From 3a211474719915c81b3ca0a5847855c51bc7cfd9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=A3o=20Capucho?= Date: Wed, 13 Dec 2023 12:51:49 +0000 Subject: [PATCH] SM: Aula 09 --- 2ano1/SM/aula09/Alfabeto1.m | 3 + 2ano1/SM/aula09/Alfabeto2.m | 4 + 2ano1/SM/aula09/Entropia.m | 6 + 2ano1/SM/aula09/GeraMensagem.m | 12 + 2ano1/SM/aula09/HuffmanCoding.m | 5 + 2ano1/SM/aula09/MakeCodes.m | 11 + 2ano1/SM/aula09/NumeroBitsCodigo2.m | 14 + .../SM/aula09/assets/HistrogramaSimbolos.png | Bin 0 -> 16522 bytes .../assets/ex03_arith_coding.excalidraw | 3092 ++++++++++++++ 2ano1/SM/aula09/assets/ex03_decode.excalidraw | 1818 ++++++++ .../assets/ex03_huffman_tree.excalidraw | 948 +++++ .../assets/huffman_tree_assign.excalidraw | 1638 ++++++++ .../assets/huffman_tree_building.excalidraw | 3661 +++++++++++++++++ 2ano1/SM/aula09/ex00.m | 13 + 2ano1/SM/aula09/ex01.m | 14 + 2ano1/SM/aula09/ex03.m | 20 + 2ano1/SM/aula09/ex04.m | 11 + 2ano1/SM/aula09/guiao.norg | 407 ++ 2ano1/SM/map.toml | 1 + 19 files changed, 11678 insertions(+) create mode 100644 2ano1/SM/aula09/Alfabeto1.m create mode 100644 2ano1/SM/aula09/Alfabeto2.m create mode 100644 2ano1/SM/aula09/Entropia.m create mode 100644 2ano1/SM/aula09/GeraMensagem.m create mode 100644 2ano1/SM/aula09/HuffmanCoding.m create mode 100644 2ano1/SM/aula09/MakeCodes.m create mode 100644 2ano1/SM/aula09/NumeroBitsCodigo2.m create mode 100644 2ano1/SM/aula09/assets/HistrogramaSimbolos.png create mode 100644 2ano1/SM/aula09/assets/ex03_arith_coding.excalidraw create mode 100644 2ano1/SM/aula09/assets/ex03_decode.excalidraw create mode 100644 2ano1/SM/aula09/assets/ex03_huffman_tree.excalidraw create mode 100644 2ano1/SM/aula09/assets/huffman_tree_assign.excalidraw create mode 100644 2ano1/SM/aula09/assets/huffman_tree_building.excalidraw create mode 100644 2ano1/SM/aula09/ex00.m create mode 100644 2ano1/SM/aula09/ex01.m create mode 100644 2ano1/SM/aula09/ex03.m create mode 100644 2ano1/SM/aula09/ex04.m create mode 100644 2ano1/SM/aula09/guiao.norg diff --git a/2ano1/SM/aula09/Alfabeto1.m b/2ano1/SM/aula09/Alfabeto1.m new file mode 100644 index 0000000..3b240cb --- /dev/null +++ b/2ano1/SM/aula09/Alfabeto1.m @@ -0,0 +1,3 @@ +function [Simbolos] = Alfabeto1(Texto) + Simbolos = unique(Texto); +end \ No newline at end of file diff --git a/2ano1/SM/aula09/Alfabeto2.m b/2ano1/SM/aula09/Alfabeto2.m new file mode 100644 index 0000000..c733584 --- /dev/null +++ b/2ano1/SM/aula09/Alfabeto2.m @@ -0,0 +1,4 @@ +function [Simbolos, Frequencia] = Alfabeto2(Texto) + [Frequencia, Simbolos] = groupcounts(Texto'); + Frequencia = Frequencia / sum(Frequencia); +end \ No newline at end of file diff --git a/2ano1/SM/aula09/Entropia.m b/2ano1/SM/aula09/Entropia.m new file mode 100644 index 0000000..3d0be3f --- /dev/null +++ b/2ano1/SM/aula09/Entropia.m @@ -0,0 +1,6 @@ +function [H] = Entropia(Texto) + [~, Frequencia] = Alfabeto2(Texto); + + H = -sum(Frequencia .