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#
# Reference arithmetic coding
# Copyright (c) Project Nayuki
#
# https://www.nayuki.io/page/reference-arithmetic-coding
# https://github.com/nayuki/Reference-arithmetic-coding
#
import sys
python3 = sys.version_info.major >= 3
# ---- Arithmetic coding core classes ----
# Provides the state and behaviors that arithmetic coding encoders and decoders share.
class ArithmeticCoderBase(object):
# Number of bits for 'low' and 'high'. Configurable and must be at least 1.
STATE_SIZE = 32
# Maximum range during coding (trivial), i.e. 1000...000.
MAX_RANGE = 1 << STATE_SIZE
# Minimum range during coding (non-trivial), i.e. 010...010.
MIN_RANGE = (MAX_RANGE >> 2) + 2
# Maximum allowed total frequency at all times during coding.
MAX_TOTAL = MIN_RANGE
# Mask of STATE_SIZE ones, i.e. 111...111.
MASK = MAX_RANGE - 1
# Mask of the top bit at width STATE_SIZE, i.e. 100...000.
TOP_MASK = MAX_RANGE >> 1
# Mask of the second highest bit at width STATE_SIZE, i.e. 010...000.
SECOND_MASK = TOP_MASK >> 1
# Constructs an arithmetic coder, which initializes the code range.
def __init__(self):
# Low end of this arithmetic coder's current range. Conceptually has an infinite number of trailing 0s.
self.low = 0
# High end of this arithmetic coder's current range. Conceptually has an infinite number of trailing 1s.
self.high = ArithmeticCoderBase.MASK
# Updates the code range (low and high) of this arithmetic coder as a result
# of processing the given symbol with the given frequency table.
# Invariants that are true before and after encoding/decoding each symbol:
# - 0 <= low <= code <= high < 2^STATE_SIZE. ('code' exists only in the decoder.)
# Therefore these variables are unsigned integers of STATE_SIZE bits.
# - (low < 1/2 * 2^STATE_SIZE) && (high >= 1/2 * 2^STATE_SIZE).
# In other words, they are in different halves of the full range.
# - (low < 1/4 * 2^STATE_SIZE) || (high >= 3/4 * 2^STATE_SIZE).
# In other words, they are not both in the middle two quarters.
# - Let range = high - low + 1, then MAX_RANGE/4 < MIN_RANGE <= range
# <= MAX_RANGE = 2^STATE_SIZE. These invariants for 'range' essentially
# dictate the maximum total that the incoming frequency table can have.
def update(self, freqs, symbol):
# State check
low = self.low
high = self.high
if low >= high or (low & ArithmeticCoderBase.MASK) != low or (high & ArithmeticCoderBase.MASK) != high:
raise AssertionError("Low or high out of range")
range = high - low + 1
if not (ArithmeticCoderBase.MIN_RANGE <= range <= ArithmeticCoderBase.MAX_RANGE):
raise AssertionError("Range out of range")
# Frequency table values check
total = freqs.get_total()
symlow = freqs.get_low(symbol)
symhigh = freqs.get_high(symbol)
if symlow == symhigh:
raise ValueError("Symbol has zero frequency")
if total > ArithmeticCoderBase.MAX_TOTAL:
raise ValueError("Cannot code symbol because total is too large")
# Update range
newlow = low + symlow * range // total
newhigh = low + symhigh * range // total - 1
self.low = newlow
self.high = newhigh
# While the highest bits are equal
while ((self.low ^ self.high) & ArithmeticCoderBase.TOP_MASK) == 0:
self.shift()
self.low = (self.low << 1) & ArithmeticCoderBase.MASK
self.high = ((self.high << 1) & ArithmeticCoderBase.MASK) | 1
# While the second highest bit of low is 1 and the second highest bit of high is 0
while (self.low & ~self.high & ArithmeticCoderBase.SECOND_MASK) != 0:
self.underflow()
self.low = (self.low << 1) & (ArithmeticCoderBase.MASK >> 1)
self.high = ((self.high << 1) & (ArithmeticCoderBase.MASK >> 1)) | ArithmeticCoderBase.TOP_MASK | 1
