卷积运算的基本原理

     
      注意：卷积运算的结果与滤波器的大小有关，输出信号的维度会根据滤波器的大小而变化。
     

一维卷积实现

```objective-c  - (NSArray *)oneDimensionalConvolution:(NSArray *)inputArray withFilter:(NSArray *)filterArray  {    // 计算输出数组的长度    NSInteger outputLength = [inputArray length] + [filterArray length] - 1;    NSMutableArray *outputArray = [[NSMutableArray alloc] init];    // 遍历输入数组，进行卷积计算    for (NSInteger i = 0; i < [inputArray length]; i++)    {      NSInteger filterStartIndex = max(0, i - [filterArray length] + 1);      NSInteger filterEndIndex = min([inputArray length] - 1, i + [filterArray length] - 1);      NSInteger sum = 0;      for (NSInteger j = filterStartIndex; j <= filterEndIndex; j++)      {        sum += [inputArray[j] * filterArray[j - filterStartIndex]];      }      [outputArray addObject:sum];    }    return [outputArray autorelease];  }  ```
     
      说明：上述代码实现了一个简单的一维卷积算法，通过遍历输入数组并与滤波器进行点积，生成输出数组。
     

二维卷积实现

```objective-c  - (NSArray *)twoDimensionalConvolution:(NSArray *)inputMatrix withFilter:(NSArray *)filterMatrix  {    // 计算输出矩阵的尺寸    NSInteger inputRows = [inputMatrix length];    NSInteger inputColumns = [inputMatrix[0] length];    NSInteger filterRows = [filterMatrix length];    NSInteger filterColumns = [filterMatrix[0] length];    NSInteger outputRows = inputRows + filterRows - 1;    NSInteger outputColumns = inputColumns + filterColumns - 1;    NSMutableArray *outputMatrix = [[NSMutableArray alloc] init];    // 遍历输入矩阵，进行卷积计算    for (NSInteger i = 0; i < inputRows; i++)    {      for (NSInteger j = 0; j < inputColumns; j++)      {        NSInteger filterRowStart = max(0, i - filterRows + 1);        NSInteger filterRowEnd = min(inputRows - 1, i + filterRows - 1);        NSInteger filterColumnStart = max(0, j - filterColumns + 1);        NSInteger filterColumnEnd = min(inputColumns - 1, j + filterColumns - 1);        NSInteger sum = 0;        for (NSInteger fi = filterRowStart; fi <= filterRowEnd; fi++)        {          for (NSInteger fj = filterColumnStart; fj <= filterColumnEnd; fj++)          {            sum += [inputMatrix[fi][fj] * filterMatrix[fi - filterRowStart][fj - filterColumnStart]];          }        }        [outputMatrix addObject:sum];      }    }    return [outputMatrix autorelease];  }  ```
     
      说明：上述代码实现了一个简单的二维卷积算法，通过遍历输入矩阵并与滤波器进行二维点积，生成输出矩阵。
     

完整源码示例

```objective-c  #import 
     
        @interface Convolution : NSObject  - (NSArray *)oneDimensionalConvolution:(NSArray *)inputArray withFilter:(NSArray *)filterArray;  - (NSArray *)twoDimensionalConvolution:(NSArray *)inputMatrix withFilter:(NSArray *)filterMatrix;  - (NSArray *)convolve:(NSArray *)array withFilter:(NSArray *)filter;  - (NSArray *)convolve2D:(NSArray *)matrix withFilter:(NSArray *)filter;  - (void)printResult:(id)result;  - (void)printMatrix:(NSArray *)matrix;
     

@end

@implementation Convolution  - (NSArray *)oneDimensionalConvolution:(NSArray *)inputArray withFilter:(NSArray *)filterArray  {    NSInteger inputLength = [inputArray length];    NSInteger filterLength = [filterArray length];    if (inputLength == 0 || filterLength == 0)    {      return [NSArray array];    }    NSInteger outputLength = inputLength + filterLength - 1;    NSMutableArray *outputArray = [[NSMutableArray alloc] init];    for (NSInteger i = 0; i < inputLength; i++)    {      NSInteger filterStart = max(0, i - filterLength + 1);      NSInteger filterEnd = min(inputLength - 1, i + filterLength - 1);      NSInteger sum = 0;      for (NSInteger j = filterStart; j <= filterEnd; j++)      {        sum += [inputArray[j] * filterArray[j - filterStart]];      }      [outputArray addObject:sum];    }    return [outputArray autorelease];  }  - (NSArray *)twoDimensionalConvolution:(NSArray *)inputMatrix withFilter:(NSArray *)filterMatrix  {    NSInteger rows = [inputMatrix length];    if (rows == 0) return [NSArray array];    NSInteger cols = [inputMatrix[0] length];    NSInteger filterRows = [filterMatrix length];    if (filterRows == 0) return [NSArray array];    NSInteger filterCols = [filterMatrix[0] length];    NSInteger outputRows = rows + filterRows - 1;    NSInteger outputCols = cols + filterCols - 1;    NSMutableArray *outputMatrix = [[NSMutableArray alloc] init];    for (NSInteger i = 0; i < rows; i++)    {      for (NSInteger j = 0; j < cols; j++)      {        NSInteger rowStart = max(0, i - filterRows + 1);        NSInteger rowEnd = min(rows - 1, i + filterRows - 1);        NSInteger colStart = max(0, j - filterCols + 1);        NSInteger colEnd = min(cols - 1, j + filterCols - 1);        NSInteger sum = 0;        for (NSInteger fi = rowStart; fi <= rowEnd; fi++)        {          for (NSInteger fj = colStart; fj <= colEnd; fj++)          {            sum += [inputMatrix[fi][fj] * [filterMatrix[fi - rowStart][fj - colStart]]];          }        }        [outputMatrix addObject:sum];      }    }    return [outputMatrix autorelease];  }  - (void)printResult:(id)result  {    NSLog(@"结果：%@", result);  }  - (void)printMatrix:(NSArray *)matrix  {    for (NSInteger i = 0; i < [matrix length]; i++)    {      NSLog(@"行 %d：%@", i, matrix[i]);    }  }  @end