* log2(Frequencia)); +end + diff --git a/2ano1/SM/aula09/GeraMensagem.m b/2ano1/SM/aula09/GeraMensagem.m new file mode 100644 index 0000000..a4b56f5 --- /dev/null +++ b/2ano1/SM/aula09/GeraMensagem.m @@ -0,0 +1,12 @@ +function [NumBits, NumBPS] = GeraMensagem(f, CompMesg, nBits) + % Utilizamos os índices dos símbolos como o alfabeto + alfabeto = 1:length(f); + % Geramos a mensagem aleatória como uma matriz (CompMesg x 1) que + % utiliza as probabilidades que nos foram passadas e o alfabeto que + % criamos. + msg = randsrc(CompMesg, 1, [alfabeto; f]); + + NumBits = sum(nBits(msg)); + NumBPS = NumBits / CompMesg; +end + diff --git a/2ano1/SM/aula09/HuffmanCoding.m b/2ano1/SM/aula09/HuffmanCoding.m new file mode 100644 index 0000000..a39ea79 --- /dev/null +++ b/2ano1/SM/aula09/HuffmanCoding.m @@ -0,0 +1,5 @@ +function [SimbolosOrdenados, Codigos] = HuffmanCoding(Simbolos, Frequencia) + [~, idxs] = sort(Frequencia); + SimbolosOrdenados = Simbolos(idxs); +end + diff --git a/2ano1/SM/aula09/MakeCodes.m b/2ano1/SM/aula09/MakeCodes.m new file mode 100644 index 0000000..a9ae612 --- /dev/null +++ b/2ano1/SM/aula09/MakeCodes.m @@ -0,0 +1,11 @@ +function [Codigos] = MakeCodes(Texto) + [Simbolos, ~] = Alfabeto2(Texto); + + N = length(Simbolos); + + Codigos = cell(N, 1); + + for k=1:length(Simbolos) + Codigos{k} = [repmat('1', 1, k - 1) '0']; + end +end \ No newline at end of file diff --git a/2ano1/SM/aula09/NumeroBitsCodigo2.m b/2ano1/SM/aula09/NumeroBitsCodigo2.m new file mode 100644 index 0000000..8367c55 --- /dev/null +++ b/2ano1/SM/aula09/NumeroBitsCodigo2.m @@ -0,0 +1,14 @@ +function [NumBits, mensagem] = NumeroBitsCodigo2(Texto) + [Simbolos, ~] = Alfabeto2(Texto); + + Codigos = MakeCodes(Texto); + + [~, indexes] = ismember(Texto, Simbolos); + + mensagem = []; + for idx=indexes + mensagem = [mensagem Codigos{idx}]; + end + + NumBits = length(mensagem); +end \ No newline at end of file diff --git a/2ano1/SM/aula09/assets/HistrogramaSimbolos.png 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b/2ano1/SM/aula09/ex00.m new file mode 100644 index 0000000..f18b135 --- /dev/null +++ b/2ano1/SM/aula09/ex00.m @@ -0,0 +1,13 @@ +load Text.mat; + +[Simbolos, Frequencia] = Alfabeto2(Text); + +entropia = Entropia(Text); + +Codigos = MakeCodes(Text); + +numero_medio_bits = sum(Frequencia .* cellfun(@length, Codigos)); + +numero_medio_bits_1000 = 1000 * numero_medio_bits; + +fprintf("O numero medio de bits para 1000 caracteres seria %f\n", numero_medio_bits_1000); \ No newline at end of file diff --git a/2ano1/SM/aula09/ex01.m b/2ano1/SM/aula09/ex01.m new file mode 100644 index 0000000..d14052e --- /dev/null +++ b/2ano1/SM/aula09/ex01.m @@ -0,0 +1,14 @@ +Simbolos = ['A', 'B', 'C', 'D', 'E']; +Frequencia = [14 64 5 10 7] / 100; + +% a) +Entropia = -sum(Frequencia .* log2(Frequencia)); + +% b) +dicionario = huffmandict(1:5, Frequencia); + +num_bits = cellfun(@length, dicionario(:, 2))'; + +% c) +numero_medio_por_simbolo = sum(num_bits .* Frequencia); +numero_medio_bits = numero_medio_por_simbolo * 1000; diff --git a/2ano1/SM/aula09/ex03.m b/2ano1/SM/aula09/ex03.m new file mode 100644 index 0000000..505729a --- /dev/null +++ b/2ano1/SM/aula09/ex03.m @@ -0,0 +1,20 @@ +Simbolos = ['A', '!', 'L', 'P', 'O']; +Frequencia = [0.2 0.05 0.25 0.15 0.35]; + +% a) +dicionario = huffmandict(1:length(Simbolos), Frequencia); + +% b) +entropia = -sum(Frequencia .* log2(Frequencia)); +num_bits = cellfun(@length, dicionario(:, 2))'; + +numero_medio_bits = sum(num_bits .