT. Assume a CRT exponent of 2.2 unless detailed calibration measurements of this particular CRT are available.

Many modern frame buffers have lookup tables that are used to perform gamma correction, and on these systems the display exponent value should be the exponent of the lookup table and CRT combined. You may not be able to find out what the lookup table contains from within an image viewer application, so you may have to ask the user what the display system's exponent is. Unfortunately, different manufacturers use different ways of specifying what should go into the lookup table, so interpretation of the system "gamma" value is system-dependent. The Gamma Tutorial gives some examples.

The response of real displays is actually more complex than can be described by a single number (the display exponent). If actual measurements of the monitor's light output as a function of voltage input are available, the third and fourth lines of the computation above can be replaced by a lookup in these measurements, to find the actual frame buffer value that most nearly gives the desired brightness.

10.6. Decoder color handling

See Color Tutorial if you aren't already familiar with color issues.

In many cases, decoders will treat image data in PNG files as device-dependent RGB data and display it without modification (except for appropriate gamma correction). This provides the fastest display of PNG images. But unless the viewer uses exactly the same display hardware as the original image author used, the colors will not be exactly the same as the original author saw, particularly for darker or near-neutral colors. The cHRM chunk provides information that allows closer color matching than that provided by gamma correction alone.

Decoders can use the cHRM data to transform the image data from RGB to CIE XYZ and thence into a perceptually linear color space such as CIE LAB. They can then partition the colors to generate an optimal palette, because the geometric distance between two colors in CIE LAB is strongly related to how different those colors appear (unlike, for example, RGB or XYZ spaces). The resulting palette of colors, once transformed back into RGB color space, could be used for display or written into a PLTE chunk.

Decoders that are part of image processing applications might also transform image data into CIE LAB space for analysis.

In applications where color fidelity is critical, such as product design, scientific visualization, medicine, architecture, or advertising, decoders can transform the image data from source RGB to the display RGB space of the monitor used to view the image. This involves calculating the matrix to go from source RGB to XYZ and the matrix to go from XYZ to display RGB, then combining them to produce the overall transformation. The decoder is responsible for implementing gamut mapping.

Decoders running on platforms that have a Color Management System (CMS) can pass the image data, gAMA, and cHRM values to the CMS for display or further processing.

Decoders that provide color printing facilities can use the facilities in Level 2 PostScript to specify image data in calibrated RGB space or in a device-independent color space such as XYZ. This will provide better color fidelity than a simple RGB to CMYK conversion. The PostScript Language Reference manual [POSTSCRIPT] gives examples. Such decoders are responsible for implementing gamut mapping between source RGB (specified in the cHRM chunk) and the target printer. The PostScript interpreter is then responsible for producing the required colors.

Decoders can use the cHRM data to calculate an accurate grayscale representation of a color image. Conversion from RGB to gray is simply a case of calculating the Y (luminance) component of XYZ, which is a weighted sum of the R, G, and B values. The weights depend on the monitor type, i.e., the values in the cHRM chunk. Decoders may wish to do this for PNG files with no cHRM chunk. In that case, a reasonable default would be the CCIR 709 primaries [ITU-BT709]. Do not use the original NTSC primaries, unless you really do have an image color-balanced for such a monitor. Few monitors ever used the NTSC primaries, so such images are probably nonexistent these days.

10.7. Background color

The background color given by bKGD will typically be used to fill unused screen space around the image, as well as any transparent pixels within the image. (Thus, bKGD is valid and useful even when the image does not use transparency.) If no bKGD chunk is present, the viewer will need to make its own decision about a suitable background color.

Viewers that have a specific background against which to present the image (such as Web browsers) should ignore the bKGD chunk, in effect overriding bKGD with their preferred background color or background image.

The background color given by bKGD is not to be considered transparent, even if it happens to match the color given by tRNS (or, in the case of an indexed-color image, refers to a palette index that is marked as transparent by tRNS). Otherwise one would have to imagine something "behind the background" to composite against. The background color is either used as background or ignored; it is not an intermediate layer between the PNG image and some other background.

Indeed, it will be common that bKGD and tRNS specify the same color, since then a decoder that does not implement transparency processing will give the intended display, at least when no partially-transparent pixels are present.

10.8. Alpha channel processing

In the most general case, the alpha channel can be used to composite a foreground image against a background image; the PNG file defines the foreground image and the transparency mask, but not the background image. Decoders are not required to support this most general case. It is expected that most will be able to support compositing against a single background color, however.