# Called to handle the situation when the top bit of 'low' and 'high' are equal.
def shift(self):
raise NotImplementedError()
# Called to handle the situation when low=01(...) and high=10(...).
def underflow(self):
raise NotImplementedError()
# Encodes symbols and writes to an arithmetic-coded bit stream.
class ArithmeticEncoder(ArithmeticCoderBase):
# Constructs an arithmetic coding encoder based on the given bit output stream.
def __init__(self, bitout):
super(ArithmeticEncoder, self).__init__()
# The underlying bit output stream.
self.output = bitout
# Number of saved underflow bits. This value can grow without bound.
self.num_underflow = 0
# Encodes the given symbol based on the given frequency table.
# This updates this arithmetic coder's state and may write out some bits.
def write(self, freqs, symbol):
if not isinstance(freqs, CheckedFrequencyTable):
freqs = CheckedFrequencyTable(freqs)
self.update(freqs, symbol)
# Terminates the arithmetic coding by flushing any buffered bits, so that the output can be decoded properly.
# It is important that this method must be called at the end of the each encoding process.
# Note that this method merely writes data to the underlying output stream but does not close it.
def finish(self):
self.output.write(1)
def shift(self):
bit = self.low >> (ArithmeticCoderBase.STATE_SIZE - 1)
self.output.write(bit)
# Write out the saved underflow bits
for i in range(self.num_underflow):
self.output.write(bit ^ 1)
self.num_underflow = 0
def underflow(self):
self.num_underflow += 1
# Reads from an arithmetic-coded bit stream and decodes symbols.
class ArithmeticDecoder(ArithmeticCoderBase):
# Constructs an arithmetic coding decoder based on the
# given bit input stream, and fills the code bits.
def __init__(self, bitin):
super(ArithmeticDecoder, self).__init__()
# The underlying bit input stream.
self.input = bitin
# The current raw code bits being buffered, which is always in the range [low, high].
self.code = 0
for i in range(ArithmeticCoderBase.STATE_SIZE):
self.code = self.code << 1 | self.read_code_bit()
# Decodes the next symbol based on the given frequency table and returns it.
# Also updates this arithmetic coder's state and may read in some bits.
def read(self, freqs):
if not isinstance(freqs, CheckedFrequencyTable):
freqs = CheckedFrequencyTable(freqs)
# Translate from coding range scale to frequency table scale
total = freqs.get_total()
if total > ArithmeticCoderBase.MAX_TOTAL:
raise ValueError("Cannot decode symbol because total is too large")
range = self.high - self.low + 1
offset = self.code - self.low
value = ((offset + 1) * total - 1) // range
assert value * range // total <= offset
assert 0 <= value < total
# A kind of binary search. Find highest symbol such that freqs.get_low(symbol) <= value.
start = 0
end = freqs.get_symbol_limit()
while end - start > 1:
middle = (start + end) >> 1
if freqs.get_low(middle) > value:
end = middle
else:
start = middle
assert start + 1 == end
symbol = start
assert freqs.get_low(symbol) * range // total <= offset < freqs.get_high(symbol) * range // total
self.update(freqs, symbol)
if not (self.low <= self.code <= self.high):
raise AssertionError("Code out of range")
return symbol
def shift(self):
self.code = ((self.code << 1) & ArithmeticCoderBase.MASK) | self.read_code_bit()
def underflow(self):
self.code = (self.code & ArithmeticCoderBase.TOP_MASK) | ((self.code << 1) & (ArithmeticCoderBase.MASK >> 1)) | self.read_code_bit()
# Returns the next bit (0 or 1) from the input stream. The end
# of stream is treated as an infinite number of trailing zeros.
def read_code_bit(self):
temp = self.input.read()
if temp == -1:
temp = 0
return temp
# ---- Frequency table classes ----
# A table of symbol frequencies. The table holds data for symbols numbered from 0
# to get_symbol_limit()-1. Each symbol has a frequency, which is a non-negative integer.
# Frequency table objects are primarily used for getting cumulative symbol
# frequencies. These objects can be mutable depending on the implementation.
class FrequencyTable(object):
# Returns the number of symbols in this frequency table, which is a positive number.
def get_symbol_limit(self):
raise NotImplementedError()
# Returns the frequency of the given symbol. The returned value is at least 0.
def get(self, symbol):
raise NotImplementedError()
# Sets the frequency of the given symbol to the given value.
# The frequency value must be at least 0.
def set(self, symbol, freq):
raise NotImplementedError()
# Increments the frequency of the given symbol.
def increment(self, symbol):
raise NotImplementedError()
# Returns the total of all symbol frequencies. The returned value is at
# least 0 and is always equal to get_high(get_symbol_limit() - 1).
def get_total(self):
raise NotImplementedError()
# Returns the sum of the frequencies of all the symbols strictly
# below the given symbol value. The returned value is at least 0.
def get_low(self, symbol):
raise NotImplementedError()
# Returns the sum of the frequencies of the given symbol
# and all the symbols below. The returned value is at least 0.
def get_high(self, symbol):
raise NotImplementedError()
# An immutable frequency table where every symbol has the same frequency of 1.
# Useful as a fallback model when no statistics are available.
class FlatFrequencyTable(FrequencyTable):
# Constructs a flat frequency table with the given number of symbols.
def __init__(self, numsyms):
if numsyms < 1:
raise ValueError("Number of symbols must be positive")
self.numsymbols = numsyms # Total number of symbols, which is at least 1
# Returns the number of symbols in this table, which is at least 1.
def get_symbol_limit(self):
return self.numsymbols
# Returns the frequency of the given symbol, which is always 1.
def get(self, symbol):
self._check_symbol(symbol)
return 1
# Returns the total of all symbol frequencies, which is
# always equal to the number of symbols in this table.
def get_total(self):
return self.numsymbols
# Returns the sum of the frequencies of all the symbols strictly below
# the given symbol value. The returned value is equal to 'symbol'.
def get_low(self, symbol):
self._check_symbol(symbol)
return symbol
# Returns the sum of the frequencies of the given symbol and all
# the symbols below. The returned value is equal to 'symbol' + 1.
def get_high(self, symbol):
self._check_symbol(symbol)
return symbol + 1
# Returns silently if 0 <= symbol < numsymbols, otherwise raises an exception.
def _check_symbol(self, symbol):
if 0 <= symbol < self.numsymbols:
return
else:
raise ValueError("Symbol out of range")
# Returns a string representation of this frequency table. The format is subject to change.
def __str__(self):
return "FlatFrequencyTable={}".format(self.numsymbols)
# Unsupported operation, because this frequency table is immutable.
def set(self, symbol, freq):
raise NotImplementedError()
# Unsupported operation, because this frequency table is immutable.
def increment(self, symbol):
raise NotImplementedError()
# A mutable table of symbol frequencies. The number of symbols cannot be changed
# after construction. The current algorithm for calculating cumulative frequencies
# takes linear time, but there exist faster algorithms such as Fenwick trees.
class SimpleFrequencyTable(FrequencyTable):
# Constructs a simple frequency table in one of two ways:
# - SimpleFrequencyTable(sequence):
# Builds a frequency table from the given sequence of symbol frequencies.
# There must be at least 1 symbol, and no symbol has a negative frequency.
# - SimpleFrequencyTable(freqtable):
# Builds a frequency table by copying the given frequency table.
def __init__(self, freqs):
if isinstance(freqs, FrequencyTable):
numsym = freqs.get_symbol_limit()
self.frequencies = [freqs.get(i) for i in range(numsym)]