* Frequencia); + +% c) +msg = 'OLA!'; +[~, idxs] = ismember(msg, Simbolos); +bitstream = huffmanenco(idxs, dicionario); + +% d) +reverse_idxs = huffmandeco(bitstream, dicionario); +reverse = Simbolos(reverse_idxs); diff --git a/2ano1/SM/aula09/ex04.m b/2ano1/SM/aula09/ex04.m new file mode 100644 index 0000000..35f34c1 --- /dev/null +++ b/2ano1/SM/aula09/ex04.m @@ -0,0 +1,11 @@ +Simbolos = ['A', '!', 'L', 'P', 'O']; +Frequencia = [20 5 25 15 35]; + +% c) +msg = 'OLA!'; +[~, idxs] = ismember(msg, Simbolos); +bitstream = arithenco(idxs, Frequencia); + +% d) +reverse_idxs = arithdweco(bitstream, Frequencia, length(msg)); +reverse = Simbolos(reverse_idxs); \ No newline at end of file diff --git a/2ano1/SM/aula09/guiao.norg b/2ano1/SM/aula09/guiao.norg new file mode 100644 index 0000000..eb0b246 --- /dev/null +++ b/2ano1/SM/aula09/guiao.norg @@ -0,0 +1,407 @@ +@document.meta +title: Aula 09 +description: Codificação Eficiente de Informação +author: João Capucho +@end + +* Codificação Eficiente de Informação + +** Exercício 1 + + > O alfabeto de um esquema de codificação apresenta o seguinte histograma + típico de ocorrência dos seus símbolos: + + @embed image + ./assets/HistrogramaSimbolos.png + @end + + Podemos converter este histograma em dois arrays do Matlab com os símbolos + e as suas frequências: + + @code matlab + Simbolos = ['A', 'B', 'C', 'D', 'E']; + Frequencia = [14 64 5 10 7] / 100; + @end + +*** Alínea A + + > Determine a entropia da informação codificada com esse alfabeto. + + A entropia é dada pela seguinte fórmula: + + @math + H(X) = - \sum_{x \in X} p(x) \log_2 \left( p(x) \right) + @end + + Onde $X$ é o conjunto de todos os símbolos, e $p(x)$ é a + probabilidade/frequência de um dado símbolo $x$ pertencente a $X$. + + Isto pode ser implementado com o seguinte código Matlab: + + @code matlab + Entropia = -sum(Frequencia .* log2(Frequencia)); + @end + + Executando o código com os valores dados obtemos uma entropia de + $\approx 1.6260$ bits. + +*** Alínea B + + > Usando o código de Huffman, atribua um código binário a cada símbolo + deste alfabeto. + + A construção do código de Huffman permite-nos atribuir um código binário a + cada símbolo com a propriedade que nenhum código é prefixo de outro código + (por exemplo, `11` e `110` não são códigos de Huffman válidos porque um é + prefixo do outro). + + Esta propriedade é bastante útil na descodificação porque dado um conjunto + de valores só existe uma descodificação válida sem serem necessários dados + extra para determinar onde começa e termina um símbolo. + + O código de Huffman pode ser construído em duas etapas, primeiro + construímos uma representação em árvore binária dos símbolos com base nas + suas probabilidades. Esta etapa é efetuada através dos seguintes passos: + + ~ Os símbolos são todos ordenados por ordem crescente de probabilidade + ~ Os dois primeiros símbolos são unidos num /metacaractere/, este representa + um nó na árvore com ambos os símbolos como folhas, a sua probabilidade + corresponde a soma das probabilidades dos dois símbolos que foram unidos. + ~ Os símbolos e /metacaracteres/ são ordenados pela sua probabilidade. + ~ O passo 2 é repetido até só restar um símbolo ou /metacaractere/. + + Se aplicarmos este processo ao histograma que nos foi dado obtemos o + seguinte resultado: + + @embed image + ./assets/huffman_tree_building.excalidraw + @end + + Agora que já temos a árvore binária, podemos passar a segunda fase que + consiste em