How to clip the BackdropFilter with smooth edges?

I was able to achieve it but it is kind of way out as there is no way to directly implement it as of now. Make this function for creating the effect. This function will recieve a list of widgets that you want to blur.

List<Widget> buildBlurredImage(List<Widget> l) {
    List<Widget> list = [];
    list.addAll(l);
    double sigmaX = 0;
    double sigmaY = 0.1;
    for (int i = 100; i < 350; i += 5) {
     // 100 is the starting height of blur
     // 350 is the ending height of blur
      list.add(Positioned(
        top: i.toDouble(),
        bottom: 0,
        left: 0,
        right: 0,
        child: ClipRect(
          child: BackdropFilter(
            filter: ImageFilter.blur(
              sigmaX: sigmaX,
              sigmaY: sigmaY,
            ),
            child: Container(
              color: Colors.black.withOpacity(0),
            ),
          ),
        ),
      ));
      sigmaX += 0.1;
      sigmaY += 0.1;
    }
    return list;
  }

then inside your widget class inside stack use the function like this

@override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(
        child: Stack(
          alignment: Alignment.bottomCenter,
          // children: <Widget>[],
          children: buildBlurredImage([
            Container(
              height: 500,
              child: Image.network(
                'https://www.thewowstyle.com/wp-content/uploads/2015/02/Beautiful-Wallpapers-14.jpg',
                fit: BoxFit.cover,
              ),
            ),
          ]),
        ),
      ),
    );
  }

Your final Widget class will look like this

class MyApp extends StatelessWidget {
  List<Widget> buildBlurredImage(List<Widget> l) {
    List<Widget> list = [];
    list.addAll(l);
    double sigmaX = 0;
    double sigmaY = 0.1;
    for (int i = 100; i < 350; i += 5) {
      // 100 is the starting height of blur
      // 350 is the ending height of blur
      list.add(Positioned(
        top: i.toDouble(),
        bottom: 0,
        left: 0,
        right: 0,
        child: ClipRect(
          child: BackdropFilter(
            filter: ImageFilter.blur(
              sigmaX: sigmaX,
              sigmaY: sigmaY,
            ),
            child: Container(
              color: Colors.black.withOpacity(0),
            ),
          ),
        ),
      ));
      sigmaX += 0.1;
      sigmaY += 0.1;
    }
    return list;
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(
        child: Stack(
          alignment: Alignment.bottomCenter,
          // children: <Widget>[],
          children: buildBlurredImage([
            Container(
              height: 500,
              child: Image.network(
                'https://www.thewowstyle.com/wp-content/uploads/2015/02/Beautiful-Wallpapers-14.jpg',
                fit: BoxFit.cover,
              ),
            ),
          ]),
        ),
      ),
    );
  }
}

The best approach for me would be to used a ShaderMask above the BackdropFilter.

Unfortunately that stop working due to a change in the engine: link

I created a issue on flutter, and I hope they solve the bug soon.