【图像压缩】基于matlab GUI FFT图像压缩【含Matlab源码 843期】

一、简介

FFT(Fast Fourier transform)：快速傅里叶变换，是DFT的工程化实现方法。
DFT直接求解太过于复杂，FFT方法根据DFT求解过程中旋转因子的性质并引入分治算法思想，大大简化计算过程，被广泛应用在频谱分析的工程实践中，如matlab，C，C++，CUDA等底层实现

1 DFT简介
频谱分析是信号处理中的重要环节，从傅里叶变换FT，到拉普拉斯变换LT，离散时间傅里叶变换DTFT，Z变换ZT，到我们所讲的离散傅里叶变换DFT（他们之间的联系和区别见我的其他博客）。
相比于其他变换，DFT被广泛应用的原因是其输入的时域信号是离散的，输出的频域结果也是离散的。这就极大方便了我们进行基于计算机的频谱计算，存储和分析，没办法数字信号处理是大趋势。
DFT变换的公式为：

但为了分析方便，在FFT的计算过程中，我们依然使用k = 0 ∼ N − 1 k = 0 \sim N-1k=0∼N−1的选取策略。也即，如下：

2 旋转因子WWW的性质

3 FFT蝶形计算证明

4 FFT计算过程

二、源代码

function varargout = fft_encoding(varargin)
% FFT_ENCODING M-file for fft_encoding.fig
%      FFT_ENCODING, by itself, creates a new FFT_ENCODING or raises the existing
%      singleton*.
%
%      H = FFT_ENCODING returns the handle to a new FFT_ENCODING or the handle to
%      the existing singleton*.
%
%      FFT_ENCODING('CALLBACK',hObject,eventData,handles,...) calls the local
%      function named CALLBACK in FFT_ENCODING.M with the given input arguments.
%
%      FFT_ENCODING('Property','Value',...) creates a new FFT_ENCODING or raises the
%      existing singleton*.  Starting from the left, property value pairs are
%      applied to the GUI before fft_encoding_OpeningFunction gets called.  An
%      unrecognized property name or invalid value makes property application
%      stop.  All inputs are passed to fft_encoding_OpeningFcn via varargin.
%
%      *See GUI Options on GUIDE's Tools menu.  Choose "GUI allows only one
%      instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES% Edit the above text to modify the response to help fft_encoding% Last Modified by GUIDE v2.5 18-Jun-2009 20:05:02% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name',       mfilename, ...'gui_Singleton',  gui_Singleton, ...'gui_OpeningFcn', @fft_encoding_OpeningFcn, ...'gui_OutputFcn',  @fft_encoding_OutputFcn, ...'gui_LayoutFcn',  [] , ...'gui_Callback',   []);
if nargin & isstr(varargin{1})gui_State.gui_Callback = str2func(varargin{1});
endif nargout[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
elsegui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT% --- Executes just before fft_encoding is made visible.
function fft_encoding_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
% varargin   command line arguments to fft_encoding (see VARARGIN)
cr = 0.5;
I = imread('lena.bmp');
axes(handles.axes1);
imshow(I);
I = double(I)/255;
fftcoe = blkproc(I,[8 8],'fft2(x)');
coevar = im2col(fftcoe,[8 8],'distinct');
coe = coevar;
[y,ind] = sort(coevar);
[m,n] = size(coevar);
snum = 64-64*cr;
for i = 1:ncoe(ind(1:snum),i) = 0;
end
B2 = col2im(coe,[8 8],[512 512],'distinct');
I2 = blkproc(B2,[8,8],'ifft2(x)');
axes(handles.axes2);
imshow(I2);
e = double(I) - double(I2);
[m,n]=size(e);
erms = sqrt(sum(e(:).^2)/(m*n));
set(handles.erms_edit,'string',erms);
set(handles.cr_edit,'string',0.5);
% Choose default command line output for fft_encoding
handles.output = hObject;% Update handles structure
guidata(hObject, handles);% UIWAIT makes fft_encoding wait for user response (see UIRESUME)
% uiwait(handles.figure1);% --- Outputs from this function are returned to the command line.
function varargout = fft_encoding_OutputFcn(hObject, eventdata, handles)
% varargout  cell array for returning output args (see VARARGOUT);
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)% Get default command line output from handles structure
varargout{1} = handles.output;% --- Executes during object creation, after setting all properties.