atribuir os códigos aos nós das árvores, esta etapa é bastante + mais simples: + + ~ Para cada nó que não seja folha atribuímos 0 ao ramo da esquerda e 1 ao + ramo da direita. + ~ O código do símbolo é equivalente aos valores percorridos desde a raiz + até chegar ao nó que contêm o símbolo. + + @embed image + ./assets/huffman_tree_assign.excalidraw + @end + + Obtemos então os seguintes códigos: + + : A1 : Símbolo + : A2 : Código + : A3 : Número de bits ($l_i$) + : B1 : A + : B2 : `00` + : B3 : 2 + : C1 : B + : C2 : `1` + : C3 : 1 + : D1 : C + : D2 : `0110` + : D3 : 4 + : E1 : D + : E2 : `010` + : E3 : 3 + : F1 : E + : F2 : `0111` + : F3 : 4 + +**** Matlab + + É possível também utilizar o Matlab para atribuir um código a cada símbolo + através da função + {https://www.mathworks.com/help/comm/ref/huffmandict.html}[`huffmandict`], + esta recebe os arrays de símbolos como primeiro argumento e as suas + frequências como segundo argumento. + + Em vez de utilizar os símbolos diretamente, vamos utilizar os seus índices, + isto simplifica a codificação de mensagens. + + @code matlab + dicionario = huffmandict(1:5, Frequencia); + @end + + E obtemos o seguinte dicionário: + + : A1 : Símbolo + : A2 : Código + : A3 : Número de bits ($l_i$) + : B1 : A + : B2 : `11` + : B3 : 2 + : C1 : B + : C2 : `0` + : C3 : 1 + : D1 : C + : D2 : `1001` + : D3 : 4 + : E1 : D + : E2 : `101` + : E3 : 3 + : F1 : E + : F2 : `1000` + : F3 : 4 + + Os códigos diferem dos que obtemos manualmente, isto não significa que os + nossos códigos estão errados, existem diversos códigos de Huffman para os + mesmos símbolos e probabilidades. Neste caso os `0` e `1` estão trocados + em relação ao código que fizemos manualmente. + +*** Alínea C + + > Qual seria o número médio de bits por símbolo que esperaria obter numa + mensagem composta por 1000 símbolos deste alfabeto? + + O número médio de bits pode ser calculado através do número de bits + associado a um símbolo a multiplicar pela sua probabilidade. + + @math + \begin{align*} + \sum_{i=1}^{\left\lvert X \right\rvert} \; l_i \cdot p(X_i) &= + 2 \cdot 0.14 + 1 \cdot 0.64 + 4 \cdot 0.05 + 3 \cdot 0.10 + 4 \cdot 0.07 \\ + &= 1.7 + \end{align*} + @end + +**** Matlab + + Mais uma vez podemos utilizar o Matlab para realizar as contas: + + @code matlab + num_bits = cellfun(@length, dicionario(:, 2))'; + numero_medio_por_simbolo = sum(num_bits .* Frequencia); + @end + +** Exercício 2 + + > Desenvolva, agora, a função: + + @code matlab + [NumBits, NumBPS] = GeraMensagem(f, CompMesg, nBits) + @end + + > que gera uma mensagem aleatória, com um número total de símbolos dado por + `CompMesg`, usando um alfabeto de símbolos cuja frequência de ocorrência é + especificada (em percentagem) no vetor `f`, de dimensão $N \times 1$. Na + geração da mensagem, a probabilidade com que ocorre cada símbolo deve + seguir o valor especificado em `f`. A função recebe ainda o vetor `nBits`, + também de dimensão $N \times 1$, em que cada elemento indica o número de + bits com que é codificado o símbolo correspondente. A função retorna, em + `NumBits`, o número total de bits que a mensagem gerada aleatoriamente + requer para ser representada, e retorna também, em `NumBPS`, o número + médio de bits por símbolo verificado na mensagem gerada. + + Primeiro vamos considerar