The equation for computing a composited sample value is

   output = alpha * foreground + (1-alpha) * background

where the alpha value and the input and output sample values are expressed as fractions in the range 0 to 1. This computation should be performed with intensity samples (not gamma-encoded samples). For color images, the computation is done separately for R, G, and B samples.

The following code illustrates the general case of compositing a foreground image over a background image. It assumes that you have the original pixel data available for the background image, and that output is to a frame buffer for display. Other variants are possible; see the comments below the code. The code allows the sample depths and gamma values of foreground and background images to be different, and not necessarily suited to the display system. Don't assume everything is the same without checking.

This code is standard C, with line numbers added for reference in the comments below.

   01  int foreground[4];  /* image pixel: R, G, B, A */
   02  int background[3];  /* background pixel: R, G, B */
   03  int fbpix[3];       /* frame buffer pixel */
   04  int fg_maxsample;   /* foreground max sample */
   05  int bg_maxsample;   /* background max sample */
   06  int fb_maxsample;   /* frame buffer max sample */
   07  int ialpha;
   08  float alpha, compalpha;
   09  float gamfg, linfg, gambg, linbg, comppix, gcvideo;
   
       /* Get max sample values in data and frame buffer */
   10  fg_maxsample = (1 << fg_sample_depth) - 1;
   11  bg_maxsample = (1 << bg_sample_depth) - 1;
   12  fb_maxsample = (1 << frame_buffer_sample_depth) - 1;
       /*
        * Get integer version of alpha.
        * Check for opaque and transparent special cases;
        * no compositing needed if so.
        *
        * We show the whole gamma decode/correct process in
        * floating point, but it would more likely be done
        * with lookup tables.
        */
   13  ialpha = foreground[3];
   
   14  if (ialpha == 0) {
           /*
            * Foreground image is transparent here.
            * If the background image is already in the frame
            * buffer, there is nothing to do.
            */
   15      ;
   16  } else if (ialpha == fg_maxsample) {
           /*
            * Copy foreground pixel to frame buffer.
            */
   17      for (i = 0; i < 3; i++) {
   18          gamfg = (float) foreground[i] / fg_maxsample;
   19          linfg = pow(gamfg, 1.0/fg_gamma);
   20          comppix = linfg;
   21          gcvideo = pow(comppix, 1.0/display_exponent);
   22          fbpix[i] = (int) (gcvideo * fb_maxsample + 0.5);
   23      }
   24  } else {
           /*
            * Compositing is necessary.
            * Get floating-point alpha and its complement.
            * Note: alpha is always linear; gamma does not
            * affect it.
            */
   25      alpha = (float) ialpha / fg_maxsample;
   26      compalpha = 1.0 - alpha;
   
   27      for (i = 0; i < 3; i++) {
               /*
                * Convert foreground and background to
                * floating point, then undo gamma encoding.
                */
   28          gamfg = (float) foreground[i] / fg_maxsample;
   29          linfg = pow(gamfg, 1.0/fg_gamma);
   30          gambg = (float) background[i] / bg_maxsample;
   31          linbg = pow(gambg, 1.0/bg_gamma);
               /*
                * Composite.
                */
   32          comppix = linfg * alpha + linbg * compalpha;
               /*
                * Gamma correct for display.
                * Convert to integer frame buffer pixel.
                */
   33          gcvideo = pow(comppix, 1.0/display_exponent);
   34          fbpix[i] = (int) (gcvideo * fb_maxsample + 0.5);
   35      }
   36  }

Variations:

  1. If output is to another PNG image file instead of a frame buffer, lines 21, 22, 33, and 34 should be changed to be something like
       /*
        * Gamma encode for storage in output file.
        * Convert to integer sample value.
        */
       gamout = pow(comppix, outfile_gamma);
       outpix[i] = (int) (gamout * out_maxsample + 0.5);
    

    Also, it becomes necessary to process background pixels when alpha is zero, rather than just skipping pixels. Thus, line 15 will need to be replaced by copies of lines 17-23, but processing background instead of foreground pixel values.

  2. If the sample depths of the output file, foreground file, and background file are all the same, and the three gamma values also match, then the no-compositing code in lines 14-23 reduces to nothing more than copying pixel values from the input file to the output file if alpha is one, or copying pixel values from background to output file if alpha is zero. Since alpha is typically either zero or one for the vast majority of pixels in an image, this is a great savings. No gamma computations are needed for most pixels.