function image_pop_menu_CreateFcn(hObject, eventdata, handles)
% hObject    handle to image_pop_menu (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called% Hint: popupmenu controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispcset(hObject,'BackgroundColor','white');
elseset(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));
end% --- Executes on selection change in image_pop_menu.
function image_pop_menu_Callback(hObject, eventdata, handles)
% hObject    handle to image_pop_menu (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
val = get(hObject,'value');
str = get(hObject,'string');
cr = str2num(get(handles.cr_edit,'string'));
switch str{val}case 'Lena'I = imread('lena.bmp');case 'Cameraman'I = imread('cameraman.tif');case 'Peppers'I = imread('peppers.bmp');case 'Fingerprint'I = imread('fingerprint.jpg');case 'Licenceplate'I = imread('licenceplate.jpg');case 'Cloudy'I = imread('cloudy.tif');
end
axes(handles.axes1);
imshow(I);
I = double(I)/255;
fftcoe = blkproc(I,[8 8],'fft2(x)');
coevar = im2col(fftcoe,[8 8],'distinct');
coe = coevar;
[y,ind] = sort(coevar);
[m,n] = size(coevar);
snum = 64-64*cr;
for i = 1:ncoe(ind(1:snum),i) = 0;
end
B2 = col2im(coe,[8 8],size(I),'distinct');
I2 = blkproc(B2,[8,8],'ifft2(x)');
axes(handles.axes2);
imshow(I2);
e = double(I) - double(I2);
[m,n]=size(e);
erms = sqrt(sum(e(:).^2)/(m*n));
set(handles.erms_edit,'string',erms);
clc;
% Hints: contents = get(hObject,'String') returns image_pop_menu contents as cell array
%        contents{get(hObject,'Value')} returns selected item from image_pop_menu% --- Executes during object creation, after setting all properties.
function cr_edit_CreateFcn(hObject, eventdata, handles)
% hObject    handle to cr_edit (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called% Hint: edit controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispcset(hObject,'BackgroundColor','white');
elseset(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));
endfunction cr_edit_Callback(hObject, eventdata, handles)
% hObject    handle to cr_edit (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)% Hints: get(hObject,'String') returns contents of cr_edit as text
%        str2double(get(hObject,'String')) returns contents of cr_edit as a double% --- Executes on button press in apply_button.
function apply_button_Callback(hObject, eventdata, handles)
% hObject    handle to apply_button (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
cr = str2num(get(handles.cr_edit,'string'));
I = getimage(handles.axes1);
I = double(I)/255;
fftcoe = blkproc(I,[8 8],'fft2(x)');
coevar = im2col(fftcoe,[8 8],'distinct');
coe = coevar;
[y,ind] = sort(coevar);
[m,n] = size(coevar);
snum = 64-floor(64*cr);
for i = 1:ncoe(ind(1:snum),i) = 0;
end
B2 = col2im(coe,[8 8],size(I),'distinct');
I2 = blkproc(B2,[8,8],'ifft2(x)');
axes(handles.axes2);
imshow(abs(I2));
e = double(I) - double(I2);
[m,n]=size(e);
erms = sqrt(sum(e(:).^2)/(m*n));
set(handles.erms_edit,'string',erms);
% --- Executes on button press in close_button.
function close_button_Callback(hObject, eventdata, handles)
% hObject    handle to close_button (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
close(fft_encoding);

三、运行结果

四、matlab版本及参考文献

1 matlab版本
2014a

2 参考文献
[1] 蔡利梅.MATLAB图像处理——理论、算法与实例分析[M].清华大学出版社，2020.
[2]杨丹,赵海滨,龙哲.MATLAB图像处理实例详解[M].清华大学出版社，2013.
[3]周品.MATLAB图像处理与图形用户界面设计[M].清华大学出版社，2013.
[4]刘成龙.精通MATLAB图像处理[M].清华大学出版社，2015.