como gerar a mensagem. Esta é suposto utilizar um + alfabeto predefinido com diferentes probabilidades para cada símbolo, o + Matlab fornece a função + {https://www.mathworks.com/help/comm/ref/randsrc.html}[`randsrc`] que + permite gerar uma matriz aleatória com um tamanho controlado por nós e + opcionalmente utilizar um alfabeto e probabilidades também definidas por + nós. + + No entanto, não nos é fornecido o alfabeto, apenas as suas probabilidades. + Mas o alfabeto em específico, não nos é relevante, podemos apenas utilizar + os índices correspondentes a cada símbolo como o alfabeto: + + @code matlab + % Utilizamos os índices dos símbolos como o alfabeto + alfabeto = 1:length(f); + % Geramos a mensagem aleatória como uma matriz (CompMesg x 1) que + % utiliza as probabilidades que nos foram passadas e o alfabeto que + % criamos. + msg = randsrc(CompMesg, 1, [alfabeto; f]); + @end + + A seguir precisamos de converter cada símbolo da mensagem para o seu + número de bits, isto é facilmente feito, pois como o nosso alfabeto é o índice + do símbolo podemos indexar diretamente em `nBits` com a mensagem gerada + para obter o número de bits para cada símbolo. + + Finalmente basta calcular `NumBits`, que corresponde a soma de todos os + números de bits de uma mensagem, e `NumBPS` que é a média do número de bits + da mensagem, o que nos deixa com a seguinte função completa: + + @code matlab + function [NumBits, NumBPS] = GeraMensagem(f, CompMesg, nBits) + % Utilizamos os índices dos símbolos como o alfabeto + alfabeto = 1:length(f); + % Geramos a mensagem aleatória como uma matriz (CompMesg x 1) que + % utiliza as probabilidades que nos foram passadas e o alfabeto que + % criamos. + msg = randsrc(CompMesg, 1, [alfabeto; f]); + + NumBits = sum(nBits(msg)); + NumBPS = NumBits / CompMesg; + end + @end + + + > Teste esta função usando o alfabeto da questão anterior. + + @code matlab + [NumBits, NumBPS] = GeraMensagem(Frequencia, 1000, num_bits) + @end + + O `NumBits` e `NumBPS` variam devido a natureza aleatória da função, mas o + valor de `NumBPS` aproxima-se do valor calculado na alínea C do exercício 1 + como era de esperar. + +** Exercício 3 + + > Considere a seguinte Tabela + + : A1 : Símbolo + : A2 : Probabilidade + : B1 : A + : B2 : $0.20$ + : C1 : ! + : C2 : $0.05$ + : D1 : L + : D2 : $0.25$ + : E1 : P + : E2 : $0.15$ + : F1 : O + : F2 : $0.35$ + + Começamos por converter a tabela para arrays de Matlab: + + @code matlab + Simbolos = ['A', '!', 'L', 'P', 'O']; + Frequencia = [0.2 0.05 0.25 0.15 0.35]; + @end + +*** Alínea A + + > Usando o código de Huffman, atribua um código binário a cada símbolo + deste alfabeto. + + Utilizamos mais uma vez a função `huffmandict` para obter o dicionário dos + códigos de huffman. + + @code matlab + dicionario = huffmandict(1:length(Simbolos), Frequencia); + @end + + : A1 : Símbolo + : A2 : Código + : A3 : Número de bits ($l_i$) + : B1 : A + : B2 : `10` + : B3 : 2 + : C1 : ! + : C2 : `111` + : C3 : 3 + : D1 : L + : D2 : `01` + : D3 : 2 + : E1 : P + : E2 : `110` + : E3 : 3 + : F1 : O + : F2 : `00` + : F3 : 2 + +*** Alínea B + + > Calcule a entropia e o número médio de bits deste código. + + A entropia pode ser calculada com o mesmo código que utilizamos no + exercício 1.a): + + @code matlab + entropia = -sum(Frequencia .