else: # Assume it is a sequence type
self.frequencies = list(freqs) # Make copy
# 'frequencies' is a list of the frequency for each symbol.
# Its length is at least 1, and each element is non-negative.
if len(self.frequencies) < 1:
raise ValueError("At least 1 symbol needed")
for freq in self.frequencies:
if freq < 0:
raise ValueError("Negative frequency")
# Always equal to the sum of 'frequencies'
self.total = sum(self.frequencies)
# cumulative[i] is the sum of 'frequencies' from 0 (inclusive) to i (exclusive).
# Initialized lazily. When it is not None, the data is valid.
self.cumulative = None
# Returns the number of symbols in this frequency table, which is at least 1.
def get_symbol_limit(self):
return len(self.frequencies)
# Returns the frequency of the given symbol. The returned value is at least 0.
def get(self, symbol):
self._check_symbol(symbol)
return self.frequencies[symbol]
# Sets the frequency of the given symbol to the given value. The frequency value
# must be at least 0. If an exception is raised, then the state is left unchanged.
def set(self, symbol, freq):
self._check_symbol(symbol)
if freq < 0:
raise ValueError("Negative frequency")
temp = self.total - self.frequencies[symbol]
assert temp >= 0
self.total = temp + freq
self.frequencies[symbol] = freq
self.cumulative = None
# Increments the frequency of the given symbol.
def increment(self, symbol):
self._check_symbol(symbol)
self.total += 1
self.frequencies[symbol] += 1
self.cumulative = None
# Returns the total of all symbol frequencies. The returned value is at
# least 0 and is always equal to get_high(get_symbol_limit() - 1).
def get_total(self):
return self.total
# Returns the sum of the frequencies of all the symbols strictly
# below the given symbol value. The returned value is at least 0.
def get_low(self, symbol):
self._check_symbol(symbol)
if self.cumulative is None:
self._init_cumulative()
return self.cumulative[symbol]
# Returns the sum of the frequencies of the given symbol
# and all the symbols below. The returned value is at least 0.
def get_high(self, symbol):
self._check_symbol(symbol)
if self.cumulative is None:
self._init_cumulative()
return self.cumulative[symbol + 1]
# Recomputes the array of cumulative symbol frequencies.
def _init_cumulative(self):
cumul = [0]
sum = 0
for freq in self.frequencies:
sum += freq
cumul.append(sum)
assert sum == self.total
self.cumulative = cumul
# Returns silently if 0 <= symbol < len(frequencies), otherwise raises an exception.
def _check_symbol(self, symbol):
if 0 <= symbol < len(self.frequencies):
return
else:
raise ValueError("Symbol out of range")
# Returns a string representation of this frequency table,
# useful for debugging only, and the format is subject to change.
def __str__(self):
result = ""
for (i, freq) in enumerate(self.frequencies):
result += "{}\t{}\n".format(i, freq)