  3. When the sample depths and gamma values all match, it may appear attractive to skip the gamma decoding and encoding (lines 28-31, 33-34) and just perform line 32 using gamma-encoded sample values. Although this doesn't hurt image quality too badly, the time savings are small if alpha values of zero and one are special-cased as recommended here.

  4. If the original pixel values of the background image are no longer available, only processed frame buffer pixels left by display of the background image, then lines 30 and 31 need to extract intensity from the frame buffer pixel values using code like
       /*
        * Convert frame buffer value into intensity sample.
        */
       gcvideo = (float) fbpix[i] / fb_maxsample;
       linbg = pow(gcvideo, display_exponent);
    

    However, some roundoff error can result, so it is better to have the original background pixels available if at all possible.

  5. Note that lines 18-22 are performing exactly the same gamma computation that is done when no alpha channel is present. So, if you handle the no-alpha case with a lookup table, you can use the same lookup table here. Lines 28-31 and 33-34 can also be done with (different) lookup tables.

  6. Of course, everything here can be done in integer arithmetic. Just be careful to maintain sufficient precision all the way through.

Note: in floating-point arithmetic, no overflow or underflow checks are needed, because the input sample values are guaranteed to be between 0 and 1, and compositing always yields a result that is in between the input values (inclusive). With integer arithmetic, some roundoff-error analysis might be needed to guarantee no overflow or underflow.

When displaying a PNG image with full alpha channel, it is important to be able to composite the image against some background, even if it's only black. Ignoring the alpha channel will cause PNG images that have been converted from an associated-alpha representation to look wrong. (Of course, if the alpha channel is a separate transparency mask, then ignoring alpha is a useful option: it allows the hidden parts of the image to be recovered.)

Even if the decoder author does not wish to implement true compositing logic, it is simple to deal with images that contain only zero and one alpha values. (This is implicitly true for grayscale and truecolor PNG files that use a tRNS chunk; for indexed-color PNG files, it is easy to check whether tRNS contains any values other than 0 and 255.) In this simple case, transparent pixels are replaced by the background color, while others are unchanged.

If a decoder contains only this much transparency capability, it should deal with a full alpha channel by converting it to a binary alpha channel, either by treating all nonzero alpha values as fully opaque or by dithering. Neither approach will yield very good results for images converted from associated-alpha formats, but it's better than doing nothing. Dithering full alpha to binary alpha is very much like dithering grayscale to black-and-white, except that all fully transparent and fully opaque pixels should be left unchanged by the dither.

10.9. Progressive display

When receiving images over slow transmission links, decoders can improve perceived performance by displaying interlaced images progressively. This means that as each pass is received, an approximation to the complete image is displayed based on the data received so far. One simple yet pleasing effect can be obtained by expanding each received pixel to fill a rectangle covering the yet-to-be-transmitted pixel positions below and to the right of the received pixel. This process can be described by the following pseudocode:

   Starting_Row [1..7] =  { 0, 0, 4, 0, 2, 0, 1 }
   Starting_Col [1..7] =  { 0, 4, 0, 2, 0, 1, 0 }
   Row_Increment [1..7] = { 8, 8, 8, 4, 4, 2, 2 }
   Col_Increment [1..7] = { 8, 8, 4, 4, 2, 2, 1 }
   Block_Height [1..7] =  { 8, 8, 4, 4, 2, 2, 1 }
   Block_Width [1..7] =   { 8, 4, 4, 2, 2, 1, 1 }
   
   pass := 1
   while pass <= 7
   begin
       row := Starting_Row[pass]
   
       while row < height
       begin
           col := Starting_Col[pass]
   
           while col < width
           begin
               visit (row, col,
                      min (Block_Height[pass], height - row),
                      min (Block_Width[pass], width - col))
               col := col + Col_Increment[pass]
           end
           row := row + Row_Increment[pass]
       end
   
       pass := pass + 1
   end

Here, the function visit(row,column,height,width) obtains the next transmitted pixel and paints a rectangle of the specified height and width, whose upper-left corner is at the specified row and column, using the color indicated by the pixel. Note that row and column are measured from 0,0 at the upper left corner.