* log2(Frequencia)); + @end + + Obtendo um valor para a entropia de $\approx 2.1211$ bits. + + O cálculo do número médio de bits utiliza o código da alínea C: + + @code matlab + num_bits = cellfun(@length, dicionario(:, 2))'; + numero_medio_bits = sum(num_bits .* Frequencia); + @end + + Que nos dá um número médio de bits de $2.2$. + +*** Alínea C + + > Suponha agora que a mensagem a codificar é "OLA!". Qual a sequência de + bits para a codificar usando o código de Huffman? + + O Matlab disponibiliza a função + {https://www.mathworks.com/help/comm/ref/huffmanenco.html}[`huffmanenco`] + que produz a mensagem codificada com um dado dicionário de códigos de + huffman. + + No entanto, antes de passarmos a mensagem vamos primeiro precisar de a + converter nos índices dos símbolos correspondentes, isto pode ser feito com + a função + {https://www.mathworks.com/help/matlab/ref/double.ismember.html}[`ismember`] + do Matlab. + + @code + msg = 'OLA!'; + [~, idxs] = ismember(msg, Simbolos); + bitstream = huffmanenco(idxs, dicionario); + @end + + Obtemos então a seguinte mensagem codificada: `000110111`. + +*** Alínea D + + > Faça agora o processo inverso, i.e. descodifique a mensagem binária a + partir da árvore de codificação. + + Começamos por converter o nosso dicionário na sua representação em árvore: + + @embed image + ./assets/ex03_huffman_tree.excalidraw + @end + + Para descodificar a mensagem começamos na raiz e seguimos o ramo que é + igual ao bit que estamos a processar, repetimos até chegar a uma folha, a + este ponto já temos um símbolo para adicionar ao resultado descodificado. + Recomeçamos na raiz da árvore e repetimos o processo até processarmos a + mensagem codificada na sua totalidade. + + @embed image + ./assets/ex03_decode.excalidraw + @end + +** Exercício 4 + + > Considerando a tabela anterior, codifique a mensagem "OLA!" usando + codificação aritmética. + + A codificação aritmética baseia-se na subdivisão de um intervalo com base + nas probabilidades do alfabeto até ser possível obter uma fração que + represente de forma não ambígua a mensagem. + + O funcionamento do algoritmo de codificação pode ser descrito nos seguintes + passos: + + ~ O intervalo é inicializado a $[0; 1[$. + ~ O intervalo é subdividido em partes proporcionais as probabilidades de + cada símbolo. + ~ É escolhido o intervalo do símbolo que estamos a processar + ~ Repete-se o passo 2 até todos os símbolos serem processados + + No final vamos obter um intervalo que representa a nossa mensagem, no + entanto, não é necessário o intervalo todo para descodificar a mensagem, + apenas um valor que pertença a este intervalo (também é necessário + informação fora de banda para saber quando o descodificador deve parar, tal + como o tamanho do código ou um marcador final). + + Aplicando agora ao nosso alfabeto e mensagem, obtemos o seguinte resultado: + + @embed image + ./assets/ex03_arith_coding.excalidraw + @end + + O intervalo final da nossa mensagem é $[0.741; 0.741875[$, logo o código da + mensagem poderia ser $0.741$. diff --git a/2ano1/SM/map.toml b/2ano1/SM/map.toml index 3bf7298..1c2c0d5 100644 --- a/2ano1/SM/map.toml +++ b/2ano1/SM/map.toml @@ -1,2 +1,3 @@ [children."Guiões".children] +"Aula 09" = "aula09/guiao.norg" "Aula 10" = "aula10/guiao.norg"