return result
# A wrapper that checks the preconditions (arguments) and postconditions (return value) of all
# the frequency table methods. Useful for finding faults in a frequency table implementation.
class CheckedFrequencyTable(FrequencyTable):
def __init__(self, freqtab):
# The underlying frequency table that holds the data
self.freqtable = freqtab
def get_symbol_limit(self):
result = self.freqtable.get_symbol_limit()
if result <= 0:
raise AssertionError("Non-positive symbol limit")
return result
def get(self, symbol):
result = self.freqtable.get(symbol)
if not self._is_symbol_in_range(symbol):
raise AssertionError("ValueError expected")
if result < 0:
raise AssertionError("Negative symbol frequency")
return result
def get_total(self):
result = self.freqtable.get_total()
if result < 0:
raise AssertionError("Negative total frequency")
return result
def get_low(self, symbol):
if self._is_symbol_in_range(symbol):
low = self.freqtable.get_low (symbol)
high = self.freqtable.get_high(symbol)
if not (0 <= low <= high <= self.freqtable.get_total()):
raise AssertionError("Symbol low cumulative frequency out of range")
return low
else:
self.freqtable.get_low(symbol)
raise AssertionError("ValueError expected")
def get_high(self, symbol):
if self._is_symbol_in_range(symbol):
low = self.freqtable.get_low (symbol)
high = self.freqtable.get_high(symbol)
if not (0 <= low <= high <= self.freqtable.get_total()):
raise AssertionError("Symbol high cumulative frequency out of range")
return high
else:
self.freqtable.get_high(symbol)
raise AssertionError("ValueError expected")
def __str__(self):
return "CheckFrequencyTable (" + str(self.freqtable) + ")"
def set(self, symbol, freq):
self.freqtable.set(symbol, freq)
if not self._is_symbol_in_range(symbol) or freq < 0:
raise AssertionError("ValueError expected")
def increment(self, symbol):
self.freqtable.increment(symbol)
if not self._is_symbol_in_range(symbol):
raise AssertionError("ValueError expected")
def _is_symbol_in_range(self, symbol):
return 0 <= symbol < self.get_symbol_limit()
# ---- Bit-oriented I/O streams ----
# A stream of bits that can be read. Because they come from an underlying byte stream,
# the total number of bits is always a multiple of 8. The bits are read in big endian.
class BitInputStream(object):
# Constructs a bit input stream based on the given byte input stream.
def __init__(self, inp):
# The underlying byte stream to read from
self.input = inp
# Either in the range [0x00, 0xFF] if bits are available, or -1 if end of stream is reached
self.currentbyte = 0
# Number of remaining bits in the current byte, always between 0 and 7 (inclusive)
self.numbitsremaining = 0
# Reads a bit from this stream. Returns 0 or 1 if a bit is available, or -1 if
# the end of stream is reached. The end of stream always occurs on a byte boundary.
def read(self):
if self.currentbyte == -1:
return -1
if self.numbitsremaining == 0:
temp = self.input.read(1)
if len(temp) == 0:
self.currentbyte = -1
return -1
self.currentbyte = temp[0] if python3 else ord(temp)
self.numbitsremaining = 8
assert self.numbitsremaining > 0
self.numbitsremaining -= 1
return (self.currentbyte >> self.numbitsremaining) & 1
# Reads a bit from this stream. Returns 0 or 1 if a bit is available, or raises an EOFError
# if the end of stream is reached. The end of stream always occurs on a byte boundary.
def read_no_eof(self):
result = self.read()
if result != -1:
return result
else:
raise EOFError()
# Closes this stream and the underlying input stream.
def close(self):
self.input.close()
self.currentbyte = -1
self.numbitsremaining = 0
# A stream where bits can be written to. Because they are written to an underlying
# byte stream, the end of the stream is padded with 0's up to a multiple of 8 bits.
# The bits are written in big endian.
class BitOutputStream(object):
# Constructs a bit output stream based on the given byte output stream.
def __init__(self, out):
self.output = out # The underlying byte stream to write to
self.currentbyte = 0 # The accumulated bits for the current byte, always in the range [0x00, 0xFF]
self.numbitsfilled = 0 # Number of accumulated bits in the current byte, always between 0 and 7 (inclusive)
# Writes a bit to the stream. The given bit must be 0 or 1.
def write(self, b):
if b not in (0, 1):
raise ValueError("Argument must be 0 or 1")
self.currentbyte = (self.currentbyte << 1) | b
self.numbitsfilled += 1
if self.numbitsfilled == 8:
towrite = bytes((self.currentbyte,)) if python3 else chr(self.currentbyte)
self.output.write(towrite)
self.currentbyte = 0
self.numbitsfilled = 0
# Closes this stream and the underlying output stream. If called when this
# bit stream is not at a byte boundary, then the minimum number of "0" bits
# (between 0 and 7 of them) are written as padding to reach the next byte boundary.
def close(self):
while self.numbitsfilled != 0:
self.write(0)
self.output.close()
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