If the decoder is merging the received image with a background image, it may be more convenient just to paint the received pixel positions; that is, the visit() function sets only the pixel at the specified row and column, not the whole rectangle. This produces a "fade-in" effect as the new image gradually replaces the old. An advantage of this approach is that proper alpha or transparency processing can be done as each pixel is replaced. Painting a rectangle as described above will overwrite background-image pixels that may be needed later, if the pixels eventually received for those positions turn out to be wholly or partially transparent. Of course, this is a problem only if the background image is not stored anywhere offscreen.

10.10. Suggested-palette and histogram usage

For viewers running on indexed-color hardware trying to display a truecolor image, or an indexed-color image whose palette is too large for the framebuffer, the encoder may have provided one or more suggested palettes in sPLT chunks. If one of them is found to be suitable, based on its size and perhaps its name, the decoder can use that palette. Note that suggested palettes with a sample depth different from what the decoder needs can be converted using sample depth rescaling (See Recommendations for Decoders: Sample depth rescaling).

When the background is a solid color, the decoder should composite the image and the suggested palette against that color, then quantize the resulting image to the resulting RGB palette. When the image uses transparency and the background is not a solid color, no suggested palette is likely to be useful.

For truecolor images, a suggested palette might also be provided in a PLTE chunk. If the image has a tRNS chunk and the background is a solid color, the viewer can adapt the suggested paletted for use with this background color. To do this, replace the palette entry closest to the tRNS color with the background color, or just add a palette entry for the background color if the viewer can handle more colors than there are palette entries.

For images of color type 6 (truecolor with alpha channel), any PLTE chunk should have been designed for display of the image against a uniform background of the color specified by bKGD. Viewers should probably ignore the palette if they intend to use a different background, or if the bKGD chunk is missing. Viewers can use the suggested palette for display against a different background than it was intended for, but the results may not be very good.

If the viewer presents a transparent truecolor image against a background that is more complex than a single color, it is unlikely that the PLTE chunk will be optimal for the composite image. In this case it is best to perform a truecolor compositing step on the truecolor PNG image and background image, then color-quantize the resulting image.

In truecolor PNG files, if both PLTE and sPLT appear, the decoder can choose from among the palettes suggested by both, bearing in mind the different transparency semantics mentioned above.

The frequencies in sPLT and hIST chunks are useful when the viewer cannot provide as many colors as are used in the palette. If the viewer is short only a few colors, it is usually adequate to drop the least-used colors from the palette. To reduce the number of colors substantially, it's best to choose entirely new representative colors, rather than trying to use a subset of the existing palette. This amounts to performing a new color quantization step; however, the existing palette and frequencies can be used as the input data, thus avoiding a scan of the image data.

If no suggested palettes are provided, a decoder can develop its own, at the cost of an extra pass over the image data. Alternatively, a default palette (probably a color cube) can be used.

See also Recommendations for Encoders: Suggested palettes.

10.11. Text chunk processing

If practical, decoders should have a way to display to the user all tEXt and zTXt chunks found in the file. Even if the decoder does not recognize a particular text keyword, the user might be able to understand it.

PNG text is not supposed to contain any characters outside the ISO 8859-1 (Latin-1) character set (that is, no codes 0-31 or 127-159), except for the newline character (decimal 10). But decoders might encounter such characters anyway. Some of these characters can be safely displayed (e.g., TAB, FF, and CR, decimal 9, 12, and 13, respectively), but others, especially the ESC character (decimal 27), could pose a security hazard because unexpected actions may be taken by display hardware or software. To prevent such hazards, decoders should not attempt to directly display any non-Latin-1 characters (except for newline and perhaps TAB, FF, CR) encountered in a tEXt or zTXt chunk. Instead, ignore them or display them in a visible notation such as "\nnn". See Security considerations.

Even though encoders are supposed to represent newlines as LF, it is recommended that decoders not rely on this; it's best to recognize all the common newline combinations (CR, LF, and CR-LF) and display each as a single newline. TAB can be expanded to the proper number of spaces needed to arrive at a column multiple of 8.

Decoders running on systems with non-Latin-1 character set encoding should provide character code remapping so that Latin-1 characters are displayed correctly. Some systems may not provide all the characters defined in Latin-1. Mapping unavailable characters to a visible notation such as "\nnn" is a good fallback. In particular, character codes 127-255 should be displayed only if they are printable characters on the decoding system. Some systems may interpret such codes as control characters; for security, decoders running on such systems should not display such characters literally.

Decoders should be prepared to display text chunks that contain any number of printing characters between newline characters, even though encoders are encouraged to avoid creating lines in excess of 79 characters.


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